Multivariable Calculus

Unit aims

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

Unit description

The course develops multivariable calculus from Calculus 1. The main focus of the course is on developing differential vector calculus, tools for changing coordinate systems and major theorems of integral calculus for functions of more than one variable.

Relation to other units

This unit feeds into applied mathematics units such as Fluid Dynamics.  Applied Partial Differential Equations and Mathematical Methods also use the material.

Learning objectives

At the end of the course the student should:

  • understand the definition of the derivative for multivariable functions.
  • be comfortable with vector identities in differential calculus, and differential operators in curvilinear coordinate systems.
  • 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.


  • Differentiation of vector functions, operations on maps. Jacobians, inverses, implicit function theorem. Multi-dimensional Taylor's theorem.
  • Differential vector calculus:  Grad, div, curl.  Identities. Summation Convention, Levi-Cevita symbol. Differential operators in curvilinear coordinate systems. Scale factors.
  • Integration in vector calculus. Line integrals of vector fields. Surface integrals. Stokes' theorem. Three-dimensional integrals. The divergence theorem, Green's theorem in the plane, Green's identities.

Reading and References

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

Unit code: MATH20901
Level of study: I/5
Credit points: 10
Teaching block (weeks): 1 (1-6)
Lecturer: Dr Richard Porter


Linear Algebra and Geometry, Calculus 1, Analysis 1A and Analysis 1B



Methods of teaching

Lectures, problems classes, homework and solutions.

Methods of Assessment

The pass mark for this unit is 40.

The final mark is calculated as follows:

  • 100% from a 1 hour 30 minute exam in January.

NOTE: Calculators are NOT allowed in the examination.

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.

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