Department of Mathematical Sciences



Advanced Calculus and Linear Algebra

The course material is divided into two equal sections, namely:

Section A:  Advanced Calculus

Section A follows on the Calculus of the first year, and the contents consists of curves in the plane and in space, vector calculus, functions of several variables, partial and directed derivatives and extreme value problems.  Thereafter follows a discussion of multiple integrals, line and surface integrals with applications, together with Stokes’and Green’s theorems.

Section B:  Linear Algebra

  • Euclidean vector spaces
  • General vector spaces
  • Eigenvalues and eigenvectors.

Module Information

  • 21539 214 (16) Mathematics 214
  • Year 2, semester 1 of the Programme in the Mathematical Sciences
  • Lecturing load: 4 lectures and one tutorial of 2 hours per week
  • Pass prerequisite modules: Mathematics 114 and 144
  • Language specification : A


  • Prof S Mouton:  Office 1023B, Industrial Psychology/Mathematics Building (e-mail:
  • Prof L van Wyk:  Office 1023E, Industrial Psychology/Mathematics Building (e-mail:

Learning Material

Prescribed texts:

  • Advanced Calculus:  ”Calculus”, 7th Edition.  James Stewart, Brooks/Cole Cengage Learning, 2012.
  • Linear Algebra:  ”Elementary Linear Algebra:  Applications Version”,  11th Edition.  Anton & Rorres. John Wiley,  2015.

Module Contents

See Summary.

Learning Opportunities

  • The learning material is covered during the lecture periods.
  • During the tutorial periods problems are solved under supervision.


Your class mark will be the average of your marks for the two midsemester tests.  There will be no opportunity to rewrite a test (i.e. there will be no “valskermtoetse”).  To obtain permission to write the exams, you require a class mark of at least 40%.  To pass the course, you need a final mark of at least 50%.


The module forms part of the Programme for Mathematical Sciences, and supplies the basic training in Mathematics necessary for the succesful completion of other modules in this programme and in others. It is a core module and a prerequisite for several other modules.


A student who has passed this module can be expected to command the following knowledge and skills:

Section A:

  • Understand vector functions and space curves.
  • Be able to draw graphs of simple multivariable functions, such as quadric surfaces.
  • Understands limits, continuity, partial derivatives, tangent planes, the chain rule, directional derivatives, and be able to determine the minimum and maximum value(s) of a function of two variables.
  • Understand the techniques of integration in multiple integrals.
  • Know the techniques of line and surface integrals and the concepts of curl and divergence of vector fields.
  • Know the fundamental theorems of vector calculus and can apply them.

Section B:

  • Skill in the use of matrix notation: The student can interpret the basic algorithms of linear algebra in terms of matrix multiplication and matrix factorisation.
  • Skill in the use of elementary matrix manipulations for more advanced applications: The student can find bases for general vector spaces and determine dependencies in a set of vectors.
  • Skill in the use of eigenvalues and eigenvectors.

The above knowledge of Advanced Calculus and Linear Algebra is essential for the study of subjects such as Statistics, Physics, Chemistry and Engineering as well as other applied subjects.

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