# Undergraduate Catalogue of Courses 2012/2013

## MATHEMATICS

MA 2507

ADVANCED CALCULUS

CREDIT POINTS 15

*Course Co-ordinator:* Dr Jean-Baptiste Gramain

*Pre-requisite(s):*
MA 2005

A) Several variables - Continuity and Partial Differentiation.

- Functions of several variables, graphical surface representations, limits and continuity, partial derivatives, higher order partials, plane tangent, linear approximation, small errors.
- Chain rule, polar coordinates, applications to some elementary PDEs.
- Critical points, second derivative test, some discussion of global global max/min.
- Taylor series and quadratic approximation.

B) Several variables - Multiple Integrals.

- Revision of definite integral as area under curve, approximated by rectangles. Double integral as volume, approximated by rectangular pillars.
- Iteration formulae for rectangles and for more general regions, change of order of integration via double integral.
- Change of variable in double integrals (with emphasis on polars).
- Triple integrals, cylindrical and spherical polar coordinates.
- Applications to volumes, moments and centres of mass.

C) Ordinary differential equations.

- Basic terminology, general solution, integral curves, initial and boundary conditions.
- First order ODEs: linear equations, separable equations, brief treatment of homogeneous and Bernoulli equations, applications (eg. population problems and mixing problems).
- Second order linear ODEs: basic theory for solution of equations with constant coefficients via CF + PI, applications; reduction of order and variation of parameters.

D) Introduction to computing software (4 practical sessions).

- Simple arithmetic, operations, variables, booleans, conditionals, functions, procedures, plotting, functions in two variables, contour plots, parametric plots, basic algebra, differentiation, integration, programming, loops.

12 week course - 3 one-hour lectures per week, 1 one-hour tutorial per week and 4 one-hour practicals over the 12 weeks. To be arranged.

1st Attempt: 1 two-hour written examination (80%); in-course assessment (20%).

Resit: 1 two-hour written examination paper, maximum resit (100%) and resit (80%) with in-course assessment (20%).