Course Catalogue

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This course on multivariate calculus gives a practical introduction to differentiating and integrating in multiple dimensions, and to fundamental concepts found in diverse fields such as geometry and physics. It is an essential course for intending honours students. The emphasis is on methods and applications.

2 x lectures per week. There may be extra lectures in teaching week 1. Fortnightly tutorials.

Mathematics 1 at at least grade D3.

None.

One degree examination (80%) (1 hour 30 mins); coursework (20%).

Reassessment will not be available for continuous assessment. Immediate feedback will be given to students at the assignment deadline, and it is not practical to set multiple versions of each of these exercises.

In total 80% of the assessment can be repeated for students not reaching the threshold grade.

Note that students are required to submit 75% of the assessments in order to receive a grade for the course. Accordingly, students who have a substantial period of illness and have all eAssignments and tutorial group work MV'ed, will not be in danger of failing to receive credits.

This course covers fundamental topics in multivariable calculus. The aim of the first part of the course is to teach students how to effectively compute partial derivatives of multivariate functions. This is used in the analysis of local minima and maxima of functions. The aim of the second part of the course is to understand the concepts div, grad and curl and learn to use the identities they satisfy to solve concrete problems. The third part of the course aims to teach students how to compute double and triple integrals. In particular, we aim to understand how to perform line and surface integrals by using appropriate change of variables. The final aim of the course is to understand Green's theorem and the divergence theorem and learn how to apply these theorems in concrete situations.

By the end of the course, students will be able to:

■ Sketch three-dimensional surfaces using cross-sections and contours; apply the chain rule to partial derivatives;

■ Determine the stationary points of functions of several variables; classify these stationary points using first principles and the Hessian criterion; apply the method of Lagrange multipliers to solve practical extreme value problems.

■ Parametrise curves in two- and three-dimensions; state the definitions of the divergence, gradient and curl operators; interpret and apply these operators to solve suitable problems.

■ Compute the (signed) volume under a surface and compute double integrals; determine limits for a given domain; change the order of integration; determine Jacobian matrices for given functions and apply these in a change of variables.

■ Compute line integrals; prove path-independence of line integrals for conservative vector fields; give a parametric description of a surface and compute surface integrals.

■ Apply Green's theorem to evaluating closed line integrals; apply the divergence theorem to evaluate closed surface integrals.

■ State the definitions, results and formulae presented in lectures; apply and adapt these to solve suitable problems.

Students must submit at least 75% by weight of the components (including examinations) of the course's summative assessment.