## Applied Mathematics AMAT

Instruction offered by members of the Department of Mathematics and Statistics in the Faculty of Science.

Department Head - T. Bisztriczky

Note: For listings of related courses, see Actuarial Science, Mathematics, Pure Mathematics and Statistics.

## Junior Courses

### Applied Mathematics 217 H(3-1T-1.5)

#### Calculus for Engineers and Scientists

Functions, limits, continuity, derivatives, Mean Value Theorem, integrals, Fundamental Theorem of Calculus, applications to the physical sciences.

**
Prerequisites:
** A grade of 70% or higher in Pure Mathematics 30 and credit in Mathematics 31; or admission to the Faculty of Engineering including credit in Pure Mathematics 30 and Mathematics 31.

**
Note:
** Credit for more than one of Mathematics 249, 251, 281 or Applied Mathematics 217 will not be allowed.

### Applied Mathematics 219 H(3-1T-1.5)

#### Multivariable Calculus for Engineers

Techniques of integration, double and triple integrals, partial derivatives, applications.

**
Prerequisites:
** Applied Mathematics 217; or Mathematics 249 or 251 or 281 plus 117; or consent of Applied Mathematics Division.

**
Note:
** Credit for more than one of Mathematics 253, 263, 283 or Applied Mathematics 219 will not be allowed.

## Senior Courses

### Applied Mathematics 307 H(3-1.5T)

#### Differential Equations for Engineers

Definition, existence and uniqueness of solutions, first and second order equations with applications, series solutions about regular points and singular points, special functions. Laplace transform, systems of equations.

**
Prerequisites:
** Mathematics 211 or 221; and Applied Mathematics 219 or both Mathematics 253 or 263 or 283 plus 114.

**
Note:
** Credit for both Applied Mathematics 307 and 311 will not be allowed.

### Applied Mathematics 309 H(3-1.5T)

#### Vector Calculus for Engineers

Functions of several variables, chain rule and differentials. Vector calculus, line, surface and volume integrals, Green's, Gauss' and Stokes' theorems. Students will complete a project using a computer algebra system.

**
Prerequisites:
** Applied Mathematics 219 or both Mathematics 253 or 263 or 283 plus 114.

**
Note:
** Credit for more than one of Mathematics 353, 381, and Applied Mathematics 309 will not be allowed.

### Applied Mathematics 311 H(3-1T)

#### Differential Equations I

Classification of ordinary differential equations, first and second order equations with applications, series solutions about regular points and singular points, special functions, Laplace transform.

**
Prerequisites:
** Mathematics 253 or 263 or 283 or Applied Mathematics 219.

**
Note:
** Credit for both Applied Mathematics 307 and 311 will not be allowed.

### Applied Mathematics 411 H(3-1T)

#### Differential Equations II

Existence and uniqueness theorems, comparison and oscillation theorems, Green's functions, Sturm-Liouville problems, systems of equations, phase portraits, stability.

**
Prerequisites:
** One of Applied Mathematics 311 or 307, and one of Mathematics 331, 353, 381, Applied Mathematics 309, or consent of the Division.

**
Note:
** It is recommended that students complete Pure Mathematics 435 or 455 before taking this course.

### Applied Mathematics 413 H(3-1T)

#### Introduction to Partial Differential Equations

Orthogonal sets of functions, Fourier series, solution of potential equation, heat equation and wave equation. Numerical solution of partial differential equations.

**
Prerequisites:
** One of Mathematics 353, 381, Applied Mathematics 309, Mathematics 331; or consent of the Division.

**
Note:
** Credit for both Applied Mathematics 413 and 407 will not be allowed.

### Applied Mathematics 415 H(3-1T)

#### Mathematical Methods

Mathematical analysis of linear systems. Fourier and Laplace transforms, applications and numerical methods. Functions of a complex variable and applications.

**
Prerequisites:
** One of Applied Mathematics 311, 307, Mathematics 331, 353, 381, or Applied Mathematics 309.

**
Note:
** Credit for both Applied Mathematics 415 and 433 will not be allowed for Applied Mathematics and General Mathematics programs.

### Applied Mathematics 425 H(3-1T)

#### Introduction to Optimization

Examples of optimization problems. Quadratic forms, minimum energy and distance. Least squares, generalized inverse. Location and classification of critical points. Variational treatment of eigenvalues. Lagrange multipliers. Linear programming.

**
Prerequisites:
** Mathematics 311; and Mathematics 353 or 381 or Applied Mathematics 309 or Mathematics 331.

### Applied Mathematics 433 H(3-1T)

#### Mathematical Methods in Physics

Complex analysis and residue integrals. Fourier analysis. Vector spaces. Eigenvalues and eigenvectors. Extensive physical applications.

**
Prerequisites:
** One of Applied Mathematics 307 or 311; one of Applied Mathematics 309 or Mathematics 353 or 381 or 331; Mathematics 221 or 211.

**
Note:
** Credit for both Applied Mathematics 415 and 433 will not be allowed for Applied Mathematics and General Mathematics programs.

### Applied Mathematics 481 H(3-1T)

#### Introduction to Mathematical Finance

Introduction to financial markets and derivatives, asset price random walks, Black-Scholes option pricing model, American options and other generalizations.

### Applied Mathematics 483 H(3-1T)

#### Computational Methods in Mathematical Finance

Review of financial models, Monte-Carlo simulation, binomial and trinomial trees, finite-difference method, aspects of time series and parameter estimation, volatility modelling and estimation.

**
Prerequisites:
** Applied Mathematics 481 and 491.

**
Corequisites:
** Applied Mathematics 493.

### Applied Mathematics 491 H(3-1T)

#### Numerical Analysis I

Interpolation and approximation, numerical integration, numerical methods for the solution of nonlinear equations, systems of linear equations and the eigenvalue problem.

**
Prerequisites:
** Mathematics 311; 381 or {349 and (353 or Applied Mathematics 309)}; and Computer Science 231 or 217; or consent of the Division.

**
Note:
** Not open to students with credit in Computer Science 491.

### Applied Mathematics 493 H(3-1T)

#### Numerical Analysis II

Numerical differentiation, numerical solution of ordinary and partial differential equations.

**
Prerequisites:
** Applied Mathematics 311, 413, and 491 or Computer Science 491.

### Applied Mathematics 501 H(3-0)

#### Seminar in Applied Mathematics

Topics will be chosen according to the interests of instructors and students and could include analysis of optimization algorithms, approximation theory, control theory, differential equations, mathematical physics.

**
Prerequisites:
** Consent of the Division.

**
MAY BE REPEATED FOR CREDIT
**

### Applied Mathematics 503 H(3-1T)

#### The Mathematics of Wavelets, Signal and Image Processing

Continuous and discrete Fourier transforms, the Fast Fourier Transform, wavelet transforms, multiresolution analysis and orthogonal wavelet bases, and applications.

**
Prerequisites:
** Applied Mathematics 491 or Computer Science 491.

### Applied Mathematics 505 H(3-0)

#### Calculus on Manifolds

Integral and differential calculus on manifolds including tensor fields, covariant differentiation, Lie differentiation, differential forms, Frobenius' theorem, Stokes' theorem, flows of vector fields.

**
Prerequisites:
** Pure Mathematics 445 or 545; and one of Applied Mathematics 311 or 307; or consent of the Division.

### Applied Mathematics 507 H(3-0)

#### Introduction to Relativity Theory

Mathematical theories of space and time. Special Relativity. Electro-dynamics. General Relativity.

**
Prerequisites:
** Applied Mathematics 505 or consent of the Division.

### Applied Mathematics 509 H(3-0)

#### Analytical Dynamics

Symplectic geometry, Hamilton's equation, Hamilton-Jacobi theory, constraints and reduction.

**
Prerequisites:
** Applied Mathematics 505 or consent of the Division.

### Applied Mathematics 581 H(3-0)

#### Advanced Futures and Options

Stochastic calculus and the dynamics of asset prices, martingale theory and risk-neutral valuation, interest rate models, energy and commodity markets, value-at-risk and risk management.

**
Prerequisites:
** Applied Mathematics 483 and Statistics 407.

**
Corequisites:
** Statistics 409.

## Graduate Courses

In addition to the prerequisites listed below, consent of the Applied Mathematics Division is a prerequisite for all graduate courses in Applied Mathematics.

### Applied Mathematics 601 H(3-0)

#### Topics in Applied Mathematics

Topics will be chosen according to the interests of instructors and students.

**
Prerequisites:
** Consent of the Division.

**
MAY BE REPEATED FOR CREDIT
**

### Applied Mathematics 605 H(3-0)

#### Differential Equations III

Linear systems, classification. Nonlinear systems: Existence and uniqueness. Flow and one parameter groups of transformations. Stability theory. Hyperbolicity, Unstable/Stable/Center manifold theorems. Poincare-Bendixson.

**
Prerequisites:
** Applied Mathematics 411 and Pure Mathematics 445 or 545 or equivalents.

### Applied Mathematics 613 H(3-0)

#### Partial Differential Equations II

Fundamental solutions, integral equations, eigenvalue problems, non-linear problems.

**
Prerequisites:
** Consent of the Division.

### Applied Mathematics 617 H(3-0)

(formerly Pure Mathematics 617)

#### Analysis IV

Analysis in abstract spaces. Function spaces.

**
Prerequisites:
** Pure Mathematics 545.

### Applied Mathematics 621 Q(2S-0)

#### Research Seminar

Reports on studies of the literature or of current research.

**
Note:
** All graduate students in Mathematics and Statistics are required to participate in one of Applied Mathematics 621, Pure Mathematics 621, Statistics 621 each semester.

**
MAY BE REPEATED FOR CREDIT
**

**
NOT INCLUDED IN GPA
**

### Applied Mathematics 643 H(3-0)

#### Perturbation Theory

Perturbation problems for ordinary differential equations, matrices and more general operators. Applications. Methods will be motivated by discussion of physical problems.

**
Prerequisites:
** Familiarity with complex variables, linear algebra and differential equations.

### Applied Mathematics 671 H(3-0)

#### Numerical Linear Algebra

Iterative and elimination methods for linear systems of equations, determination of eigenvalues, linear and convex programming.

**
Prerequisites:
** Applied Mathematics 441 or Mathematics 411; and Applied Mathematics 491.

### Applied Mathematics 673 H(3-0)

#### Approximation Theory

Existence, uniqueness of minimal solutions, Haar systems, characterization by alternation, Remez algorithm, monotone operators, spline approximation.

**
Prerequisites:
** Applied Mathematics 491; and Pure Mathematics 435 or 455.

### Applied Mathematics 677 H(3-0)

#### Numerical Solution of Partial Differential Equations

Explicit and implicit methods for PDE, difference equations.

In addition to the numbered and titled courses shown above, the department offers a selection of advanced level graduate courses specifically designed to meet the needs of individuals or small groups of students at the advanced doctoral level. These courses are numbered in the series 800.01 to 899.99. Such offerings are, of course, conditional upon the availability of staff resources.