# Mathematics

### Courses for First-Year Students

Depending on the program, completion of Calculus III or IV satisfies the basic mathematics requirement. Normally students who have taken an AP Calculus course begin with either Calculus II or Calculus III. Refer to the AP guidelines on page 14 for placement information. The sequence ends with *MATH UN2030: Ordinary differential equations*.

Students who wish to transfer from one calculus course to another are allowed to do so beyond the date specified on the Academic Calendar. They are considered to be adjusting their level, not changing their program. They must, however, obtain the approval of the new instructor and the Center for Student Advising before reporting to the Registrar.

**MATH UN1101x or y Calculus, I**

*3pts. Lect: 3.*

Functions, limits, derivatives, introduction to integrals.

**MATH UN1102x or y Calculus, II**

*3 pts. Lect: 3.*

Prerequisite: MATH UN1101 or equivalent. Methods of integration, applications of integral, Taylor’s theorem, infinite series.

**MATH UN1201x or y Calculus, III**

*3 pts. Lect: 3.*

Prerequisite: MATH UN1101 with a grade of B or better or UN1102 or equivalent. Vectors in dimensions 2 and 3, complex numbers and the complex exponential function with applications to differential equations, Cramer’s rule, vector-valued functions of one variable, scalar-valued functions of several variables, partial derivatives, gradients, surfaces, optimization, the method of Lagrange multipliers.

**MATH UN1202x or y Calculus, IV**

*3 pts. Lect: 3.*

Prerequisite: MATH UN1102, UN1201, or equivalent. Multiple integrals, Taylor’s formula in several variables, line and surface integrals, calculus of vector fields, Fourier series.

**MATH UN1207x-UN1208y Honors math A-B**

*4 pts. Lect and recit. Professors Gallagher, Thaddeus, and Hansen.*

Prerequisite: Score of 5 on the Advanced Placement BC calculus exam. The second term of this course may not be taken without the first. Multivariable calculus and linear algebra from a rigorous point of view.

**MATH UN2010 x or y Linear algebra**

3 pts. Lect: 3. Professors Bayer, Maulik, Stein, and Khovanov.

Prerequisite: MATH UN1201 or equivalent. Matrices, vector spaces, linear transformations, eigenvalues and eigenvectors, canonical forms, applications.

**MATH UN2030x or y Ordinary differential equations**

3 pts. Lect: 3. Professors Chang-Lara, Chen, and Gorskity.

Prerequisite: MATH UN1102-1201 or the equivalent. Special differential equations of order one. Linear differential equations with constant and variable coefficients. Systems of such equations. Transform and series solution techniques. Emphasis on applications.

**MATH UN2500x or y Analysis and optimization**

*3 pts. Lect: 3. Professors Dubedat, Maulik, and Jeon.*

Prerequisites: MATH UN1102-1201 or equivalent, and UN2010. Mathematical methods for economics. Quadratic forms, Hessian, implicit functions. Convex sets, convex functions. Optimization, constrained optimization, Kuhn-Tucker conditions. Elements of the calculus of variations and optimal control.

**MATH UN3007y Complex variables**

*3 pts. Lect: 3. Professor Gautam.*

Prerequisite: MATH UN1202. An elementary course in functions of a complex variable. Fundamental properties of the complex numbers, differentiability, Cauchy-Riemann equations, Cauchy integral theorem, Taylor and Laurent series, poles, and essential singularities. Residue theorem and conformal mapping.

**MATH UN3027x Ordinary differential equations**

*3 pts. Lect: 3. Professor Daskalopoulos.*

Prerequisite: MATH UN1201 or equivalent. Corequisite: MATH UN2010. Equations of order one, systems of linear equations, second-order equations, series solutions at regular and singular points, boundary value problems, selected applications.

**MATH UN3028y Partial differential equations**

*3 pts. Lect: 3. Professor Savin.*

Prerequisite: MATH UN3027 and UN2010 or equivalent. Introduction to partial differential equations. First-order equations. Linear second-order equations, separation of variables, solution by series expansions. Boundary value problems.

**MATH GU4032x Fourier analysis**

*3 pts. Lect: 3. Professor Savin.*

Prerequisites: three terms of calculus and linear algebra or four terms of calculus. Fourier series and integrals, discrete analogues, inversion and Poisson summation, formulae, convolution, Heisenberg uncertainty principle. Stress on the application of Fourier analysis to a wide range of disciplines.

**MATH GU4041x-GU4042y Introduction to modern algebra**

*3 pts. Lect: 3. Professors Friedman, Gallagher, Harris, Khovano, and Thaddeus.*

The second term of this course may not be taken without the first. Prerequisites: MATH UN1102-1202 and UN2010 or equivalent. Groups, homomorphisms, rings, ideals, fields, polynominals, and field extensions, Galois theory.

**MATH GU4061x-GU4062y Introduction to modern analysis**

*3 pts. Lect: 3. Professors De Silva, Guo, Zhang, and Chang-Lora.*

The second term of this course may not be taken without the first. Prerequisite: MATH UN1202 or equivalent and UN2010. Real numbers, metric spaces, elements of general topology. Continuous and differentiable functions. Implicit functions. Integration, change of variables. Function spaces.

**MATH GU4065x Honors complex variables**

*3 pts. Lect: 3. Professor Dubedat or Urban.*

Prerequisite: MATH UN1207 and UN1208, or GU4061. A theoretical introduction to analytic functions. Holomorphic functions, harmonic functions, power series, Cauchy-Riemann equations, Cauchy’s integral formula, poles, Laurent series, residue theorem. Other topics as time permits: elliptic functions, the gamma and zeta functions, the Riemann mapping theorem, Riemann surfaces, Nevanlinna theory.