# The Written Comprehensive Exams

### General Information

All graduate mathematics students are required to take the written comprehensive examination in the three following subjects: Advanced Calculus, Complex Variable, and Linear Algebra.

Past examinations are a recommended guide to the level and typical content of the examinations. A complete set of examinations is available at the departmental office, room 623, Warren Weaver Hall.

The examinations are given twice a year, in late August or early September and in early January, just before the beginning of the fall and spring terms, respectively. They are scheduled on two consecutive days, with three hours allotted to each subject examination. All students must take the examinations in order to be allowed to register for coursework beyond 36 points of credit; it is recommended that students attempt the examinations well before this deadline. Graduate Assistants are required to take the examinations during their first year of study.

Students are required to apply for the September examination by the end of the previous spring term, and for the January examination by the end of the previous fall term; specific application deadlines, pertaining to each academic year, are found in the departmental Academic Year Calendar, distributed to students prior to each fall term (and also available at both fall and spring registration periods). Application forms are available at the Mathematics Department Office; they will be mailed to students upon request.

Passing the written comprehensive examinations is part of one of the alternatives for fulfilling the requirements for the master of science degree in mathematics. Passing with the grade of A is a prerequisite to the oral preliminary examination for the Ph.D. degree in mathematics. Students may take the examination twice without special permission; a third try will require the permission of the Director of Graduate Studies. In the fall term, the Department offers a series of workshops, taught by an advanced graduate assistant, to help students prepare for the written examinations.

### The Examinations

The following lists suggest the scope of the examinations but are not necessarily complete:

**Advanced Calculus**

Real numbers. Functions of one variable: continuity, mean-value theorems, convergence, differentiability, maxima and minima, integrals, fundamental theorem of calculus, inequalities, estimation of sums and integrals, elementary functions and their power series. Functions of several variables: partial derivatives, chain rule, MacLaurin expansion, critical points, Lagrange multipliers, inverse and implicit function theorems, Jacobian, divergence and curl, area and volume integrals, Green and Stokes theorems.

**Complex Variables**

Complex numbers, analytical functions, Cauchy-Riemann equations, Cauchy's integral and applications, power series, maximum principle, Liouville's theorem, elementary functions and their conformal maps, bilinear transformation, classification of singularities, residue theorem and contour integration, Laurent series, Rouche's theorem, number of zeros and poles.

**Linear Algebra**

Vector spaces, linear dependence, basis, dimension, inner product, linear transformation, systems of linear equations, matrices, determinants, ranks, eigenvalues, diagonalization of matrices, quadratics forms, symmetric and orthogonal transformations.

Cooperative preparation is encouraged, as it is for all examinations. You may also find the following books helpful:

- Buck,
*Advanced Calculus* - Courant and John,
*Introduction to Calculus and Analysis* - Strang,
*Linear Algebra* - Churchill,
*Complex Variables and Applications*