# Unit information: Multivariable Calculus in 2015/16

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Unit name Multivariable Calculus MATH20901 10 I/5 Teaching Block 1A (weeks 1 - 6) Professor. Robbins Not open None None School of Mathematics Faculty of Science

## Description

This unit extends elementary calculus to vector-valued functions of several variables to the point where the major theorems (Green's, Stokes' and the divergence theorem) can be presented. The emphasis is on basic ideas and methods; theorems will be stated rigorously and the theory will be carefully developed, but the style is closer to first year calculus than to analysis.

Aims:

To develop an understanding of multivariable calculus including the major theorems of vector calculus.

Syllabus

• Differential calculus in R^n: Matrix norm. Continuity. Differentiability. Relation to partial derivatives. Equality of mixed partials. Higher-order derivatives. Taylor's theorem. Criteria for local minima/maxima.
• Differential vector calculus: Grad, div, curl. Identities. Levi-Cevita symbol (from 2011).
• Integration in vector calculus. Line integrals of vector fields. Surface integrals. Stokes' theorem. Three-dimensional integrals. Gauss' theorem.

Relation to Other Units

This unit comprises the first half of MATH 20900 Calculus 2. It is provided primarily for joint honours students looking for a 10cp maths unit at level 2 and wanting to extend their calculus capabilities. Students wanting to take units such as MATH 20402 Applied Partial Differential Equations and MATH 30800 Mathematical Methods must do Calculus 2 instead, as these units currently require parts of Calculus 2 not included in this unit. There is no option to take the second half of Calculus 2 later.

## Intended learning outcomes

At the end of the course the student should:

• understand the definition of the derivative for multivariable functions
• be able to calculate Taylor series and identify local maxima/minima for multivariable functions
• understand and be able to evaluate line, surface and volume integrals
• understand the main integral theorems of vector calculus

Transferable Skills:

Clear logical thinking, problem solving, assimilation of abstract ideas and application to particular problems.

## Teaching details

Lectures (15 in all), problems classes, homework and solutions (issued later).

## Assessment Details

The final mark for Multivariable Calculus is calculated from a 1 ½ -hour written examination in April consisting of THREE questions. A candidate's best TWO answers will be used for assessment. Calculators are NOT permitted.