## Course Description by year from Rice University General Announcements

### 1912/13

No specific course information

### 1913/14

No specific course information

### 1914/15

**Mathematics 100.** Trigonometry, Analytic Geometry, Introduction to Calculus, constituting the Freshman course in mathematics which is required of all students in the Institute.

**Mathematics 200.** Differential and Integral Calculus. This course, including derivatives, integrals, differentials with their applications to geometry and mechanics, infinite series, Taylor's theorem, partial derivatives, is the foundation of theoretical physics and advanced mathematics, and the ideas introduced are of great importance in many branches of modern thought.

**Mathematics 210.** Differential and Integral Calculus. This course covers the ground of course 200) in mathematics, and in addition the subject of integration of functions of several variables. It will be more complete than the preceding course, and is intended for students who have greater facility in mathematical reasoning. It is a sufficient introduction to courses 300 and 310 in mathematics, and is open to students who obtain high grades in mathematics 100 or otherwise satisfy the instructor of their fitness to take the course.

**Mathematics 220.** Modern Geometry and Algebra. Introduction to modern methods in geometry and algebra, including abridged notation and the theory of transformation and invariants.

**Mathematics 300.** Advanced Calculus and Differential Equations. Differentiation and integration of functions of several variables; surface and volume integrals, introduction to the theory of differential equations. This course or mathematics 310 should be taken by students whose major interest lies in physics or engineering; it is open to those who pass successfully in courses 200 or 210 in mathematics.

**Mathematics 310.** Theory of Functions of a Real Variable. Surface and volume integrals ; introduction to the theory of differential equations and integral equations, with applications of the theory of sets of points; the Lebesgue integral; trigonometric series. Open to those who satisfy the instructor of their fitness to take the course.

**Mathematics 320.** Theory of Functions of a Complex Variable. This course is given in alternate years with course 310 to those students who satisfy the instructor that they are prepared to take the course. It has the same requirements for admission, but is not to be given in 1914-15.

**Applied Mathematics 200.** Theoretical Mechanics. Mathematical theory of the fundamental principles with applications to machines and structures.

**Applied Mathematics 300.** Mechanics of Elastic Bodies and Fluids. The bending and vibrations of elastic bodies, the motion of fluids and waves considered by means of harmonic functions. Open to those who satisfy the instructor that they are prepared to take the course.

**Applied Mathematics 310.** Mechanics of Rigid Bodies and General Analytical Dynamics. In this course the method of generalized co-ordinates, Lagrange's equations, and Hamilton's Principle will be applied to the problems of rotating rigid bodies, celestial bodies and to general theoretical physics. This course is given in alternate years with course 300 in applied mathematics to those students who satisfy the instructor of their fitness to take the course.

### 1915/16

**Mathematics 100**. Trigonometry, Analytic Geometry, and Advanced Algebra, constituting the freshman course in mathematics which is required of all students in the Institute.

**Mathematics 200**. Differential and Integral Calculus. This course, including the study of derivatives, indefinite and definite integrals, infinite series, and Taylor's theorem, is the foundation of theoretical physics and advanced mathematics, and the ideas introduced are, as ideas, of fundamental importance in many branches of modern thought.

**Mathematics 210.** Differential and Integral Calculus. This course covers the ground of course 200, but is more complete and goes further. It is intended for students who have greater facility in mathematical reasoning. It is a sufficient introduction to Mathematics 310, 320, and 330, and is open to students who obtain high grades in Mathematics 100 or otherwise satisfy the instructor of their fitness to take the course.

**Mathematics 220.** Modern Geometry and Algebra. Introduction to modern methods in geometry and algebra, including abridged notation, and the theory of transformations and invariants. This course will not be given in 1915-16.

**Mathematics 230.** History of Geometry. Introduction to Non-Euclidean geometry. This course is designed for those who are interested in the teaching of mathematics, and also for those who desire to investigate the development of our ideas with regard to space and time.

**Mathematics 300.** Advanced Calculus and Differential Equations. Differentiation and integration of functions of several variables; surface and volume integrals; introduction to the theory of differential equations. This course or Mathematics 310 should be taken by students whose major interest lies in physics or engineering; it is open to those who pass successfully in courses 200 or 210 in mathematics.

**Mathematics 310.** Theory of Functions of a Real Variable. This course consists of the theory of sets of points; multiple, curvilinear, and improper integrals; the Lebesgue integral; Fourier series; and an introduction to the theory of differential equations. Open to those who satisfy the instructor of their fitness to take the course.

**Mathematics 320.** Theory of Functions of a Complex Variable. An introductory course in the general theory of functions of a complex variable. Open to students who satisfy the instructor that they are prepared to take the course.

**Mathematics 330.** Differential Geometry. A study of the properties of curves and surfaces in the neighborhood of a point; curves in the plane; analytical geometry of space; curves and surfaces in space. Open to students who have had course 210 or a third year course in mathematics.

**Applied Mathematics 200.** Theoretical Mechanics. A mathematical study of the fundamental principles, with applications to machines and structures. It includes elementary statics, dynamics, and hydraulics. This course is a necessary part of the engineering course and is recommended to students of physics and mathematics.

**Applied Mathematics 300.** Mechanics of Elastic Bodies and Fluids. A study of the bending and vibrations of elastic bodies, the motion of fluids, and allied problems, by means of harmonic functions. This course is intended for advanced students in engineering, physics, and mathematics, and is open to those who satisfy the instructor that they are prepared to take the course.

**Applied Mathematics 310.** General Analytical Dynamics. In this course the methods of generalized coordinates, Lagrange's equation, and Hamilton's principle will be applied to the problems of rigid bodies in mechanics and of thermodynamics and electricity in physics. This course is intended for the same students as course 300 in applied mathematics and is subject to the same conditions. It is given in years alternating with the latter course, and is not to be given in 1915-16.

### 1916/17

**Mathematics 100.** Trigonometry, Analytic Geometry, and Advanced Algebra, constituting the freshman course in mathematics which is required of all students in the Institute.

**Mathematics 110.** Plane Analytic Geometry, Topics from Algebra and Analytic Geometry of Three Dimensions. For students who are already well grounded in trigonometry. The course will be given if a sufficient number of properly qualified students register.

**Mathematics 200.** Differential and Integral Calculus. This course, including the study of derivatives, indefinite and definite integrals, infinite series, and Taylor's theorem, is the foundation of theoretical physics and advanced mathematics, and the ideas introduced are, as ideas, of fundamental importance in many branches of modern thought.

**Mathematics 210.** Differential and Integral Calculus. This course covers the ground of course 200, but is more complete and goes further. It is intended for students who have greater facility in mathematical reasoning. It is a sufficient introduction to Mathematics 310, 320, and 330, and is open to students who obtain high grades in Mathematics 100 or otherwise satisfy the instructor of their fitness to take the course.

**Mathematics 220.** Modern Geometry and Algebra. Introduction to modern methods in geometry and algebra; abridged notation; line coordinates; reciprocal polars; cross ratio; projection; linear transformations; inversion.

**Mathematics 230.** History of Geometry. Introduction to Non-Euclidean geometry. This course is designed for those who are interested in the teaching of mathematics, and also for those who desire to investigate the development of our ideas with regard to space and time.

**Mathematics 300.** Advanced Calculus and Differential Equations. Differentiation and integration of functions of several variables; multiple integrals; introduction to the theory of differential equations. This course or Mathematics 310 should be taken by students whose major interest lies in physics or engineering; it is open to those who pass successfully in courses 200 or 210 in mathematics.

**Mathematics 310.** Advanced Calculus and Differential Equations. Applications of calculus to the study of curves and surfaces; differential equations; multiple and improper integrals. This is a more extended course than course 300, and is intended for students who have greater facility in mathematical reasoning. It is recommended to students who are specializing in mathematics, physics, and engineering.

**Mathematics 400.** Theory of Functions of a Real Variable. This course consists of the theory of sets of points; multiple, curvilinear, and improper integrals; the Lebesgue integral; Fourier series; and an introduction to the theory of differential equations. Open to those who satisfy the instructor of their fitness to take the course. Not given during the year 19 16-17.

**Mathematics 410.** Theory of Functions of a Complex Variable. An introductory course in the general theory of functions of a complex variable. Open to students who satisfy the instructor that they are prepared to take the course. Not given during the year 19 16-17.

**Mathematics 420.** Differential Equations. Ordinary and partial differential equations, with an introduction to integral equations. This course is designed to follow course 310.

**Mathematics 430.** Differential Geometry. A study of the properties of curves and surfaces in the neighborhood of a point; curves in the plane; analytical geometry of space; curves and surfaces in space. Open to students who have had course 210 or a third year course in mathematics. Not given during the year 1916-17.

**Mathematics 460.** Algebra. Introduction to higher algebra; group theory; Galois theory. This is a general course in algebra which introduces students to specialized work in higher algebra and analysis. Students should have covered the ground of course 220 before taking this course.

**Mathematics 470.** Vector Analysis. Vectors; vector fields; applications to geometry and physics; quaternions; generalized algebras. This course follows course 310.

**Applied Mathematics 200.** Mechanics. A study of the fundamental principles, with applications to machines and structures. It includes elementary statics, dynamics, and hydraulics. This course is a necessary part of the engineering course and is recommended to students of physics.

**Applied Mathematics 300.** General Dynamics. A broader study of the general principles of dynamics than Applied Mathematics 200, it is designed to take its place for students who are more advanced in mathematics. It forms a necessary foundation for further work in mathematical physics.

**Applied Mathematics 400.** The Dynamics of Elastic Bodies and Fluids. In this course the solutions of Laplace's equation and its correlatives by means of harmonic functions, circular, cylindrical, and spherical functions, will be studied and applied to physical problems. Applied Mathematics 300 is a prerequisite.

### 1917/18

**Mathematics 100.** Trigonometry, Analytic Geometry, and Advanced Algebra, constituting the freshman course in mathematics which is required of all students in the Institute.

**Mathematics 110.** Plane Analytic Geometry, Topics from Algebra and Analytic Geometry of Three Dimensions. For students who are already well grounded in trigonometry. The course will be given if applications are received from a sufficient number of properly qualified students.

**Mathematics 200.** Differential and Integral Calculus. This course, including the study of derivatives, indefinite and definite integrals, infinite series, and Taylor's theorem, is the foundation of theoretical physics and advanced mathematics, and the ideas introduced are, as ideas, of fundamental importance in many branches of modern thought.

**Mathematics 210.** Differential and Integral Calculus. This course covers the ground of Course 200, but is more complete and goes further. It is intended for students who have greater facility in mathematical reasoning. It is a sufficient introduction to Mathematics 310, 320, and 330, and is open to students who obtain high grades in Mathematics 100 or otherwise satisfy the instructor of their fitness to take the course.

**Mathematics 220.** Modern Geometry and Algebra. Introduction to modern methods in geometry and algebra; abridged notation; line coordinates; reciprocal polars; cross ratio; projection; linear transformations; inversion.

**Mathematics 300.** Advanced Calculus and Differential Equations. Differentiation and integration of functions of several variables; multiple integrals; introduction to the theory of differential equations. This course or Mathematics 310 should be taken by students whose major interest lies in physics or engineering; it is open to those who pass successfully in Course 200 or 210 in mathematics.

**Mathematics 310.** Advanced Calculus and Differential Equations. Applications of calculus to the study of curves and surfaces; differential equations; multiple and improper integrals; Fourier's Series. This is a more extended course than Course 300, and is intended for students who have greater facility in mathematical reasoning. It is recommended to students who are specializing in mathematics, physics, and engineering.

**Mathematics 400.** Theory of Functions of a Real Variable. This course consists of the theory of sets of points, the Lebesgue and Stieltjes integrals; integral equations; divergent series, and their applications to analysis.

**Mathematics 410.** Theory of Functions of a Complex Variable. An introductory course in the general theory of functions of a complex variable. Open to students who satisfy the instructor that they are prepared to take the course. Not offered in 1917-18.

**Mathematics 420.** Differential Equations. Ordinary and partial differential equations, with an introduction to integral equations. This course is designed to follow Course 310.

**Mathematics 430.** Line Geometry. A study of the geometry in which the line is the fundamental element. Open to students who satisfy the instructor that they are prepared to take the course.

**Applied Mathematics 200.** Mechanics. A study of the fundamental principles, with applications to machines and structures. It includes elementary statics, dynamics, and hydraulics. This course is a necessary part of the engineering course and is recommended to students of physics.

**Applied Mathematics 310.** Statistical Economics. An analysis of statistics as applied to economics and biology, theory of probability, mathematical theory of investment.

**Applied Mathematics 410.** Aerodynamics and Ballistics. This course investigates the dynamics of aeroplanes and projectiles, in particular, problems of resistance, stability, and trajectory.

### 1918/19

**Mathematics 100.** Elementary Analysis. A study of Algebra, Trigonometry and Analytic Geometry, constituting the Freshman course in mathematics which is required of all students in the Institute.

**Mathematics 200.** Differential and Integral Calculus. This course, including the study of derivatives, indefinite and definite integrals, infinite series, and Taylor's theorem, is the foundation of theoretical physics and advanced mathematics, and the ideas introduced are, as ideas, of fundamental importance in many branches of modern thought.

**Mathematics 210.** Differential and Integral Calculus. This course covers the ground of Course 200, but is more complete and goes further. It is intended for students who have considerable facility in mathematical reasoning. It is a sufficient introduction to Mathematics 310 and 430, and is open to students who obtain high grades in Mathematics 100 or otherwise satisfy the instructor of their fitness to take the course.

**Mathematics 220.** Modern Geometry and Algebra. Introduction to modern methods in geometry and algebra; abridged notation; line coordinates; reciprocal polars; cross ratio; projection; linear transformations; inversion.

**Mathematics 300.** Advanced Calculus and Differential Equations. Differentiation and integration of functions of several variables; multiple integrals; introduction to the theory of differential equations. This course or Mathematics 310 should be taken by students whose major interest lies in physics or engineering; it is open to those who pass successfully in Course 200 or 210 in mathematics.

**Mathematics 310.** Advanced Calculus and Differential Equations. Applications of calculus to the study of curves and surfaces; differential equations; multiple and improper integrals; Fourier's Series. This is a more extended course than Course 300, and is intended for students who have greater facility in mathematical reasoning. It is recommended to students who are specializing in mathematics, physics, and engineering.

**Mathematics 400.** Theory of Functions of a Real Variable. This course consists of the theory of sets of points, the Lebesgue and Stieltjes integrals; integral equations; divergent series, and their applications, to analysis. (Not offered in 1918-19.)

**Mathematics 410.** Theory of Functions of a Complex Variable. An introductory course in the general theory of functions of a complex variable. Open to students who satisfy the instructor that they are prepared to take the course. Hours to be arranged.

**Mathematics 420.** Differential Equations. Ordinary and partial differential equations, with an introduction to integral equations. This course is designed to follow Course 310. (Not offered in 1918-19.)

**Mathematics 430.** Differential Geometry. A study of the properties of curves and surfaces in the neighborhood of a point. Curves in the plane; analytical geometry of space; curves and surfaces in space. Open to students who have had Course 210 or a third year course in mathematics. Hours to be arranged.

**Mathematics 440.** Projective Geometry. A treatment of the geometry of position, based on a logical system of primitive concepts and postulates and developed both synthetically and analytically. (Not offered in 1918-19.)

**Mathematics 450.** Non-Euclidean Geometry. In this course the non-Euclidean metrical geometry is built up from the general projective geometry by means of the introduction of metrical concepts. The course is designed to follow immediately on Course 440. Hours to be arranged.

**Mathematics 460.** Algebra. Introduction to higher algebra; group theory; Galois theory. This is a general course in algebra which introduces students to specialized work in higher algebra and analysis. Students should have covered the ground of Course 220 before taking this course. (Not offered in 1918-19.)

**Applied Mathematics 200.** Mechanics. A study of the fundamental principles, with applications to machines and structures. It includes elementary statics, dynamics, and hydraulics. This course is a necessary part of the engineering course and is recommended to students of physics.

**Applied Mathematics 300.** General Analytical Dynamics. In this course the methods of generalized coordinates, Lagrange's equation, and Hamilton's principle will be applied to the problems of rigid bodies in mechanics and of thermodynamics and electricity in physics. This course is intended for advanced students in engineering, physics, and mathematics, and is open to those who satisfy the instructor that they are prepared for it. (Not offered in 1918-19.)

**Applied Mathematics 310.** Statistical Economics. An analysis of statistics as applied to economics and biology, theory of probability, mathematical theory of investment.

**Applied Mathematics 320.** The Dynamics of Elastic Bodies and Fluids. In this course the solutions of Laplace's equation and its correlatives by means of harmonic functions, circular, cylindrical, and spherical functions, will be studied and applied to physical problems. The course is intended for the same students as Course 300 in applied mathematics and is subject to the same conditions. (Not offered in 1918-19.)

**Applied Mathematics 400.** Theories of Radiation, Motion of Electrons, Gravitation. A study of some of the more modern hypotheses in theoretical physics. Hours to be arranged.

**Applied Mathematics 410.** Aerodynamics and Ballistics. This course investigates the dynamics of aeroplanes and projectiles, and in particular problems of resistance, stability, and trajectory. (Not offered in 1918-19.)

### 1919/20

**Mathematics 100.** Elementary Analysis. A study of Algebra, Trigonometry and Analytic Geometry, constituting the Freshman course in mathematics which is required of all students in the Institute.

**Mathematics 200.** Differential and Integral Calculus. This course, including the study of derivatives, indefinite and definite integrals, infinite series, and Taylor's theorem, is the foundation of theoretical physics and advanced mathematics, and the ideas introduced are, as ideas, of fundamental importance in many branches of modern thought.

**Mathematics 210.** Differential and Integral Calculus. This course covers the ground of Course 200, but is more complete and goes further. It is intended for students who have considerable facility in mathematical reasoning. It is a sufficient introduction to Mathematics 310 and 430, and is open to students who obtain high grades in Mathematics 100 or otherwise satisfy the instructor of their fitness to take the course.

**Mathematics 300.** Advanced Calculus and Differential Equations. Differentiation and integration of functions of several variables; multiple integrals;introduction to the theory of differential equations. This course or Mathematics 310 should be taken by students whose major interest lies in physics or engineering; it is open to those who pass successfully in Course 200 or 210 in mathematics

**Mathematics 310.** Advanced Calculus and Differential Equations. Applications of calculus to the study of curves and surfaces; differential equations; multiple and improper integrals; Fourier's Series. This is a more extended course than Course 300, and is intended for students who have greater facility in mathematical reasoning. It is recommended to students who are specializing in mathematics, physics, and engineering.

**Mathematics 340.** Modern Geometry and Algebra. Introduction to modern methods in geometry and algebra; abridged notation; line coordinates; reciprocal polars; cross ratio; projection; linear transformations; inversion.

**Mathematics 400.** Theory of Functions of a Real Variable. This course consists of the theory of sets of points, the Lebesgue and Stieltjes integrals; integral equations; divergent series, and their applications to analysis. Not offered in 1919-20.

**Mathematics 410.** Theory of Functions of a Complex Variable. An introductory course in the general theory of functions of a complex variable. Open to students who satisfy the instructor that they are prepared to take the course. Not offered in 1919-20.

**Mathematics 420.** Differential Equations. Ordinary and partial differential equations, with an introduction to integral equations. This course is designed to follow Course 310. Hours to be arranged.

**Mathematics 430.** Differential Geometry. A study of the properties of curves and surfaces in the neighborhood of a point. Curves in the plane; analytical geometry of space; curves and surfaces in space. Open to students who have had Course 210 or a third year course in mathematics. Not offered in 1919-20.

**Mathematics 440.** Projection Geometry. A treatment of the geometry of position, based on a logical system of primitive concepts and postulates and developed both synthetically and analytically. Not offered in 1919-20.

**Mathematics 450.** Non-Euclidean Geometry. In this course the non-Euclidean metrical geometry is built up from the general projective geometry by means of the introduction of metrical concepts. The course is designed to follow immediately on Course 440. Not offered in 1919-20.

**Mathematics 460.** Algebra. Introduction to higher algebra; group theory; Galois theory. This is a general course in algebra which introduces students to specialized work in higher algebra and analysis. Students should have covered the ground of Course 320 before taking this course. Not offered in 1919-20.

**Applied Mathematics 200.** Mechanics. A study of the fundamental principles, with applications to machines and structures. It includes elementary statics, dynamics, and hydraulics. This course is a necessary part of the engineering course and is recommended to students of physics.

**Applied Mathematics 300.** General Analytical Dynamics. In this course the methods of generalized coordinates, Lagrange's equation, and Hamilton's principle will be applied to the problems of rigid bodies in mechanics and of thermodynamics and electricity in physics. This course is intended for advanced students in engineering, physics, and mathematics, and is open to those who satisfy the instructor that they are prepared for it. Not offered in 1919-20.

**Applied Mathematics 310.** Statistical Economics. An analysis of statistics as applied to economics and biology, theory of probability, mathematical theory of investment.

**Applied Mathematics 320.** The Dynamics of Elastic Bodies and Fluids. In this course the solution of Laplace's equation and its correlatives by means of harmonic functions, circular, cylindrical, and spherical functions, will be studied and applied to physical problems. The course is intended for the same students as Course 300 in applied mathematics and is subject to the same conditions. Not offered in 1919-20.

**Applied Mathematics 400.** Theories of Radiation, Motion of Electrons, Gravitation. A study of some of the more modern hypotheses in theoretical physics. Hours to be arranged.

**Applied Mathematics 410.** Aerodynamics and Ballistics. This course investigates the dynamics of aeroplanes and projectiles, and in particular problems of resistance, stability and trajectory. Not offered in 1919-20.

### 1920/21

**Mathematics 100.** Elementary Analysis. This course includes Trigonometry and Analytic Geometry and constitutes the required Freshman course in mathematics. With the second term a section is formed for students who have considerable facility in mathematical reasoning.

**Mathematics 200.** Differential and Integral Calculus. This course is a continuation of Mathematics 100, and deals with derivatives, integrals and series, with their applications. One section is designed for students who have considerable facility in mathematical reasoning.

**Mathematics 300.** Advanced Calculus and Differential Equations. Differentiation and integration of functions of several variables; multiple integrals; introduction to the theory of differential equations. This course or Mathematics 310 should be taken by students whose major interest lies in physics or engineering; it is open to those who pass successfully in Course 200 or 210 in mathematics

**Mathematics 310.** Advanced Calculus and Differential Equations. Applications of calculus to the study of curves and surfaces; differential equations; multiple and improper integrals; Fourier's Series. This is a more extended course than Course 300, and is intended for students who have greater facility in mathematical reasoning. It is recommended to students who are specializing in mathematics, physics, and engineering.

**Mathematics 340.** Modern Geometry and Algebra. Introduction to modern methods in geometry and algebra; abridged notation; line coordinates; reciprocal polars; cross ratio; projection; linear transformations; inversions. This course should be taken by students specially interested in mathematics as early as the sophomore year

**Mathematics 400.** Theory of functions, real and complex variable. The important functions of analysis and modern general methods. Hours to be arranged.

**Mathematics 420.** Differential Equations. Ordinary and partial differential equations, with an introduction to integral equations. This course is designed to follow Course 310. Not offered in 1920-21.

**Mathematics 450.** Non-Euclidean Geometry. In this course the non-Euclidean metrical geometry is built up from the general projective geometry by means of the introduction of metrical concepts. Not offered in 1920-21.

**Mathematics 500.** Metrical, projective and non- Euclidean geometry. An advanced course. Hours to be arranged.

**Applied Mathematics 200.** Mechanics. A study of the fundamental principles, with applications to machines and structures. It includes elementary statics, dynamics, and hydraulics. This course is a necessary part of the engineering course and is recommended to students of physics.

**Applied Mathematics 300.** General Analytical Dynamics. In this course the methods of generalized coordinates, Lagrange's equation, and Hamilton's principle will be applied to the problems of rigid bodies in mechanics and of thermodynamics and electricity in physics. This course is intended for advanced students in engineering, physics, and mathematics, and is open to those who satisfy the instructor that they are prepared for it.

**Applied Mathematics 310.** Statistical Economics. An analysis of statistics as applied to economics and biology, theory of probability, mathematical theory of investment. Not offered in 1920-21.

**Applied Mathematics 320.** Theoretical Economics. Work of Cournot, Fisher, Walras. Not offered in 1920-21.

**Applied Mathematics 410.** Aerodynamics and Ballistics. This course investigates the dynamics of aeroplanes and projectiles, and in particular problems of resistance, stability and trajectory. Not offered in 1920-21.

**Applied Mathematics 500.** Theories of Radiation, Motion of Electrons, Gravitation. A study of some of the more modern hypotheses in theoretical physics. Hours to be arranged.

### 1930/31

Mathematics 0 begins 1927/28 school year

**Mathematics 100.** Elementary Analysis. Trigonometry, analytic geometry, and introduction to calculus. This course is required for Freshmen because it forms a necessary introduction to work in mathematics and pure and applied science, and assists the students in developing habits of self criticism in thinking and writing. As one of the most modern of sciences and, at the same time, one of the most ancient of humanities, mathematics is regarded as an integral part in any general education.

**Mathematics 0.** Elementary Algebra. This course begins about March first. It does not count towards a degree, since it contains nothing which is not a part of the requirement for entrance to the Institute. This course is intended and required for any student who has to drop Mathematics 100 through lack of knowledge of high school mathematics. Successful completion of the course is necessary in order that such a student may register again in Mathematics 100.

**Mathematics 200.** Differential and Integral Calculus. Elements of differential equations, differentials, definite integrals, infinite series, and their applications, especially to mechanics. Prescribed for engineers who do not take Mathematics 210. This course continues the work of Mathematics 100 in calculus and analytic geometry, with systematic applications to Newton's laws of motion and calculation of moments of forces and of inertia, centers of gravity, etc. Students who have considerable facility in mathematical reasoning should register for Mathematics 210.

**Mathematics 210.** Differential and Integral Calculus. This course covers the ground of Mathematics 200 but is more complete and goes further. It is open to students who obtain high grades in Mathematics 100, or otherwise satisfy the instructor of their fitness to take the course. A feature of this course is the writing of theses on the applications of mathematics to science, engineering, and philosophy, so that the student shall have practice in expressing himself in clear English.

**Mathematics 220.** Algebra and Mechanics. Solutions of equations, vectors, invariants, determinants, and interpolation; systematic statics and parts of dynamics. This course, required for engineers, fits the student with the algebraic technique necessary for the applications, and concerns itself with the fundamental principles of mechanics, and applications to machines and structures. It may be counted as a junior course if the student makes studies of additional thesis and problem subjects.

**Mathematics 300.** Advanced Calculus and Dynamics. Differentiation and integration of functions of several variables, differential equations, Fourier series, systematic dynamics. This course or Mathematics 310 should be taken by students whose major interest lies in science or engineering; it is open to those who have passed Mathematics 200 and 220, or otherwise satisfy the instructor of their fitness to take it.

**Mathematics 310.** Advanced Calculus and Dynamics. Students with considerable facility in mathematical reasoning should take this course rather than Mathematics 300, the ground of which it covers. Such students may take Mathematics 220 during the same year. Opportunity to write theses is given.

**Mathematics 320.** Geometry. A survey of elementary projective geometry using both synthetic and analytic methods; algebraic forms and their invariants. Metrical geometry; development of properties of space of distance relations, with applications to analytic and axiomatical geometry.

**Mathematics 400.** Theory of functions, real and complex variable. The important functions of analysis and modern general methods.

**Mathematics 420.** Differential and Integral Equations. Boundary value problems. Groups. Hours to be arranged.

**Mathematics 500.** Theory of functions of a complex variable. The algebraic functions and their integrals, functions of two or more complex variables and differential equations. Hours to be arranged.

**Mathematics 510.** Theory of functions of a real variable. Summable functions, Lebesgue and Stieltjes integrals, general integrals, functions of point sets and of plurisegments; Fourier series. Hours to be arranged.

**Mathematics 520.** Theory of Dimension and Curve Theory. Modern methods. Hours to be arranged.

**Applied Mathematics 310.** Finance, statistics and probability. Mathematical theory of investment, analysis of statistics as applied to economics and biology, theory of probability.

**Applied Mathematics 320.** Mathematical Introduction to Economics. A study of a unified sequence of economic problems by means of the elementary methods of the calculus. Mathematics 200 or 210 is a prerequisite.

**Applied Mathematics 500.** Advanced Mechanics and Relativity. This course assumes some knowledge of differential geometry, and gives the theory of Einstein and Weyl, based on the absolute calculus of Ricci and Levi-Civita. (Not offered in 1930-31.)

**Applied Mathematics 510.** Calculus of Variations. Problem of Plateau, minimal surfaces. (Not offered in 1930-31.)

**Applied Mathematics 520.** Celestial Mechanics and Cosmogony. Planetary motion, forms of equilibrium of rotating and radiating masses, and the evolution of stellar systems. (Not offered in 1930-31.)

**Seminar in Mathematics.** The Seminar meets every other week in order to allow the exposition of original investigations by its members.

**Seminar in Mathematical Physics.** A course in the mathematical methods of modern physics, given cooperatively by members of the Seminar. Hours to be arranged.

### 1940/41

Mathematics 0 gone 1935/36 school year

**Mathematics 100**. Elementary Analysis. Trigonometry and analytic geometry. This course is required for Freshmen because it forms a necessary introduction to work in mathematics and pure and applied science, and assists the students in developing habits of self criticism in thinking and writing. As one of the most modern of sciences and, at the same time, one of the most ancient of humanities, mathematics is regarded as an integral part of any general education. Engineering sections meet in three two- hour periods.

**Mathematics 200.** Differential and Integral Calculus. Derivatives, differentials, definite integrals, infinite series, and their applications, especially to mechanics. Prescribed for engineers who do not take Mathematics 210. This course continues the work of Mathematics 100 in calculus and analytic geometry, with applications to Newton's laws of motion and calculation of moments of forces and of inertia, centers of gravity, etc. Students who have considerable facility in mathematical reasoning should register for Mathematics 210.

**Mathematics 210.** Differential and Integral Calculus. This course covers the ground of Mathematics 200 but is more complete and goes further. It is open to students who obtain high grades in Mathematics 100, or otherwise satisfy the instructor of their fitness to take the course. A feature of this course is the writing of theses on the applications of mathematics to science, engineering, and philosophy.

**Mathematics 220.** Algebra and Mechanics. Solutions of equations, vectors, invariants, determinants, and interpolation; systematic statics and parts of dynamics. The second half deals with statics and parts of dynamics. The algebraic technique necessary for the mechanical applications is provided in the work of the first half. All engineering students are required to take the first half of Mathematics 220 or 230.

**Mathematics 220A.** Algebra. The first half of Mathematics 220. Open to all engineering students.

**Mathematics 230.** Algebra and Geometry. The work of the first half-year is algebra, the same as the work of Mathematics 220A. In the second half-year, general algebraic methods are applied to plane and solid analytic geometry and to the projective study of conies. This course is especially recommended to students who are preparing to teach mathematics in high school. It may be counted as a Junior course if the student makes studies of additional thesis and problem subjects. (Not offered 1940-41.)

**Mathematics 230A.** Algebra. The first half of Mathematics 230 and the same as Mathematics 220A. Open to all engineering students. (Not offered 1940-41.)

**Mathematics 300.** Advanced Calculus and Differential Equations. Multiple integrals, infinite series, and partial differentiation, with many applications, and the geometry of three dimensions; differential equations. This course, or Mathematics 310, is prescribed for electrical engineering students; civil and mechanical engineers are required to take the first half of it. Open to those who have passed Mathematics 200 or 210 or otherwise satisfy the instructor of their fitness to take the course.

**Mathematics 300A.** The first half of Mathematics 300. Open to civil and mechanical engineers.

**Mathematics 310.** Advanced Calculus and Dynamics. Students with considerable facility in mathematical reasoning should take this course rather than Mathematics 300, the ground of which it covers. Such students may take Mathematics 220 during the same year. Opportunity to write theses is given.

**Mathematics 400.** Theory of Functions, real and complex variable. The important functions of analysis and modern general methods.

**Mathematics 410.** Differential Geometry. The problem of area; subharmonic functions; the problem of Plateau.

**Mathematics 420.** Differential Equations and Introduction to the Calculus of Variations.

**Mathematics 500.** Theory of Functions of a Complex Variable. Harmonic functions, subharmonic functions, normal families, conformal mapping, automorphic functions, meromorphic functions. (Not offered 1940-41.)

**Mathematics 510.** Theory of Functions of a Real Variable. Summable functions, Lebesgue and Stieltjes integrals, general integrals, functions of point sets and of plurisegments; Fourier series. Hours to be arranged.

**Mathematics 520.** Series expansions in terms of orthogonal systems of functions. Trigonometric series. The course is based upon Mathematics 510. (Not offered 1940-41.)

**Mathematics 530.** Theory of Groups. (Not offered 1940-41.)

**Mathematics 540.** Introduction to Modern Algebra. The purpose of this course is to provide the student with a knowledge of the foundations of modern algebra. The topics to be presented will include the theory of sets, groups, rings and fields, galois theory, polynomials, algebraic numbers and ideals. The course will be open to graduate students and, with the consent of the instructor, to well-qualified Seniors. (Not offered 1940-41.)

**Mathematics 550.** Seminar on Continued Fractions. This seminar meets once a week for a two-hour period. It is open to graduate students who satisfy the instructor of their fitness for the course. Hours to be arranged.

**Mathematics 590.** Thesis.

**Applied Mathematics 310.** Finance, Statistics, and Probability. Mathematical theory of investment, analysis of statistics as applied to economics and biology, theory of probability. Hours to be arranged.

**Applied Mathematics 510.** Integral Equations; Potential Theory. Laplace's and related equations. Boundary value problems. (Not offered 1940-41.)

**Mathematical Colloquium.** The colloquium meets every other week in order to allow the exposition of original investigations by its members.

### 1950/52

No Applied Mathematics Classes listed beginning this General Announcement

**Mathematics 100.** Elementary Analysis. Trigonometry, analytic geometry, and elementary calculus. This course is required for Freshmen because it forms a necessary introduction to work in mathematics and pure and applied science, and assists the students in developing habits of self-criticism in thinking and writing. As one of the most modern of sciences and, at the same time, one of the most ancient of humanities, mathematics is regarded as an integral part of any general education. Engineering sections meet in three two-hour periods. Staff

**Mathematics 200.** Differential and Integral Calculus. Derivatives, differentials, definite integrals, infinite series, and their applications, especially to mechanics. This course continues the work of Mathematics 100 in calculus and analytic geometry, with applications to Newton's laws of motion and calculation of moments of forces and of inertia, centers of gravity, etc. Prescribed for all science-engineering majors who do not take Mathematics 210. Students who have considerable facility in mathematical reasoning should register for Mathematics 210. Staff

**Mathematics 210.** Differential and Integral Calculus. This course covers the ground of Mathematics 200 but is more complete and goes further. It is open to students who obtain high grades in Mathematics 100, or otherwise satisfy the instructor of their fitness to take the course. Mr. Bray or Mr. Calkin

**Mathematics 300.** Advanced Calculus and Differential Equations. Multiple integrals, infinite series, and partial differentiation, with many applications, and the geometry of three dimensions; differential equations. This course, or Mathematics 310, is prescribed for civil, electrical, and mechanical engineering students. Open to those who have passed Mathematics 200 or 210, or otherwise satisfy the instructor of their fitness to take the course. Messrs. Bray, Brunk, Calkin, and MacLane

**Mathematics 310.** Advanced Calculus and Differential Equations. Students with considerable facility in mathematical reasoning should take this course rather than Mathematics 300, the ground of which it covers. Opportunity to write theses is given. Mr. Bray or Mr. Brunk

**Mathematics 320.** Analytical Mechanics. Vector analysis; reduction of systems of forces and conditions for equilibrium. Dynamics of systems of particles; rigid bodies. Prerequisites: Mathematics 200 and 300. (The latter may be taken concurrently.) Mr. Calkin or Mr. Ulrich

**Mathematics 320a.** Analytical Mechanics. The first part of Mathematics 320. Mr. Calkin or Mr. Ulrich

**Mathematics 330.** Introduction to Higher Algebra. Properties of determinants and matrices. Theory of linear dependence. Bilinear and quadratic forms. Polynomials. Invariants. Lambda matrices and applications.

**Mathematics 340.** Differential Geometry. Theory of curves and surfaces. Geodesies. Mapping of surfaces. The absolute geometry of a surface.

**Mathematics 400.** Theory of Functions of a Complex Variable. This course is fundamental in analysis. Besides giving an introduction to basic concepts of analysis, it includes the study of analytic functions of a complex variable, the Cauchy-Riemann equations, Cauchy's Integral Theorem, Taylor's series, calculus of residues, and conformal mapping. Prerequisite: Mathematics 310. Mr. Ulrich

**Mathematics 410.** Differential Equations and Introduction to the Calculus of Variations. Prerequisite: Mathematics 300. Mr. Calkin or Mr. MacLane

**Mathematics 420.** Mechanics. Topics selected from the following: Dynamics of systems. Principle of d'Alembert. General equations of analytical dynamics. Principles of Hamilton. Hydrostatics. General theorems on perfect fluids. Theory of elasticity, elastic equilibrium, interior motions. Equations of the motion of a viscous fluid.

**Mathematics 430.** Introduction to Modern Geometry. Synthetic and algebraic geometry. The group of projective transformations and certain subgroups of the group of projective transformations. The geometries defined by these groups. Projective correspondences. Projective theory of conies.

**Mathematics 440a.** Topology. Postulates on open sets. Various topological structures. Continuous functions defined in a topological space and taking values in another topological space. Metric spaces. (First half-year.)

**Mathematics 440b.** Introduction to Modern Algebra. Groups, rings, and fields. The theory of ideals. The real and complex number systems. Polynomials. Matrix algebra; quadratic forms. (Second half-year.)

**Mathematics 500a.** Theory of Functions of a Complex Variable. Normal families of functions; theorems of Montel; theorems of Stieltjes and Vitali; theorems of Picard, Schottky, Landau, and Caratheodory; theorems of Julia and Ostrowski. (First half-year.) Mr. Mandelbrojt

**Mathematics 500b.** Theory of Functions of a Complex Variable. Theory of the distribution of values. (Second half-year.) Mr. Ulrich

**Mathematics 501.** Theory of Functions of a Complex Variable. A study of special analytic functions of importance in mathematical physics. The course is usually given as a seminar. Mr. Ulrich

**Mathematics 502a.** Topological Groups. (First half-year.)

**Mathematics 510.** Theory of Functions of a Real Variable. Theory of real numbers. Summable functions, Lebesgue and Stieltjes integrals, general integrals, functions of point sets and of plurisegments, Fourier series. Mr. Bray

**Mathematics 515.** Probability and Statistics. Mr. Brunk

**Mathematics 520.** Trigonometric Series and Related Topics. Series expansions in terms of orthogonal systems of functions. Trigonometric series. Fourier transforms and integrals. The course is based upon Mathematics 510. Mr. Bray or Mr. Brunk

**Mathematics 530.** Laplace Transformations. Theory of the Laplace transformation with particular reference to the properties of the transform as a function of a complex variable. Applications to the solution of difference equations, integral equations of the convolution type, and ordinary differential systems. Boundary value problems. Certain Sturm-Liouville systems. Abelian and Tauberian theorems. Asymptotic representations. Mr. Ulrich

**Mathematics 535a.** Fourier Transforms in the Complex Domain. Properties of the class of Fourier transforms of functions of class (L). Properties of the solution of the integral equation of convolution type with special reference to the Fourier transforms of the kernel, together with a study of the complex transform of the solutions. General Tauberian theorems. The Paley-Wiener theorem. Applications. (First half-year.) Mr. Mandelbrojt

**Mathematics 535b.** Analytic Continuation and Infinitely Differentiable Functions. Topics selected from the following: regularization of sequences, problem of equivalence of classes, quasi-analyticity, Watson's problem, applications to Fourier series, singularities of Taylor series, relationship between singularities of Taylor series and quasi-analyticity. The course will be based on a general theory of asymptotic series. (Second half-year.)

**Mathematics 536a.** Theory of Composition. Properties of functions defined by composition of convolution type in relation to the component functions. Applications to the study of functions defined by Taylor series and Dirichlet series. Applications to asymptotic series and quasi-analyticity. (First half-year.) Mr. Mandelbrojt

**Mathematics 536b.** General Problem of Moments. The Stieltjes, Hausdorff, and Hamburger problems. Connections with the theory of Stieltjes continued fractions. Connection with the theory of functions holomorphic in a half-plane. Applications. General related problems. (Second half-year.)

**Mathematics 540.** Mathematical Foundations of Linear Physics. Matrix algebra; coupled systems and normal coordinates. Differential and integral equations; orthogonal functions; vibrating systems. The Fourier integral and the problem of heat flow. The Schrodinger equation. Mr. Calkin

**Mathematics 545b.** Hydrodynamics. Selected topics in the theory of incompressible fluid motion. Introduction to the problems of compressible flow. (First half-year.) Mr. Calkin

**Mathematics 550.** Advanced Theory of Riemann Surfaces: topological properties, theory of entire and meromorphic functions, problem of type. Mr. Ulrich

**Mathematics 562.** Theory of Linear Vector Spaces and Its Applications to Analysis. Function spaces. The theory of Hilbert space and its applications. Prerequisite: Mathematics 510. Mr. Calkin

**Mathematics 590.** Thesis.

**Mathematical Colloquium.** The colloquium usually meets one afternoon every other week in order to allow the exposition of original investigations by its members.

### 1960/61

**Mathematics 100.** Elementary Analysis Calculus and analytic geometry. The idea of the calculus are introduced by considering the rate and area problems. The course includes the differentiation of the elementary function and some of the simpler integration formulae, with applications. Analytic geometry, through a study of the conic sections and the reduction of the general equation of second degree, is treated. This course is required of all freshmen because it forms a necessary introduction to work in mathematics and pure and applied science, and assists the student in developing habits of self criticism in thinking and writing. As one of the most modem of the sciences and at the same time, one of the most ancient of the humanities, mathematics is regarded as an integral part of any general education. Science-engineering sections meet four hours per week. Staff

**Mathematics 101.** Fundamental Concepts of Mathematics A course designed expressly for students in the academic division and intended to convey an appreciation of the edifice of mathematical ideas, the topics treated being largely chosen for the light they shed on the nature and role of mathematics. The elements of algebraic theory of ruler and compass constructions. The ideas of the calculus are introduced in connection with the analysis of planetary motion. An important part of the program consists of a critical study of the number systems. The course begins with a brief introduction to the concepts and notation of logic. Staff

**Mathematics 200.** Differential and Integral Calculus. Systematic integration, definite integral, improper integrals, infinite series, analytic geometry in three dimensions, algebra of vectors and multiple integrals. Applications of physical problems. Prescribed for all science-engineering majors who do not take Mathematics 210. Students who have considerable facility in mathematical reasoning should register in Mathematics 210. Staff

**Mathematics 210.** Differential and Integral Calculus. This course has the same scope as Mathematics 200 but is more complete and rigorous. It is open to students who have passed Mathematics 100 with high standing, or otherwise satisfy the instructor of their fitness to take the course. Staff

**Mathematics 300.** Advanced Calculus and Differential Equations. Partial differentiation with applications to geometry of three dimensions, vector analysis and differential equations. This course, or Mathematics 310, is prescribed for all science-engineering students. Open also to other students who have passed Mathematics 200 or 210, or otherwise satisfy the instructor of their fitness to take the course. Staff

**Mathematics 310.** Advanced Calculus and Differential Equations. This course is designed for students with considerable facility in mathematical reasoning. The scope is essentially that of Mathematics 300 but the development is more systematic and rigorous. It is open to students who have passed Mathematics 200 or 210 with high standing or otherwise satisfy the instructor of their fitness to take the course. Staff

**Mathematics 320.** Analytical Mechanics. Vector analysis, reduction of systems of forces and conditions for equilibrium, dynamics of systems of particles, rigid bodies. Prerequisites: Mathematics 200 and 300. (The latter may be taken concurrently.) Mr. MacLane or Mr. Ulrich

**Mathematics 330.** Introduction to Higher Algebra. Properties of determinants and matrices, theory of linear dependence, bi-linear and quadratic form, polynomials, invariants, lambda matrices and applications. Staff

**Mathematics 360.** An Introduction to Mathematical Probability and Statistics. Topics covered will include: conditional probability, Bernoulli's Theorem, law of large numbers, distributions, central limit theorem, correlation, large and small sample theory, goodness of fit, testing statistical hypotheses, and the design of experiments. Insofar as possible, the mathematical foundations wll be emphasized. Prerequisite: Mathematics 300 (may be taken concurrently). Enrollment with permission of instructor. Mr. Douglas

**Mathematics 400.** Theory of Functions of a Complex Variable. This course is fundamental in analysis. Besides giving an introduction to basic concepts of analysis, it includes the study of analytic functions of a complex variable, the Cauchy-Riemann equations, Cauchy's Integral Theorem, Taylor's series, calculus of residues, and conformal mapping. Mr. Ulrich

**Mathematics 410.** Differential Equations and an Introduction to the Calculus of Variations. Geometry of the integral curves and the classification of the singularities of equations of first order, existence theorems, theory of integrating factors and integration by elementary means, general theory of second order linear equations, oscillation and comparison theorems, fuchsian theory of regular singular points. Eigenvalue problems, general partial differential equations of first order, boundary value problems for certain second order linear systems, and as much calculus of variations as time permits. Mr. MacLane or Mr. Ulrich

**Mathematics 420.** Differential Geometry. Theory of curves and surfaces, geodesies, mapping of surfaces, the absolute geometry of a surface.

**Mathematics 430.** Introduction to Modern Geometry. Synthetic and algebraic geometry, the group of projective transformations and certain subgroups of the group of projective transformations, the geometries defined by these groups, projective correspondences, projective theory of conics.

**Mathematics 440.** Algebra and Topology. Groups, rings, fields, vector spaces, topological spaces, fundamentals of homology theory, homotopy, and covering spaces, classification of surfaces, Riemann surfaces. Mr. Brown or Mr. MacLane

**Mathematics 450.** Number Theory. The fundamental theorem of arithmetic, residue class rings and congruences, quadratic residues and reciprocity law. Numerical functions. Algebraic number fields, factorization and ideals. Mr. Durst

**Mathematics 460.** Numerical Analysis. Approximate integration and differentiation by finite differences, interpolation, functional approximation, linear and non-linear algebraic equations, eigenvalues, approximate solution of ordinary differential equations and of some simple partial differential equations. A digital computer is available for laboratory use. Prerequisite: Mathematics 300 or 310. Mr. Douglas

**Mathematics 500.** Theory of Normal Families of Functions. Equicontinuity and Ascoli's Lemma, limiting oscillation of Ostrowski and Caratheodory's continuous convergence, Orsove's theorem on normal families of potential functions, theorems of Vitali and Montel and Mandelbrojt's theory of kernals, location and description of singularities of families of analytic functions. Mr. Johnson

**Mathematics 501.** Theory of Functions of a Complex Variable. A study of special analytic functions of importance in mathematical physics. The course is usually given as a seminar. Mr. Ulrich

**Mathematics 505a.** Selected Topics from the Theory of Functions of a Complex Variable. The subject matter of this course varies from year to year. In past years the following topics have been among those presented: singularities of a function defined by a Taylor series, elementary theory of Dirichlet series, approximation theory, and constructive theory of functions. Mr. Mandelbrojt

**Mathematics 510.** Theory of Functions of a Real Variable. Theory of real numbers, limits and continuity, Lebesgue and Stieljes integrals, general integrals, the theory of differentiation, Fourier series, function spaces, selected topics. The student should be familiar with some of the material of Mathematics 440. These courses could be taken concurrently. Mr. Brown

**Mathematics 520.** Trigonometric Series and Related Topics. Series expansions in terms of orthogonal systems of functions. Trigonometric series. Fourier transforms and integrals. The course is based upon Mathematics 510. Mr. Bray

**Mathematics 525.** Modern Theory of Meromorphic Functions. Riemann surfaces and covering surfaces, harmonic measure and logarithmic capacity, the Nevanlinna theory of meromorphic functions and the defect relation, boundary behavior and the theory of cluster sets. Mr. Lohwater

**Mathematics 530.** Laplace Transformations. Theory of the Laplace transformation with particular reference to the properties of the transform as a function of a complex variable. Applications to the solution of difference equations, integral equations of the convolution type, and ordinary differential systems. Boundary value problems. Certain Sturm-Liouville systems. Abelian and Tauberian theorems. Asymptotic representations. Mr. Ulrich

**Mathematics 535.** Partial Differential Equations. Theorems of Couchy-Kowalewski and Holmgren, classification of partial differential equations. Cauchy problem for first order hyperbolic systems and the wave equation, boundary value problems for second order hyperbolic, elliptic and parabolic equations, numerical solution of partial differential equations and systems. Prerequisites: Mathematics 410 or 400; preferably both, Mr. Douglas

**Mathematics 540.** Topological Linear Algebra. Vector spaces. The elementary geometric and algebraic properties of Banach and Hilbert spaces. Normed rings. Operators and spectral theory. Applications and topics of related interest. Prerequisite: Mathematics 510. Mr. Brown

**Mathematics 545.** Theory of Algebraic Functions. Theory of elliptic functions. Properly discontinuous groups of linear transformations. Automorphic functions. Uniformization of algebraic functions, Mr. MacLane or Mr. Ulrich

**Mathematics 550.** Advanced Theory of Riemann Surfaces. Topological properties, theory of entire and meromorphic functions, problem of type. Mr. MacLane or Mr. Ulrich

**Mathematics 555.** Recent Developments in the Theory of Riemann Surfaces. Mr. MacLane or Mr. Ulrich

**Mathematics 560.** Potential Theory. Integral theorems of potential theory, Riesz's theorem on potentials of negative mass and subharmonic functions, a flux integral for functions which have harmonic support, boundary value problems, Poisson Integral and Green's function, exceptional points of the boundary, theorems of Kellogg and Evans, recent researches on boundary topologies. Mr. Johnson or Mr. Lohwater

**Mathematics 570a.** Selected Topics from Advanced Analysis. The subject matter of this course varies from year to year. In past years the following topics have been among those presented: Fourier transforms in the complex domain, analytic continuation and infinitely differentiable functions, theory of composition, general Tauberian theorems, general problem of moments, closure theorems, general asymptotic representations, zeta-function of Riemann, and analytic theory of numbers, ergodic theory, and monogenic and isogenic functionals and harmonic analysis. Mr. Mandelbrojt

**Mathematics 600.** Thesis

**Mathematical Colloquium.** The colloquium usually meets one afternoon every other week in order to allow the exposition of original investigations by its members.

### 1970/71

Mathematical Sciences begins 1968/69

**Mathematics 101a, 102b.** Elementary Analysis). Limits, differentiation, and integration are introduced early in the year, and applications are discussed. Other topics include a careful definition of trigonometric and exponential functions, analytic geometry, partial differentiation and vector methods. The course is designed to give the student an introduction not only to the applications of the calculus but also to the techniques of mathematical reasoning; it (or Mathematics 121a, 122b) is the basic course in mathematics and is required of most majors in the Science-Engineering Division. This course, however, is open to all students.

**Mathematics 103a.** Introduction to Calculus and Its Applications. Develops techniques of differential and integral calculus with emphasis on problem solving and applications rather than the rigorous underpinnings. Intended for non-Mathematics-Science students and not allowed for Mathematics majors.

**Mathematics 104b.** Finite Mathematics. Topics selected from the elementary propositional calculus, sets and subsets, partitions and counting, probability theory on finite sample spaces, finite Markov- chains, the gambler's win. Not open for Mathematics majors.

**Mathematics 107a.** The Role of Mathematics in Civilization. Intended for students interested in the nature and impact of research mathematics but who do not need mathematics as a tool in their area of specialization. Number systems, positional notation, efficiency of computation, inequalities, navigation, cartography, algebrization of geometry, calculus, and differential geometry are interwoven to exhibit the mathematical nature of major practical advances.

**Mathematics 108b.** Elementary Number Theory. Properties of numbers depending on the notion of divisibility. No prior mathematical knowledge is required.

**Mathematics 121a, 122b.** Analysis. An honors course for Freshmen. Registration by permission of the department. Selection is made on the basis of either the CEEB Advanced Placement Examination on analytic geometry and calculus or a qualifying examination given by the Mathematics Department at the beginning of the school year. The students are expected to be familiar with techniques of differentiation, integration, areas, volumes, max- min problems, etc., so that emphasis can be placed on the theoretical aspects.

**Mathematics 201a.** Linear Algebra. Linear transformations and matrices, the solution of linear equations, the eigenvalue problem, and quadratic forms. The topics covered are chosen from those applicable to the multi-variable calculus. No prerequisites.

**Mathematics 202b.** Advanced Analysis. Topology of Rn , the differential of a function, partial derivatives, chain rule, max-min problems, quadratic forms, Taylor's series, multiple integrals, curvilinear coordinates and change of variables in integration, Green's Theorem, Stokes' Theorem.

**Mathematics 205a, 206b.** Advanced Analysis. The course has the same scope as Mathematics 201a, 202b but is more complete and rigorous. Prerequisite: Written permission of the department.

**Mathematics 221a, 222b.** Honors Analysis. The important topics of Mathematics 201a, 202b are covered with greater generality and completeness. Additional topics selected from implicit function theorem, generalized Stokes' theorem, partial differential equations of mathematical physics, Fourier integral and series.

**Mathematics 301a.** Ordinary Differential Equations. Integration of first order equations by elementary methods, geometry of integral curves, existence and uniqueness theorems for first order differential equations, systems of equations, initial and boundary value problems, properties of solutions of linear equations, separation and oscillation theorems, theory of regular singular points, Sturm-Liouville systems, expansion problems.

**Mathematics 302b.** Partial Differential Equations. First order linear equations, characteristic curves, method of Lagrange, Cauchy problem for quasi-linear first order equations, existence and uniqueness theorems, classification of higher order equations, reduction to normal form, harmonic functions, Dirichlet, Neumann and mixed boundary value problem, Cauchy problem, initial and boundary value problems for parabolic and hyperbolic equations.

**Mathematics 321a.** Real Analysis. Lebesgue and Daniell theory of measure and integration.

**Mathematics 322b.** Complex Analysis. Power series, line integrals in the plane, Cauchy-Riemann equations, Cauchy's Theorem and its consequences, Schwarz' Lemma, Maximum Principle, singularities, calculus of residues and computation of integrals, harmonic functions, Dirichlet's problem and the Poisson integral, infinite series and products of meromorphic functions. Topics such as the Riemann Mapping Theorem, Runge's Theorem, Bergman kernel function, and elementary elliptic function theory will be covered as time permits.

**Mathematics 341a.** General Topology. An introduction to the basic notions of point set topology. Topics include elements of set theory, the well ordering principle, general topological spaces, continuity, compactness, connectedness, elementary separation axioms, metric spaces, product spaces, Tychonoff theorem, function spaces, the compact open topology, plus additional topics as time allows, such as Tietze extension theorem, a metrization theorem, arcs, paracompactness. Prerequisite: One of Mathematics 202b, 206b, 222b, or consent of instructor.

**Mathematics 342b.** Geometrical Topology. An introduction to algebraic methods in topology and differential topology. Topics include the fundamental group and covering spaces plus additional topics varying from year to year, such as 2-manifolds, elementary dimension theory, simplicial complexes, cohomology theory of manifolds, the DeRham theorem, elementary bundle theory, elementary differential topology. Prerequisite: Mathematics 341a or consent of instructor.

**Mathematics 361a, 362b.** Algebra. An introduction to the basic structures of algebraic systems: groups, rings, fields, and their morphisms. Vector spaces are studied extensively, including matrices, determinants, characteristics values, canonical forms, multilinear algebra. Basis theorem of abelian groups and modules is established. Prerequisite: Mathematics 202b, 206b, or 222b.

**Mathematics 371a, 372b.** Honors Algebra. An honors course in algebra including the material of Mathematics 361a, 362b, finite group theory, and Galois theory. Prerequisite: Mathematics 222b.

**Mathematics 401a, 402b.** Differential Geometry. Differentiable manifolds. Stokes' theorem and deRham's theorem. Fundamental theorem of local Riemannian geometry, manifolds in Euclidean spaces, Lie groups, vector space bundles, theory of affine connections.

**Mathematics 421a, 422b.** Ordinary Differential Equations. Existence and uniqueness theorems; linear systems; boundary value problems on an interval; self-adjoint problems; Green's function; Weyl's limit circle and limit point theory; Poincare-Bendixson theory. Prerequisites: Mathematics 321a, 322b or permission of the instructor.

**Mathematics 423a, 424b.** Partial Differential Equations. Cauchy-Kowalewski theorem, classification of partial differential equations, first- order hyperbolic systems, harmonic functions and potential theory, Dirichlet and Neumann problems, the Dirichlet principle, integral equations and the Fredholm alternative, hyperbolic equations, energy estimates, parabolic equations. Properties of solutions of elliptic and parabolic equations.

**Mathematics 425a, 426b.** Probability Theory. While topics may vary from year to year, a typical selection would be classical distributions, laws of large numbers, conditional expectation, renewal theory, independence and dependence, Kolmogrov construction of a stochastic process, martingales, Markov chains. Not offered in 1970-71. Prerequisite: Mathematics 321a.

**Mathematics 431a.** Topics in Complex Analysis. Content varies from year to year. Prerequisite: Mathematics 322b.

**Mathematics 432b.** Topics in Real Analysis. Content varies from year to year. Prerequisite: Mathematics 321a.

**Mathematics 435a.** Classical Numerical Analysis. Computationally oriented studies of numerical techniques: polynomial and other approximations, interpolation, finite difference methods, numerical solution of ordinary differential equations, solution of non-linear algebraic equations, iteration. Intended primarily for non-mathematics majors.

**Mathematics 437a.** Numerical Analysis: Numerical Linear Algebra and Ordinary Differential Equations. Matrix and vector operations: solution of linear systems, determination of eigensystems; condition and rounding errors. Single and multistep methods for initial value problems, finite difference and Galerkin methods for two-point boundary value problems.

**Mathematics 438b.** Numerical Analysis: Partial Differential Equations and Approximation Theory. Finite difference and Galerkin methods for partial differential equations, solution of related linear systems using approximate inverse methods: relaxation, alternating direction and factoring. Existence, uniqueness and closeness of solutions to linear approximation problems; splines, Hermite interpolation.

**Mathematics 441a, 442b.** Algebraic Topology. This course develops homotopy theory, theory of fiber spaces, singular homology and cohomology. Theorems of Hurewicz and Whitehead are established. Spectral sequences are studied and used to analyze fiber spaces. Serre C-theory is developed. Geometrical applications are made in studying fixed-point theory, imbedding problems. Prerequisites: Mathematics 342b and one of Mathematics 122b, 222b, 362b,372b.

**Mathematics 461a, 462b.** Topics in Algebra. Content varies from year to year. Prerequisite: Mathematics 362b or 372b.

**Mathematics 501a, 502b.** Advanced Topics in Geometry. Content varies. Enrollment with permission of instructor.

**Mathematics 521a, 522b.** Several Complex Variables. Typical topics from the local theory of analytic varieties are: local parameterization, the Remmert-Stein theorem, the proper mapping theorem, uniformization, Chow's theorem, and the algebraic dependence theorem. Typical topics in the global theory of complex manifolds are: holomorphic convexity and pseudo-convexity, the E. E. Levi problem, existence and approximation results for the operator, and analogous results for coherent analytic sheaves.

**Mathematics 523a, 524b.** Functional Analysis. Topological linear spaces, theory of distributions, Banach algebras, harmonic analysis.

**Mathematics 525a, 526b.** Advanced Topics in Analysis. Content varies. Enrollment with permission of instructor.

**Mathematics 527a, 528b.** Advanced Topics in Probability. 1970-71: Stochastic Processes. Basic concepts of large-scale probability processes. Fourier transforms (characteristic functions). Stochastic integrals and derivatives. Mr. Bochner

**Mathematics 535a, 536b.** Advanced Numerical Analysis. Typical topics include the Laplace transform and its application to problems in differential equations and complex variable theory, special functions of mathematical physics, methods of mathematical physics, calculus of variations, numerical analysis.

**Mathematics 541a, 542b.** Advanced Topics in Topology. Content varies. Enrollment with permission of instructor.

**Mathematics 561a, 562b.** Advanced Topics in Algebra. Content varies. Enrollment with permission of instructor.

**Mathematics 601a, 602b.** Thesis

### 1979/81

**M101a,b; 102a,b.** Differential and Integral Calculus for Functions of One Variable. Includes careful discussion of continuity; sequences, series, and power series. Mathematics 102 is open to entering students with advanced placement and departmental approval. Offered in both "self-paced" and "traditional" format. Mr. Curtis, Mr. Dadok, Mr. Veech, Mr. Rachford, Mr. Stanton, Staff

**M103a, 106b.** Introduction to Calculus and Its Applications. Emphasis on problem solving and applications. Intended for non-science- engineering students. Not open to mathematics majors. Mr. Pfeiffer

**M104b.** Finite Mathematics. Topics from elementary propositional calculus, partitions and counting, linear programing. Not open to mathematics majors. Mr. Palmer

**M107, 108.** The Role of Mathematics in Civilization. Intended for students who are interested in the nature and impact of mathematics but do not need mathematics as a tool in their own field. Not offered 1980-81. Mr. Beatrous

**M121a, 122b.** Honors Analysis. Covers the material of 101, 102 with emphasis on theoretical aspects. Registration by departmental permission. Mr. Shalen

**M211a,b.** Ordinary Differential Equations. Mr. Jones, Mr. Harvey, Mr. Fegan, Staff

**M212b.** Differential and Integral Calculus for Functions of Several Variables. Mr. Beatrous, Mr.Jaco, Mr. Taylor

**M221a, 222b.** Advanced Honors Analysis. Covers the material of Mathematics 211, 212. Emphasis is on theoretical aspects. Curves, surfaces, and more general manifolds. Stokes' theorem in detail. Mr. Polking

**M312b.** Principles of Analysis. A careful treatment of: the topology of R" , convergence of sequences and series of functions, the implicit function theorem, existence theorems for ODE's, and related topics. Mr. Fegan

**M350b.** Set Theory. Ordinal and cardinal numbers and arithmetic, well ordering, the axiom of choice, and additional topics as time allows, e.g., Godel's axiom of constructibility, large cardinal numbers. Mr. Beale

**M355a.** Linear Algebra. Linear transformations and matrices. Solution of linear equations. The eigenvalue problem and quadratic forms. Mr. Stanton

**M356b.** Abstract Algebra. Groups: normal subgroups, factor groups, Abelian groups. Rings: ideals, Euclidean rings, unique factorization. Fields: algebraic extensions. Finite fields. Note: Students may not take this course and Mathematics 463. Mr. Fegan

**M365a.** Elementary Number Theory. Properties of numbers depending mainly on the notion of divisibility. Continued fractions. Staff

**M366b.** Projective Geometry. Staff

**M381a.** Analysis and Applications. Leplace transform: inverse transform, applications to constant coefficient differential equations. Boundary value problems: Fourier series, Bessel functions, Legendre polynomials. Mr. Polking

**M382b.** Complex Analysis and Applications. Complex analysis: Cauchy integral theorem. Taylor series, residues, evaluation of integrals by means of residues, conformal mapping, application to two-dimensional fluid flow. A student may not receive credit for this course and Mathematics 427. Staff

**M401a, 402b.** Differential Geometry. Differential manifolds, Stokes' Tneorem and deRham's Theorem, fundamental theorem of local Riemannian geometry. Lie groups, vector bundles, affine connections. Mr. Harvey

**M423a, 424b.** Partial Differential Equations. Cauchy-Kowalewski Theorem, first order hyperbolic systems, harmonic functions and potential theory, Dirichlet and Neumann problems, integral equations, and parabolic and elliptic equations. Mr. Rachford

**M425a.** Real Analysis. Lebesque theory of measure and integration. Staff

**M426b.** Topics in Real Analysis. Continuation of Mathematics 425. Mr. Stanton

**M427a, 428b.** Complex Analysis. Cauchy-Riemann equations, power series, Cauchy's integral formula, residue calculus, conformal mappings, special topics such as the Riemann mapping theorem, Runge's Theorem, elliptic function theory. Mr. Dadok

**M438a.** Computational Methods in Partial Differential Equations. Methods of solution: finite-element methods, collocation methods, finite difference methods, and associated algebraic problems. Also offered as Mathematical Sciences 551. Mrs. Wheeler

**M443a.** General Topology. Basic point set topology. Includes set theory, well ordering. Metrization. Mr. Shalen

**M444b.** Geometrical Topology. Introduction to algebraic methods in topology and differential topology. Elementary homotopy theory. Covering spaces. Mr.Jaco

**M463a, 464b.** Algebra. Groups, rings, fields, vector spaces. Matrices, determinants, eigenvalues, canonical forms, multilinear algebra. Structure theorem for finitely generated abelian groups. Mr. Jones

**M466b.** Topics in Algebra. Introduction to the concepts of Lie group theory done at the concrete level of matrix groups. Not offered 1980-81. Mr. Curtis

**M490.** Supervised Reading in Mathematics.

**M521a.** Special Topics in Complex Analysis. Several complex variables. Mr. Beatrous

**M523a.** Functional Analysis. Locally convex spaces, theory of distributions. Branch spaces. Hilbert spaces. Mr. Veech

**M525a, 526b.** Advanced Topics in Analysis. Nonlinear PDE. Mr. Taylor

**M537a.** Algebraic Topology. Singular homology and cohomology. Mr. Jaco

**M538b.** Algebraic Topology. Homotopy theory, Serre spectral sequence and applications. Mr. Curtis

**M601a, 602b.** Thesis.

**M700c.** Summer Research.

**M800b.** Degree Candidate Only.

### 1990/91

**M101,F/S** SINGLE VARIABLE CALCULUS I Differentiation, extrema, Newton's method, integration, fundamental theorem of calculus, area, volume, natural logarithm, exponential.

**M102,F/S** SINGLE VARIABLE CALCULUS II Techniques of integration, arc length, surface area, Simpson's rule, L'Hopital's rule. Infinite sequences and series, tests for convergence, power series, radius of convergence.

**M111,F** FUNDAMENTAL THEOREM OF CALCULUS This course and Math 1 1 2 form a slower paced course than 101-102, and do not go into great detail in their coverage of infinite series.

**M112,S** CALCULUS AND ITS APPLICATIONS See Math 111.

**M211,F/S** ORDINARY DIFFERENTIAL EQUATIONS Separable equations, first order linear equations, nth order linear equations with constant coefficients, Laplace transforms. Vector spaces, dimension, eigenvalues and eigenvectors of matrices. Systems of linear first order differential equations, exponential of a matrix. Qualitative theory of nonlinear systems. Prereq. Math 102.

**M212,F/S** MULTIVARIABLE CALCULUS Gradient, divergence, and curl. Lagrange multipliers. Multiple integrals. Spherical coordinates. Line integrals, conservative vector fields, Green's theorem, Stokes' theorem, Gauss' theorem.

**M221,F** HONORS CALCULUS III This course and Math 222 include the material of Math 212 and more. Topology of R , calculus for functions of several variables, linear and multilinear algebra, theory of determinants, inner product spaces, exterior differential calculus, integration on manifolds. Enrollment by permission of department. A student may not receive credit for Math 222 and 2 1 2.

**M222,S** HONORS CALCULUS IV. See Math 221.

**M321,F** INTRODUCTION TO MODERN ANALYSIS I A thorough treatment of basic methods of analysis such as metric spaces, compactness, sequences and series of functions. Also further topics in analysis, such as Hilbert spaces, Fourier series, Sturm-Liouville theory. Prereq.- Math 222 or permission of department.

**M322,S** INTRODUCTION TO MODERN ANALYSIS II. See Math 321.

**M355,F** LINEAR ALGEBRA Linear transformations and matrices, solution of linear equations, eigenvalues and eigenvectors, quadratic forms, rational canonical form, Jordan canonical form.

**M356,S** ABSTRACT ALGEBRA (3-0-3 Groups: normal subgroups, factor groups, Abelian groups. Rings: ideals, Euclidean rings, and unique factorization. Fields: algebraic extensions, finite fields. Students may not take this course and Math 463.

**M365,S** NUMBER THEORY Properties of numbers depending mainly on the notion of divisibility. Continued fractions. Offered alternate years. Offered 1990-91.

**M366,S** PROJECTIVE GEOMETRY Basic elements of classical projective geometry: projective spaces, subspaces, incidence relations, comparison with other geometries. Offered alternate years. Offered 1991-92.

**M381,F **ANALYSIS AND APPLICATIONS Laplace transform: inverse transform, applications to constant coefficient differential equations. Boundary value problems: Fourier series, Bessel functions, Legendre polynomials. A student may not receive credit for this course and Masc 340.

**M382,S** COMPLEX ANALYSIS Cauchy integral theorem, Taylor series, residues, evaluation of integrals by means of residues, conformal mapping, application to two-dimensional fluid flow. A student may not receive credit for this course and Math 427, or Masc 330.

**401,S** DIFFERENTIAL GEOMETRY Differentiable manifolds, Stokes' theorem and deRham's theorem, fundamental theorem of local Riemannian geometry. Lie groups, vector bundles, affine connections.

**M402,S** DIFFERENTIAL GEOMETRY See Math 401.

**M423,F** PARTIAL DIFFERENTIAL EQUATIONS Theory of distributions. Wave equation, Laplace's equation, heat equation. Fundamental solutions. Other topics include first order hyperbolic systems, Cauchy- Kowalewski theorem, potential theory, Dirichlet and Neumann problems, integral equations, elliptic equations.

**M424,S** PARTIAL DIFFERENTIAL EQUATIONS See Math 423.

**M425,F** REAL ANALYSIS Lebesgue theory of measure and integration.

**M426,S** TOPICS IN REAL ANALYSIS Topics vary. Past topics include: Fourier series, harmonic analysis, probability theory, advanced topics in measure theory, ergodic theory.

**M427,S **COMPLEX ANALYSIS Cauchy-Riemann equations, power series, Cauchy's integral formula, residue calculus, conformal mappings, special topics such as the Riemann mapping theorem, elliptic function theory.

**M428,F** COMPLEX ANALYSIS Special topics include Riemann mapping theorem, Runge's theorem, elliptic function theory, prime number theorem, Riemann surfaces.

**M443,F** GENERAL TOPOLOGY Basic point set topology. Includes set theory, well ordering. Metrization.

**M444,S** GEOMETRICAL TOPOLOGY Introduction to algebraic methods in topology and differential topology. Elementary homotopy theory. Covering spaces.

**M463,F** ALGEBRA I Groups, rings, fields, vector spaces. Matrices, determinants, eigenvalues, canonical forms, multilinear algebra. Structure theorem for finitely generated abelian groups. Galois theory.

**M464,S** ALGEBRA II See Math 463.

**M490,F/S** SUPERVISED READING

**M501,F** TOPICS IN DIFFERENTIAL GEOMETRY Topic to be announced.

**M502,S** TOPICS IN DIFFERENTIAL GEOMETRY The Atiyah-Singer theorem, secondary invariants, and related topics.

**M517,F/S** MATHEMATICAL PHYSICS

**M518,S** MATHEMATICAL PHYSICS

**M523,F** FUNCTIONAL ANALYSIS Locally convex spaces. Banach spaces. Hilbert spaces. Special topics.

**M526,S** TOPICS IN COMPLEX ANALYSIS

**M541,F/S** TOPICS IN ADVANCED TOPOLOGY

**M542,F/S** TOPICS IN ADVANCED TOPOLOGY

**M800,F/S** THESIS AND RESEARCH

### 2000/01

**MATH 101** SINGLE VARIABLE CALCULUS I. Study of differentiation, extrema, Newton's method, integration, fundamental theorem of calculus, area, volume, natural logarithm, exponential, and basic techniques of integration. May substitute MATH 111/112 or take MATH 101 after completing MATH 111.

**MATH 102** SINGLE VARIABLE CALCULUS II. Continuation of MATH 101. Includes further techniques of integration, arc length, surface area, Simpson's rule, and L'Hopital's rule, as well as infinite sequences and series, tests for convergence, power series, radius of convergence, polar coordinates, parametric equations, and arc length.

**MATH 111(F)** FUNDAMENTAL THEOREM OF CALCULUS. Study of calculus, forming with MATH 1 12 a slower-paced version of MATH 101/102. Includes less detail in the coverage of infinite series. May take MATH 111/112 followed by MATH 102 or MATH 111 followed by MATH 101/102.

**MATH 112(S)** CALCULUS AND ITS APPLICATIONS. See MATH 111.

**MATH 211** ORDINARY DIFFERENTIAL EQUATIONS AND LINEAR ALGEBRA. Study of ordinary differential equations (e.g., solutions to separable and linear first-order equations and to higher-order linear equations with constant coefficients, the properties of solutions to differential equations, and numerical solution methods) and linear algebra (e.g., vector spaces and solutions to algebraic linear equations, dimension, eigenvalues, and eigenvectors of a matrix), as well as the application of linear algebra to first-order systems of differential equations and the qualitative theory of nonlinear systems and phase portraits. Use of the computers in Owlnet as part of each homework assignment required. Prerequisite: MATH 102.

**MATH 212** MULTIVARIABLE CALCULUS. Study of gradient, divergence, and curl, Lagrange multipliers, multiple integrals, and spherical coordinates, as well as line integrals, conservative vector fields, Green's theorem, Stokes's theorem, and Gauss's theorem. May substitute MATH 221-222.

**MATH 221(F)** HONORS CALCULUS III. This course and MATH 222 include the material of MATH 212 and more. Topology of rN, calculus for functions of several variables, linear and multilinear algebra, theory of determinants, inner product spaces, exterior differential calculus, and integration on manifolds. A student may not receive credit for both MATH 212 and 222. Prerequisite: permission of department.

**MATH 222(S)** HONORS CALCULUS IV See MATH 221.

**MATH 321(F)** INTRODUCTION TO ANALYSIS I. Thorough treatment of basic methods of analysis (e.g., metric spaces, compactness, sequences, and series of functions). Includes topics in analysis (e.g., Hilbert spaces, Fourier series, and Sturm-Liouville theory). Prerequisite: MATH 212 or 222.

**MATH 322(S)** INTRODUCTION TO ANALYSIS II. See MATH 321. Includes proofs of the basic results for multivariable calculus (MATH 321 provides proofs for single-variable calculus).

**MATH 355(F)** LINEAR ALGEBRA. Study of linear transformations and matrices, the solution of linear equations, eigenvalues and eigenvectors, and quadratic, rational canonical, and Jordan canonical forms. May not receive credit for both MATH 355 and CAAM 310.

**MATH 356(S)** ABSTRACT ALGEBRA. Groups: normal subgroups, factor groups, and Abelian groups. Rings: ideals, Euclidean rings, and unique factorization. Fields: algebraic extensions and finite fields. Students may not receive credit for both MATH 356 and MATH 463.

**MATH 365(S)** NUMBER THEORY. Study of properties of numbers depending mainly on the notion of divisibility. Includes continued fractions.

**MATH 366(S)** GEOMETRY. Study of Euclidean, spherical, hyperbolic, and projective geometry, with emphasis on the similarities and differences found in them. Includes the study and use of isometrics and other transformations and a discussion of the history of the development of geometric ideas. Recommended for prospective high school teachers. Offered in alternate years.

**MATH 368(F)** INTRODUCTION TO COMBINATORICS. Study of combinatorics and discrete mathematics. Topics that may be covered include graph theory, Ramsey theory, finite geometries, combinatorial enumeration, combinatorial games. Prerequisite: MATH 211.

**MATH 381(F)** INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS. Introduction to partial differential equations, with emphasis on equations from science and engineering, using elementary methods. Includes the use of separation of variables to study various boundary value problems and an in-depth coverage of Fourier series and various special functions (e.g., Bessel functions and Legendre polynomials).

**MATH 382(S)** COMPLEX ANALYSIS. Study of the Cauchy integral theorem, Taylor series, and residues, as well as the evaluation of integrals by means of residues, conformal mapping, and application to two-dimensional fluid flow. May not receive credit for both MATH 382 and MATH 427.

**MATH 390** UNDERGRADUATE COLLOQUIUM. Lectures by undergraduates on mathematical topics not usually covered in other courses. Presentation of one lecture and attendance at all sessions required.

**MATH 401(F)** DIFFERENTIAL GEOMETRY. Study of the differential geometry of curves and surfaces in R\ Includes an introduction to the concept of curvature and thorough treatment of the Gauss-Bonnet theorem.

**MATH 402(S)** DIFFERENTIAL GEOMETRY. Introduction to Riemannian geometry. Content varies from year to year. Prerequisites: MATH 401 and either MATH 443 or permission of instructor.

**MATH 423(F)** PARTIAL DIFFERENTIAL EQUATIONS. Study of the wave equation, Laplace's equation, and heat equation. Includes first-order hyperbolic systems, the Cauchy-Kowalewski theorem, potential theory, Dirichlet and Neumann problems, integral equations, and elliptic equations.

**MATH 424(S)** PARTIAL DIFFERENTIAL EQUATIONS. Study of the theory of distributions and fundamental solutions. Content varies from year to year. Prerequisite: MATH 423.

**MATH 425(F)** INTEGRATION. Study of the Lebesgue theory of measure and integration.

**MATH 426(S)** TOPICS IN REAL ANALYSIS. Content varies from year to year. May include Fourier series, harmonic analysis, probability theory, advanced topics in measure theory, ergodic theory, and elliptic integrals.

**MATH 427(S)** COMPLEX ANALYSIS Study of the Cauchy-Riemann equation, power series, Cauchy' s integral formula, residue calculus, and conformal mappings.

**MATH 428(F)** TOPICS IN COMPLEX ANALYSIS. Content varies from year to year. May include the Riemann mapping theorem, uniformization, Runge's theorem, elliptic function theory, the prime number theorem, and Riemann surfaces.

**MATH 443(S)** GENERAL TOPOLOGY. Study of basic point set topology. Includes a treatment of cardinality and well ordering, as well as metrization. MATH 321 or permission of instructor recommended.

**MATH 444(F)** GEOMETRIC TOPOLOGY. Introduction to algebraic methods in topology and differential topology. Includes elementary homotopy theory and covering spaces. Co-/Prerequisite: MATH 356 or 463 and MATH 32 1 or 443 or permission of instructor.

**MATH 445(S)** ALGEBRAIC TOPOLOGY. Introduction to the theory of homology . Includes simplicial complexes, cell complexes, and cellular homology and cohomology, as well as manifolds, and Poincare duality. Prerequisites: MATH 444 and 356 or 463 or permission of instructor.

**MATH 463(F)** ALGEBRA I. Study of groups, rings, fields, and vector spaces. Includes matrices, determinants, eigenvalues, canonical forms, and multilinear algebra, as well as the structure theorem for finitely generated Abelian groups and the Galois theory.

**MATH 464(S)** ALGEBRA II. Continuation of MATH 463.

**MATH 490** SUPERVISED READING

**MATH 501(F)** TOPICS IN DIFFERENTIAL GEOMETRY

**MATH 502(S)** TOPICS IN DIFFERENTIAL GEOMETRY

**MATH 517(F)** MATHEMATICAL PHYSICS

**MATH 518(S)** MATHEMATICAL PHYSICS

**MATH 521(F)** ADVANCED TOPICS IN REAL ANALYSIS

**MATH 522(F)** ADVANCED TOPICS IN REAL ANALYSIS Seminar in real analysis. May include singular integral operators, maximal functions, Hardy spaces, Lipschitz spaces, and Sobolev spaces. Prerequisite: MATH 425 or permission of instructor.

**MATH 523(S)** FUNCTIONAL ANALYSIS. Seminar on locally convex spaces, Banach spaces, Hilbert spaces, and special topics.

**MATH 526(S) **TOPICS IN COMPLEX ANALYSIS

**MATH 527(S)** ERGODIC THEORY AND TOPOLOGICAL

**MATH 590** CURRENT MATHEMATICS SEMINAR. Lectures on topics of recent research in mathematics delivered by mathematics graduate students and faculty. Prerequisite: graduate student standing or permission of department.

**MATH 591** GRADUATE TEACHING SEMINAR. Discussion of teaching issues and practice lectures by participants as preparation for classroom teaching of mathematics. Prerequisite: graduate student standing or permission of department.

### 2010/11

**MATH 101** - SINGLE VARIABLE CALCULUS I

Differentiation, extrema, Newton's method, integration, fundamental theorem of calculus, area, volume, natural logarithm, exponential, arc length, surface area, Simpson's rule, L'Hopital's rule. May substitute MATH 111-112 or take MATH 101 after completing MATH 111.

**MATH 102 **- SINGLE VARIABLE CALCULUS II

Continuation of MATH 101. Includes further techniques of integration, as well as infinite sequences and series, tests for convergence, power series, radius of convergence, polar coordinates, parametric equations, and arc length.

**MATH 111** - FUNDAMENTAL THEOREM OF CALCULUS

Study of calculus, forming with MATH 112 a slower-paced version of MATH 101/102. Contains less detail in the coverage of infinite series. Students may take MATH 111/112 followed by MATH 102, or MATH 111 followed by MATH 101/102.

**MATH 112** - CALCULUS AND ITS APPLICATIONS

Continuation of the study of calculus from MATH 111.

**MATH 211** - ORDINARY DIFFERENTIAL EQUATIONS AND LINEAR ALGEBRA

Study of ordinary differential equations (e.g., solutions to separable and linear first-order equations and to higher-order linear equations with constant coefficients, the properties of solutions to differential equations, and numerical solution methods) and linear algebra (e.g., vector spaces and solutions to algebraic linear equations, dimension, eigenvalues, and eigenvectors of a matrix), as well as the application of linear algebra to first-order systems of differential equations and the qualitative theory of nonlinear systems and phase portraits. Use of the computers in Owlnet as part of each homework assignment required. Credit may not be received for both MATH 211 and MATH 213.

**MATH 212** - MULTIVARIABLE CALCULUS

Study of gradient, divergence, and curl, Lagrange multipliers, multiple integrals, as well as line integrals, conservative vector fields, Green's theorem, Stokes's theorem, and Gauss's theorem. May substitute Math 221 and 222. Equivalency: MATH 222.

**MATH 213** - BASIC MATHEMATICAL BIOLOGY

Study of systems of differential and difference equations, with special attention to modeling of basic biological processes. Phase plane analysis of ordinary differential systems. Qualitative understanding of solutions of differential equations. Credit may not be received for this course and MATH 211. Pre-requisites: MATH 102

**MATH 221** - HONORS CALCULUS III

This course and MATH 222 include the material of MATH 212 and much more. Topology of Rn, calculus for functions of several variables, linear and multilinear algebra, theory of determinants, inner product spaces, integration on manifolds.

**MATH 222** - HONORS CALCULUS IV

See MATH 221. A student may not receive credit for both MATH 222 and MATH 212. Equivalency: MATH 212.

**MATH 300** - TOPICS IN UNDERGRADUATE MATH

Treatment of topics in undergraduate mathematics with an emphasis on writing of clear, cogent compete mathematical proofs. Topics vary by year. May be repeated for credit with permission of department. Pre-requisite: MATH 102. Pre-requisites: MATH 102

**MATH 310** - MATHEMATICS OF MUSIC

This course will survey the many places in which mathematics plays a role in music. The topics and mathematical tools used are diverse: simple number theory in musical scales and tunings, differential equations in the physics of musical instruments, and symmetry in music explained in terms of mathematical group theory. Throughout the course we will use Csound, a compiler language for creating sound, to give concrete examples. Pre-requisites: MATH 211 OR MATH 213

**MATH 321** - INTRODUCTION TO ANALYSIS I

A thorough treatment of basic methods of analysis such as metric spaces, compactness, sequences and series of functions. Also further topics in analysis, such as Hilbert spaces, Fourier series, Sturm- Liouville theory. Pre-requisites: MATH 221 OR MATH 300 or permission of department

**MATH 322** - INTRODUCTION TO ANALYSIS II

See MATH 321. Includes proofs of the basic results for multivariable calculus (MATH 321 provides proofs for single-variable calculus). Pre-requisites: MATH 321 or permission of instructor

**MATH 354** - HONORS LINEAR ALGEBRA

Systems of linear equations, matrices, vector spaces, linear transformations, eigenvalues, canonical forms, inner product spaces, bilinear and quadratic forms. Content is similar to that of MATH 355, but with more emphasis on theory. The course will include instruction on how to construct mathematical proofs. This course is appropriate for potential Mathematics majors and others interested in learning how to construct rigorous mathematical arguments. Credit may not be received for both MATH 354 and MATH 355. Equivalency: MATH 355. Recommend a 200-level math class.

**MATH 355** - LINEAR ALGEBRA

Linear transformations and matrices, solution of linear equations, eigenvalues and eigenvectors, quadratic forms, Jordan canonical form. Credit may not be received for both MATH 354 and MATH 355. Equivalency: MATH 354.

**MATH 356** - ABSTRACT ALGEBRA I

Group theory: normal subgroups, factor groups, Abelian groups, permutations, matrix groups, and group action.

**MATH 365** - NUMBER THEORY

Properties of numbers depending mainly on the notion of divisibility. Continued fractions. Offered alternate years.

**MATH 366** - GEOMETRY

Topics chosen from Euclidean, spherical, hyperbolic, and projective geometry, with emphasis on the similarities and differences found in various geometries. Isometries and other transformations are studied and used throughout. The history of the development of geometric ideas is discussed. This course is strongly recommended for prospective high school teachers.

**MATH 368** - TOPICS IN COMBINATORICS

Study of combinatorics and discrete mathematics. Topics that may be covered include graph theory, Ramsey theory, finite geometries, combinatorial enumeration, combinatorial games.

**MATH 381** - INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS

Laplace transform: inverse transform, applications to constant coefficient differential equations. Boundary value problems: Fourier series, Bessel functions, Legendre polynomials.

**MATH 382** - COMPLEX ANALYSIS

Study of the Cauchy integral theorem, Taylor series, residues, as well as the evaluation of integrals by means of residues, conformal mapping, and application to two-dimensional fluid flow. May not receive credit for this course and MATH 427. Equivalency: MATH 427.

**MATH 390** - UNDERGRADUATE COLLOQUIUM

Lectures by undergraduate students on mathematical topics not usually covered in other courses. Presentation of one lecture and attendance at all sessions required.

**MATH 401** - DIFFERENTIAL GEOMETRY

Study of the differential geometry of curves and surfaces in R3. Includes an introduction to the concept of curvature and thorough treatment of the Gauss-Bonnet theorem.

**MATH 402** - DIFFERENTIAL GEOMETRY

Introduction to Riemannian geometry. Content varies from year to year. Recommended Prerequisite: MATH 401.

**MATH 410** - CALCULUS OF VARIATIONS

Study of classical and modern theories about functions having some integral expression which is maximal, minimal, or critical. Geodesics, brachistochrone problem, minimal surfaces, and numerous applications to physics. Euler-Lagrange equations, 1st and 2nd variations, Hamilton's Principle. Pre-requisites: MATH 101 AND MATH 102 AND (MATH 211 OR MATH 212) OR (MATH 221 OR MATH 222)

**MATH 423** - PARTIAL DIFFERENTIAL EQUATIONS I

First order of partial differential equations. The method of characteristics. Analysis of the solutions of the wave equation, heat equation and Laplace's equation. Integral relations and Green's functions. Potential theory, Dirichlet and Neumann problems. Asymptotic methods: the method of stationary phase, geometrical optics, regular and singular perturbation methods. Cross-list: CAAM 423.

**MATH 424** - PARTIAL DIFFERENTIAL EQUATIONS

Continuation of MATH 423. Pre-requisites: MATH 423

**MATH 425** - INTEGRATION THEORY

Lebesgue theory of measure and integration.

**MATH 426** - TOPICS IN REAL ANALYSIS

Content varies from year to year. May include Fourier series, harmonic analysis, probability theory, advanced topics in measure theory, ergodic theory, and elliptic integrals. Pre-requisites: MATH 425

**MATH 427** - COMPLEX ANALYSIS

Study of the Cauchy-Riemann equation, power series, Cauchy's integral formula, residue calculus, and conformal mappings. Emphasis on the theory. Credit may not be received for both MATH 382 and MATH 427. Equivalency: MATH 382.

**MATH 428** - TOPICS IN COMPLEX ANALYSIS

Special topics include Riemann mapping theorem, Runge's Theorem, elliptic function theory, prime number theorem, Riemann surfaces, et al. Pre-requisites: MATH 382 OR MATH 427

**MATH 435** - DYNAMICAL SYSTEMS

Existence and uniqueness for solutions of ordinary differential equations and difference equations, linear systems, nonlinear systems, stability, periodic solutions, bifurcation theory. Theory and theoretical examples are complemented by computational, model driven examples from biological and physical sciences. Cross-list: CAAM 435. Recommended Prerequisite(s): (CAAM 210 AND MATH 212) AND (CAAM 335 OR MATH 335) AND (CAAM 401 OR MATH 321).

**MATH 443** - GENERAL TOPOLOGY

Study of basic point set topology. Includes a treatment of cardinality and well ordering, as well as metrization.

**MATH 444** - GEOMETRIC TOPOLOGY

Introduction to algebraic methods in topology and differential topology. Elementary homotopy theory. Covering spaces. Pre-requisites: MATH 443

**MATH 445** - ALGEBRAIC TOPOLOGY

Introduction to the theory of homology. Includes simplicial complexes, cell complexes and cellular homology and cohomology, as well as manifolds, and Poincare duality. Pre-requisites: MATH 444

**MATH 463** - ABSTRACT ALGEBRA II

Ring theory: ideals, polynomials, factorization. Advanced linear algebra: quadratic forms, canonical forms. Field theory: extensions, Galois theory, solubility in radicals. Continuation of MATH 356. Pre-requisites: MATH 356 or permission of instructor

**MATH 464** - ABSTRACT ALGEBRA III

Continuation of MATH 463. Tensor and exterior algebra, introductory commutative algebra, structure of modules, and elements of homological algebra. Additional advanced topics may include representations of finite groups and affine algebraic geometry. Pre-requisites: MATH 463 or permission of instructor

**MATH 465** - TOPICS IN ALGEBRA

Content varies from year to year.

**MATH 466** - TOPICS IN ALGEBRA II

Content varies from year to year.

**MATH 468** - POTPOURRI

This course deals with miscellaneous special topics not covered in other courses.

**MATH 490** - SUPERVISED READING

**MATH 498** - RESEARCH THEMES IN THE MATHEMATICAL SCIENCES

A seminar course that will cover selected theme of general research in the mathematical sciences from the perspectives of mathematics, computational and applied mathematics and statistics. The course may be repeated multiple times for credit. Cross-listed with CAAM 498 and STAT 498. Cross-list: CAAM 498, STAT 498, Graduate/Undergraduate Equivalency: MATH 698.

**MATH 499** - MATHEMATICAL SCIENCES VIGRE SEMINAR

Graduate/Undergraduate Equivalency: MATH 699.

**MATH 501** - TOPICS IN DIFFERENTIAL GEOMETRY

Topic to be announced.

**MATH 502** - TOPICS IN DIFFERENTIAL GEOMETRY

Topic to be announced.

**MATH 521** - ADVANCED TOPICS IN REAL ANALYSIS

Topic to be announced.

**MATH 522** - TOPICS IN ANALYSIS

Topic to be announced.

**MATH 523** - FUNCTIONAL ANALYSIS

Topic to be announced.

**MATH 527** - ERGODIC THEORY AND TOPOLOGICAL DYNAMICS

Topic to be announced.

**MATH 528** - ERGODIC THEORY AND TOPOLOGICAL DYNAMICS

**MATH 541** - TOPICS IN TOPOLOGY

Topic to be announced.

**MATH 542** - TOPICS IN ADVANCED TOPOLOGY

Topic to be announced.

**MATH 543** - TOPICS IN LOW-DIMENSIONAL TOPOLOGY

**MATH 567** - TOPICS IN ALGEBRAIC GEOMETRY

Possible topics include rational points on algebraic varieties, moduli spaces, deformation theory, and Hodge structures. Recommended prerequisite(s): Mathematics 463 - 464.

**MATH 590** - CURRENT MATHEMATICS SEMINAR

Lectures on topics of recent research in mathematics delivered by mathematics graduate students and faculty.

**MATH 591** - GRADUATE TEACHING SEMINAR

Discussion on teaching issues and practice lectures by participants as preparation for classroom teaching of mathematics.

**MATH 698** - RESEARCH THEMES IN THE MATHEMATICAL SCIENCES

A seminar course that will cover selected theme of general research in the mathematical sciences from the perspectives of mathematics, computational and applied mathematics and statistics. The course may be repeated multiple times for credit. Cross-listed with CAAM 698 and STAT 698. Cross-list: CAAM 698, STAT 698, Graduate/Undergraduate Equivalency: MATH 498.

**MATH 699** - MATHEMATICAL SCIENCES VIGRE SEMINAR

Graduate/Undergraduate Equivalency: MATH 499.

**MATH 777** - VISITING RESEARCH TRAINEE

**MATH 800** - THESIS AND RESEARCH