Several Variable Calculus

10 credits

Syllabus, Bachelor's level, 1MA016

Education cycle
First cycle
Main field(s) of study and in-depth level
Mathematics G1F
Grading system
Pass with distinction, Pass with credit, Pass, Fail
Finalised by
The Faculty Board of Science and Technology, 2 March 2022
Responsible department
Department of Mathematics

Entry requirements

5 credits in mathematics. Participation in Single Variable Calculus. Participation in Linear Algebra and Geometry I or Algebra and Geometry.

Learning outcomes

On the completion of the course the student shall be able to:

  • define, identify, explain and exemplify the basic concepts in differential- and integral calculus of several variables;
  • account for how the concepts in the previous paragraph are theoretically connected;
  • mathematically describe and analyze curves and surfaces in low dimensions;
  • calculate derivatives and integrals of functions and vector fields;
  • analyze and solve systems of ordinary differential equations and some simple partial differential equations;
  • apply the knowledge in previous paragraphs in the solving of specific calculation exercises and of elementary abstract problems;
  • present mathematical reasoning for others.


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. ​Introduction to partial differential equations and boundary values. Laplace equation, the heat conduction and wave equation. Examples from relevant areas of application, e.g. electromagnetism, thermodynamics and fluid mechanics.


Lectures, lessons and problem solving sessions.


Written exam (8 credits). Assignments (2 credits).

If there are special reasons for doing so, an examiner may make an exception from the method of assessment indicated and allow a student to be assessed by another method. An example of special reasons might be a certificate regarding special pedagogical support from the disability coordinator of the university.

Other directives

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