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Unit information: Multivariable Calculus in 2015/16

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Unit name Multivariable Calculus
Unit code MATH20901
Credit points 10
Level of study I/5
Teaching block(s) Teaching Block 1A (weeks 1 - 6)
Unit director Professor. Robbins
Open unit status Not open




School/department School of Mathematics
Faculty Faculty of Science


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.


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


  • 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.

Reading and References

Multivariable calculus: Jerrold E. Marsden & Anthony J. Tromba, Vector Calculus, ed. 5 , W. H. Freeman and Company, 2003