# Written 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. To receive a copy of past exams, please email Melissa Vacca (vacca@cims.nyu.edu).

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 three 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. PhD students are required to take the examinations during their first year of study.

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.

## 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*

## Written Exam Workshop

The department offers a written exam workshop in the fall, spring and summer which is taught by an advanced PhD student.

The fall 2017 Written Exam Workshop will meet on the following dates:

- TBA

## Exam Dates and Application

Examination | Examination Dates | Application Deadline |
---|---|---|

August 2017 Writtens | Wednesday, 8/30/17 Thursday, 8/31/17 Friday, 9/1/17 |
Wednesday , 5/10/17 |

January 2018 Writtens | Wednesday, 1/9/18 Thursday, 1/10/18 Friday, 1/11/18 |
Wednesday, 11/8/17 |

Students should submit the application (insert PDF link) to take the written exams before the deadline to Melissa Vacca at vacca@cims.nyu.edu or to room 622.