# Mathematics Area Exam

This is a 3 hour, closed book, written examination. The exam will have nine (9) subject areas with two (2) problems in each subject area. You will have to solve eight (8) of the eighteen (18) problems, but you must chose problems in at least six (6) different subject areas. The nine subject areas are:

- Calculus
- Determinants and Matrix Algebra
- Ordinary Differential Equations
- Complex Variables
- Vector Calculus
- Fourier Analysis
- Laplace Transforms
- Partial Differential Equations
- Numerical Methods

**NOTE:**

Calculators can be used only for simple arithmetic calculations and you will be expected to abide by the Mechanical Engineering Code of Honor for ME Graduate Students taking the Applied Mathematics Area Exam.

As noted above, this is a closed book exam and no crib sheets are allowed. A table of Laplace Transforms will be provided.

The questions in subject areas 1 through 8 are set by professors in the Mathematics Department and the questions in the 9th subject area are set by Mechanical Engineering professors who teach ME 581. The people who set the questions also grade your solutions to those questions.

On the following pages are lists of topics that may be covered in each of the nine subject area questions. Also given are the Purdue classes in which the material is covered and sample references. Of course, there are many other texts that cover similar material, which may also be suitable as references.

The emphasis in the exam is on solving problems as opposed to doing proofs.

## Section 1 Calculus

### Topics

The topics covered in the calculus questions include:

- Finding maxima and minima of functions
- Methods of integration of definite and indefinite integrals including integration by parts and change of variables
- Graphical interpretation of functions
- Partial derivatives and total differentials
- Taylor series
- Double and triple integrals
- Transformation of coordinates
- The chain rule

### Relevant Purdue Courses and References

MA161, MA162, MA165, MA166, & MA261, Text: Finney, Weir, and Giordano (2001), Stewart (2015).

## Section 2 Determinants and Matrix Algebra

### Topics

The topics covered in the determinants and matrix algebra questions include:

- Solution of linear algebraic equations
- Algebraic manipulation of rectangular or square matrices such as sums, products, transpose, etc.
- Computation of a matrix inverse
- Evaluation of determinants
- Eigenvalues of a matrix
- Matrix concepts: rank, null space, row space, column space
- Vector space concepts: linear dependence, linear independence, dimension, basis
- Inner product space
- Properties of symmetric, skew-symmetric, orthogonal, hermitian, skew-hermitian and unitary matrices
- Geometry of linear transformations

### Relevant Purdue Courses and References

MA262, Text: Goode and Annin (2015); MA265, Text: Kolman and Hill (2007); MA511, Text: Strang (2006); MA527, Text: Kreyszig (2011).

## Section 3 Ordinary Differential Equations

### Topics

The topics covered in the ordinary differential equations questions include:

- Series solutions: ordinary points, regular singular points
- Determination of general and particular solutions of first order and second order differential equations
- Formulation and solution of differential equations for physical problems, e.g., falling or sliding bodies, rate equations, diffusion, vibrating systems.
- Solving systems of linear differential equations
- 2×2 autonomous systems. Stability, phase plane analysis

### Relevant Purdue Courses and References

MA262, Text: Goode and Annin (2015); MA266, MA303 & MA304, Text: Boyce and DiPrime (2008); MA527, Text: O’Neil (2011), Kreyszig (2011).

## Section 4 Complex Variables

### Topics

The topics covered in the complex variables questions include:

- Algebra of complex numbers
- Polar notation and phasor diagrams
- Extraction of roots
- Functions of complex variables
- Power series
- Cauchy integral formula and contour integrals
- Evaluation of real and complex integrals by residues
- Properties of conformal mapping

### Relevant Purdue Courses and References

MA425, Text: Churchill and Brown (2013); MA525, Text: Saff and Snider (1993); MA528, Text: O’Neil (2011), Kreyszig (2011).

## Section 5 Vector Calculus

### Topics

The topics covered in the vector analysis questions include:

- Vector algebra and calculus
- Operations involving gradient, curl and divergence
- Stoke’s theorem
- Line and surface integrals
- Conservative, irrotational and solenoidal fields
- Scalar potentials
- Divergence theorem

### Relevant Purdue Courses and References

MA410 and MA 510, Text: Marsden and Tromba (2011).

## Section 6 Fourier Analysis

### Topics

The topics covered in the Fourier analysis questions include:

- General coefficients
- Sine and cosine series of odd and even functions
- Complex Fourier series
- Parseval’s identity
- Fourier transform and its properties

### Relevant Purdue Courses and References

MA303 or MA304, Text: Boyce and DiPrime (2008); MA520, Text: Folland (2009); MA527, Text: O’Neil (2011), Kreyszig (2011).

## Section 7 Laplace Transforms

### Topics

The topics covered in the Laplace transforms questions include:

- Solution of ordinary differential equations with constant coefficients
- Method of partial fraction expansions
- Inversions of Laplace transforms: Heaviside function, delta function, convolutions

### Relevant Purdue Courses and References

MA303 or MA304, Text: Boyce and DiPrime (2008); MA527, Text: O’Neil (2011) Kreyszig (2011).

## Section 8 Partial Differential Equations

### Topics

The topics covered in the partial differential equations questions include:

- Laplace, wave and heat equations in one, two or three variables
- Solution of initial and/or boundary value problems for these equations by the method of separation of variables
- Solution of heat equation in infinite domain using heat kernel
- D’Alembert solution for one-dimensional wave equation
- Properties of heat and wave equations

### Relevant Purdue Courses and References

MA527, Text: O’Neil (2011) Kreyszig (2011); Zachmanoglou and Thoe (1987); MA520, Text: Folland (2009).

## Section 9 Numerical Methods

### Topics

The topics covered in the numerical methods questions include:

- Systems of linear equations
- Eigenproblems
- Solution of nonlinear equations
- Polynomial approximation and interpolation
- Least squares approximation
- Numerical differential and difference formulas
- Numerical integration (quadrature)
- Solution of one-dimensional initial-value problems
- Solution of one-dimensional boundary-value problems
- Elliptic partial differential equations
- Parabolic partial differential equations
- Hyperbolic partial differential equations

### Relevant Purdue Courses and References

CS414, MA514, Text: Van Loan (1999), Gautschi (2012); ME581, Text: Bradie (2006; Heath 2002).

## References

Boyce, W.E., and R.C. DiPrime. 2008. *Elementary Differential Equations and Boundary Value Problems*. 9th ed. Wiley.

Bradie, Brian. 2006. *A Friendly Introduction to Numerical Analysis*. 1st ed. Prentice-Hall.

Churchill, R., and J. Brown. 2013. *Complex Variables and Applications*. 9th ed. McGraw Hill.

Finney, R.L., M.D. Weir, and F.R. Giordano. 2001. *Thomas’ Calculus: Early Transcendentals*. 10th ed. Addison Wesley Longman.

Folland, G. 2009. *Fourier Analysis and Its Application*. American Mathematical Society.

Gautschi, W. 2012. *Numerical Methods*. 2nd ed. Birkhauser.

Goode, S., and S.A. Annin. 2015. *An Introduction to Differential Equations and Linear Algebra*. 4th ed. Pearson.

Heath, Michael. 2002. *Scientific Computing: An Introductory Survey*. 2nd ed. McGraw-Hill.

Kolman, B., and D. Hill. 2007. *Elementary Linear Algebra with Applications*. 9th ed. Pearson.

Kreyszig, E. 2011. *Advanced Engineering Mathematics*. 10th ed. Wiley.

Marsden, J.E., and A. Tromba. 2011. *Vector Calculus*. 6th ed. W. H. Freeman.

O’Neil, P. 2011. *Advanced Engineering Mathematics*. 7th ed. Cengage Learning.

Saff, E., and A. Snider. 1993. *Fundamentals of Complex Analysis for Mathematics, Science and Engineering*. 2nd ed. Prentice Hall.

Stewart, J. 2015. *Calculus: Early Transcendentals*. 8th ed. Brooks Cole.

Strang, G. 2006. *Linear Algebra and Its Applications*. 4th ed. Cengage Learning.

Van Loan, C.F. 1999. *Introduction to Scientific Computing, a Matrix Vector Approach Using MATLAB*. 2nd ed. Pearson.

Zachmanoglou, E.C., and D. Thoe. 1987. *Introduction to Partial Differential Equations with Applications*. Dover Publications.