TFVK17 - Multivariable Calculus

Course Code:	TFVK17
Confirmed:	Mar 05, 2025
Valid From:	Sep 01, 2025

Education Cycle:	First-cycle level
Disciplinary domain:	Natural sciences
Subject group:	Mathematics
Specialised in:	G1F First cycle, has less than 60 credits in first-cycle course/s as entry requirements

Intended Learning Outcomes (ILO)

On completion of the course the student shall:

Knowledge and understanding

demonstrate an understanding of the basic concepts and theorems in the differential and integral calculus in several variables

Skills and abilities

demonstrate the ability to sketch regions given by inequalities and to parametrize some standard curves and surfaces
demonstrate the ability to compute partial derivatives, linearize a function or a parametrization, find directional derivatives and identify the directions of fastest increase and decrease of a differentiable function
demonstrate the ability to identify and classify local critical points of a function; find local and global extremes with or without constraints
demonstrate the ability to set up and solve double and triple integrals
demonstrate the ability to identify conservative vector fields and determine their potentials
demonstrate the ability to compute line and surface integrals over scalar and vector fields either via parametrization, or, when possible, using Green's, Gauss' or Stokes formulas.

Content

The course presents the basics of the Calculus in several variables.

The course focuses on the following topics:

Curves and surfaces in implicit form and parameter form (in particular the quadrics)
Basic set-theoretical concepts. Polar, cylindrical and spherical coordinates
Functions of several variables, level curves and surfaces
Limits and continuity, partial derivatives
Gradient, differentiability, directional derivatives and linerization
The chain rule, change of variables, the nabla differential operator, curl and divergence. Higher order partial derivatives, the Laplace and wave PDEs
Second-order Taylor polynomial, classification of critical points and identification of local extremes
Optimization on compact domains, optimization subject to constraints, Lagrange multipliers
Double and triple integrals, Fubini evaluation, change of variables
Basic calculus of vector-valued functions, line and surface integrals, conservative fields, potentials
The Green's, Gauss' divergence and Stokes formulas.

Type of instruction

Lectures and seminars.

Language of instruction is in English.

Entry requirements

General entry requirements and completed course Single Variable Calculus, 6 credits and Linear Algebra, 6 credits or Basic Calculus, 6 credits and Linear Algebra and Optimization, 7.5 credits (or the equivalent)

COURSE SYLLABUS