Several Variable Calculus
Syllabus, Bachelor's level, 1MA016
- Code
- 1MA016
- Education cycle
- First cycle
- Main field(s) of study and in-depth level
- Mathematics G1F
- Grading system
- Pass with distinction (5), Pass with credit (4), Pass (3), Fail (U)
- 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.
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. 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.
Instruction
Lectures, lessons and problem solving sessions.
Assessment
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.
Reading list
- Reading list valid from Autumn 2025
- Reading list valid from Autumn 2022
- Reading list valid from Spring 2022
- Reading list valid from Autumn 2021
- Reading list valid from Autumn 2020
- Reading list valid from Autumn 2019
- Reading list valid from Spring 2019
- Reading list valid from Autumn 2016
- Reading list valid from Autumn 2010, version 3
- Reading list valid from Autumn 2010, version 2
- Reading list valid from Autumn 2010, version 1
- Reading list valid from Autumn 2009
- Reading list valid from Autumn 2007, version 2
- Reading list valid from Autumn 2007, version 1
- Reading list valid from Spring 2005