ADVANCED CALCULUS

Course Code

MA 2507

Credit Points

15

Course Coordinator

Dr Jean-Baptiste Gramain

Pre-requisites

MA 2005

Overview

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.

Structure

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.

Assessment

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%).

Formative Assessment

Informal assessment of weekly homework through discussions in tutorials.

Feedback

In-course assignments will normally be marked within one week and feedback provided to students in tutorials. Students will be invited to contact the Course Coordinator for feedback on the final examination.