Print Syllabus

Syllabus

MA0037 Multivariable Calculus, 10.0 Credits

Flervariabelanalys

Subjects

Mathematics/Applied Mathematics

Education cycle

Bachelor’s level

Advanced study in the main field

First cycle, has less than 60 credits in first-cycle course/s as entry requirements
Bachelor’s level (G1F)

Grading scale

Language

Swedish

Prior knowledge

- 10 hp Single-Variable Calculus 
- 5 hp Linear Algebra

Objectives

The course aims to introduce students to mathematical analysis in several variables, to provide fundamental analytical tools for systems of ordinary differential equations, Fourier series and the Fourier transform, as well as a brief introduction to partial differential equations.

After successfully completing the course, the student should be able to:

• Explain the concepts of limits, continuity, partial derivatives, and gradients.

• Compute partial derivatives and use them to identify local and global extrema, including optimization problems.

• Explain and compute double and triple integrals, and apply them to the calculation of volumes and other quantities.

• Explain the concepts of line and surface integrals.

• Explain the concepts of linear systems of ordinary differential equations (ODEs), equilibrium points, phase portraits, and stability.

• Solve homogeneous linear systems with constant coefficients.

• Solve simple non-homogeneous linear systems with constant coefficients.

• Explain the basic concepts of partial differential equations (PDEs).

• Explain the concepts of Fourier series, including orthogonality, coefficients, and even/odd extensions.

• Explain the definition and basic properties of the Fourier transform.

• Translate problems from relevant application areas into appropriate mathematical form and present their solutions clearly.

Content

Subject-related content

Partial derivatives, differentiability, gradient, directional derivative, and differential.

Higher-order derivatives. Chain rule. Jacobian. Taylor’s formula.

Optimization problems: local and global extrema, and constrained optimization.
Multiple integrals, change of variables, improper integrals, and applications.
Green’s, Stokes’, and Gauss’ theorems.

Linear systems of ordinary differential equations (ODEs): homogeneous systems with constant coefficients, phase portraits, and stability; non-homogeneous systems.
Nonlinear two-dimensional systems: equilibria, linearization via the Jacobian, local stability; model examples from mechanics and population dynamics.

Fourier series: orthogonality, coefficients, even/odd extensions, and the idea of convergence.
Introduction to partial differential equations (PDEs): the heat equation and the wave equation.

Fourier transform: definition, key properties (linearity, scaling, shifting, differentiation rule), the heat equation on the entire real line.
Modeling with systems of ODEs.
Real-world examples and applications.

Teaching formats

The course employs lectures and tutorials to support student learning and engagement.

The course emphasizes the development of the following general competences: Critical thinking; Problem solving; Scientifc methods.

Grading form

The grade requirements within the course grading system are set out in specific criteria. These criteria must be available by the course start at the latest.

Formats and requirements for examination

Approved written exam

• If a student has failed an examination, the examiner has the right to issue supplementary assignments. This applies if it is possible and there are grounds to do so.

• The examiner can provide an adapted assessment to students entitled to study support for students with disabilities following a decision by the university. Examiners may also issue an adapted examination or provide an alternative way for the students to take the exam.

• If this syllabus is withdrawn, SLU may introduce transitional provisions for examining students admitted based on this syllabus and who have not yet passed the course.

• For the assessment of an independent project (degree project), the examiner may also allow a student to add supplemental information after the deadline for submission. Read more in the Education Planning and Administration Handbook.

Other information

• The right to participate in teaching and/or supervision only applies for the course instance the student was admitted to and registered on.

• If there are special reasons, students are entitled to participate in components with compulsory attendance when the course is given again. Read more in the Education Planning and Administration Handbook.

Responsible department

Department of Energy and Technology