School of Mathematics and Statistics (SoMaS)

 Semester 1, 2011/12 10 Credits Lecturer: Prof Vladimir Bavula Timetable Reading List Aims Outcomes Assessment Full Syllabus

This course continues the study of the calculus of functions of two variables, begun in MAS170. It includes the application of partial derivatives to finding and classifying local maxima and minima. The new concept of a line integral is introduced, and related to double integrals via Green's theorem, a kind of two-dimensional Fundamental Theorem of Calculus. Returning to functions of a single variable, the important techniques of Fourier series and Fourier transforms are introduced, with a taste of some of their many applications.

Economics dual students will get a handout on Lagrange multipliers at the beginning.

Prerequisites: MAS101 (Probability, Sets and Complex Numbers); MAS170 (Practical Calculus)

The following modules have this module as a prerequisite:

 MAS205 Statistics Core MAS272 Applied Differential Equations MAS273 Statistical Modelling MAS274 Statistical Reasoning MAS275 Probability Modelling MAS279 Career Development Skills MAS279A Career Development Skills (Autumn) MAS301 Group Project MAS330 Topics in Number Theory MAS332 Complex Analysis MAS334 Combinatorics MAS336 Differential Geometry MAS341 Graph Theory MAS342 Applicable Analysis MAS348 Game Theory MAS423 Advanced Operations Research MAS441 Optics and Symplectic Geometry MAS445 Mathematics (Numerical Methods)

## Outline syllabus

• Line integrals.
• Double integrals, Green's theorem.
• Differentiation under the integral sign.
• Fourier series.
• Fourier transforms.
• Maxima and minima.

## Aims

• Learn some new concepts and techniques involving differentiation and integration.
• Understand why they work.
• Relate them to each other and to what has been seen in other courses.
• See some interesting examples.

## Learning outcomes

• Calculate line integrals along parametrised curves.
• Evaluate line integrals of exact differentials by changing the path or finding a potential function.
• Compute double integrals and use Green's theorem to evaluate line integrals and areas.
• Differentiate a function defined by an integral.
• Calculate the Fourier coefficients of a periodic function.
• Calculate and use easy Fourier transforms, probability generating functions and characteristic functions.
• Understand why the normal distribution is special.
• Compute multivariable Taylor series.
• Find and classify the critical points of functions of two (or more) variables.
• Use Lagrange multipliers to find critical points subject to constraints.

22 lectures, 5 tutorials

## Assessment

One formal 2 hour written examination.

## Full syllabus

1. Line integrals.
Review of integration as a limit of sums. Definition of line integral as a limit of sums. Work done as motivating example. Basic properties. Calculating line integrals using parametrised curves. Integrals of exact differentials, finding the potential function. Criterion for exactness, path-independence. (3 lectures)

2. Double integrals, Green's theorem.
Probability density functions, joint distributions, probability as a double integral. Sums of independent random variables, convolution of density functions. Normal regions, Green's theorem, including application to calculating areas. Use of the MAPLE commands int and plot3d. (4 lectures)
3. Differentiation under the integral sign.
(1 lecture)
4. Fourier series.
Periodic functions, period T. Superpositions of trigonometric functions. Formulas for Fourier coefficients, examples. Even and odd functions. Complex form. Application of Fourier series to the heat equation for a finite rod. (4 lectures)
5. Fourier transforms.
Letting T→∞ in complex form of Fourier series to get Fourier transform and inversion theorem. Basic examples and properties of Fourier transform (step function, Dirac delta function, scaling etc.). Fourier transform of gaussian, derived using differentiation under the integral sign. Probability generating functions, characteristic functions. Addition of independent random variables. Use of characteristic functions to justify Central Limit Theorem. (3 1/2 lectures)
6. Maxima and minima.
Review of critical points for functions of one variable. Connection with Taylor series. Critical points for functions of n variables, recipe for their classification when n=2. Taylor series for functions of n variables. Quadratic forms and classification of critical points. Constrained maxima and minima, Lagrange multipliers. Use of the MAPLE commands mtaylor, Eigenvalues and extrema. (6 1/2 lectures)