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ING3019 Multivariable Calculus

Course description for academic year 2018/2019

Contents and structure

  • Partial derivatives, multiple integrals, vector analysis. The use of software tools
  • Three-dimensional coordinate systems, vectors, the dot product and the cross product, lines and planes in space, cylinders and quadric surfaces.
  • Curves in space with velocity  and acceleration, arc length in space.
  • Functions of several variables, limits and continuity in higher dimensions, partial derivatives, the chain rule, direction derivatives and gradient vectors, tangent planes, extreme values, Lagrange multipliers.
  • Double integrals in cartesian coordinate systems, double integrals in polar form, triple integrals in rectangular coordinates, triple integrals in cylindrical and spherical coordinates, substitution in double and triple integrals.
  • Line integrals, vector fields and line integrals in vector fields, work, circulation and flux, conservative fields, potential functions, Green¿s theorem in the plane, surface integrals, Stokes¿ theorem, the divergence theorem.
  • The use of software tools.

Learning Outcome

- Knowledge:

  • The student can describe and give examples of functions of two or more variables.
  • The student can describe and give examples of principles, approximations and methods used with functions of two or more variables.
  • The student has knowledge about the use of software tools in visualization and calculation of  multivariable problems.



  • The student can apply the knowledge of multivariable mathematics to formulate, specify and solve engineering problems in a well-founded and systematic way.
  • The student can consider solutions and results critically.
  • The student can redistribute central theories and solution methods due to the subject.
  • The student can solve numerical problems using a standard software tool.


- General qualifications

  • The student has deepened and expanded the understanding of functions of one variable to multivariable functions  (two and three variables).
  • The student has achieved insight into important technical applications of multivariable functions.
  • The student has achieved the mathematical understanding necessary for further academically development at the master level.

Entry requirements


Recommended previous knowledge

Analysis and Linear Algebra, Series and Functions of Several Variables and Physics

Teaching methods

Lectures and excercises.

Compulsory learning activities



Part 1: Portfolio accounts for 30% of the final grade.

Part 2: Written exam, accounts for 70% of the final grade.

Both parts must be passed.

Grade: A - E / F (failed).

Examination support material

Simple calculator: Allowed calculator is Casio fx-82 (all varieties: ES, ES Plus, EX, Solar etc.)

More about examination support material