Multivariable Calculus
Entry Requirements
- 5 hp Linear Algebra
Course facts
MA0037, Multivariable Calculus, 10.0 Hp
Print syllabus
Syllabus
Finalized by: PN-O, 2025-12-12
Valid from : Spring semester 2027 (2027-01-18)
Level
First cycle (G1F)
Subject
Mathematics/Applied Mathematics
Grading Scale
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.
Course language
Swedish
Entry Requirements
- 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.
Examination Formats and Requirements for Passing the Course
Approved written exam
Responsible Department/Equivalent
Department of Energy and Technology
Supplementary information
Included in program
- Bioresource Systems Engineering