Several Variable Calculus

10 credits

Syllabus, Bachelor's level, 1MA016

A revised version of the syllabus is available.
Education cycle
First cycle
Main field(s) of study and in-depth level
Mathematics G1F
Grading system
Fail (U), Pass (3), Pass with credit (4), Pass with distinction (5)
Finalised by
The Faculty Board of Science and Technology, 15 March 2016
Responsible department
Department of Mathematics

Entry requirements

Single Variable Calculus together with Linear Algebra and Geometry I or Algebra and Geometry.

Learning outcomes

In order to pass the course (grade 3) the student should be able to

  • give an account of the concepts of limit, continuity, partial derivative, gradient and differentiability for functions of several variables;

  • parametrise curves and surfaces;

  • compute partial derivatives of elementary functions;

  • use partial derivatives to compute local and global extreme values - with and without constraints;

  • outline the definition of the multiple integral, compute multiple integrals and use multiple integrals to compute volumes, centres of gravity, etc.;

  • give an account of the concepts of line integral and surface integral and know how to compute such integrals;

  • use the theorems of Green, Stokes and Gauss;

  • give an account of existence and uniqueness results for solutions to ordinary differential equations, solve simple exact equations and simple linear systems of ordinary differential equations;

  • exemplify and interpret important concepts in specific cases;

  • formulate important results and theorems covered by the course;

  • express problems from relevant areas of applications in a mathematical form suitable for further analysis;

  • use the theory, methods and techniques of the course to solve mathematical problems within the course's domain;

  • present mathematical arguments to others.
  • Content

    Polar, cylindrical and spherical coordinates. Parameterisations of curves and surfaces.

    Level curves and level surfaces. Arc length. Scalar and vector valued functions of several variables. Partial derivatives, differentiability, gradient, direction derivative, differential. Derivatives of higher order. The chain rule. The Jacobian. Taylor's formula. Implicit functions. Optimisation: local and global problems, problems with equality constraints. Multiple integrals, change of variables, improper integrals, applications of multiple integrals: volume, centres of mass, etc. Line integrals and surface integrals of scalar functions and vector fields. Divergence and curl. Identities for grad, div and curl. Green's, Stokes' and Gauss's theorems. Systems of ordinary differential equations. Exact equations. Linear systems, the exponential matrix. Second order equations, variation of parameters.


    Lectures, lessons and problem solving sessions.


    Written examination at the end of the course, or two written tests each of five credit points. Moreover, compulsory assignments may be given during the course in accordance with instructions at the beginning of the course.

    Other directives

    The course cannot be included in passing degree together with the course Several Variable Calculus, limited version.