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Multivariable Calculus (MATH 20901)

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Administrative Information

  1. Unit number and title: MATH 20901 Multivariable Calculus
  2. Level: I/5
  3. Credit point value: 10 credit points
  4. Year: 12/13
  5. First Given in this form: 1995-6. Modifications 2010-2013
  6. Unit Organiser: Jens Eggers
  7. Lecturer: Prof. JG Eggers
  8. Teaching block: 1
  9. Prerequisites: MATH 11005 Linear Algebra and Geometry, MATH 11006 Analysis 1 and MATH 11007 Calculus 1

Unit aims

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

General Description of the Unit

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 rigourously and the theory will be carefully developed, but the style is closer to first year calculus than to analysis.

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.

Teaching Methods

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

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

Assessment Methods

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.

Award of Credit Points

To be awarded the credit points for this unit you must normally pass the unit, i.e. you must achieve an assessment mark of at least 40.

Transferable Skills

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

Texts

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

Syllabus

  • Differential calculus in R^n: Matrix norm. Summation convention. Continuity.

  • Differentiability. Relation to partial derivatives.  Equality of mixed partials.  Higher-order derivatives.  Taylor's theorem.
  • Differential vector calculus:  Grad, div, curl.  Identities. 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.  Gauss' theorem.

  

Unit Home Page

Course web pages:

http://www.maths.bristol.ac.uk/~majge/calculus2.html