Calculus in Several Variables, 6 credits (TAIU08)

Main field of study

Mathematics, Applied Mathematics

Course level

First cycle

Advancement level

G1X

Course offered for

• Computer Engineering, B Sc in Engineering
• Engineering Electronics
• Chemical Analysis Engineering, B Sc in Engineering
• Mechanical Engineering, B Sc in Engineering

Entry requirements

Note: Admission requirements for non-programme students usually also include admission requirements for the programme and threshold requirements for progression within the programme, or corresponding.

Prerequisites

Linear algebra and Calculus

Intended learning outcomes

The course will give basic proficiency in several-variable calculus required for subsequent studies. After completing this course, students should be able to

• define and explain basic notions from topology and concepts as function, limit, continuity, partial derivative, extremal point, and multiple integral
• cite, explain and use central theorems such as differentiability implies existence of partial derivatives, the chain rule, Taylor's formula, the characterization of stationary points, the theorem on local maxima and minima, the implicit function theorem, and the theorem on change of variables in multiple integrals
• investigate limits, continuity, differentiability, and use the chain rule for transforming and solving partial differential equations
• explain the geometric significance of directional derivatives and gradients, and determine equations for tangent lines and tangent planes
• investigate local maxima and minima
• explain the behavior of an implicitly given function, for example by Taylor expansion and implicit differentiation
• calculate multiple integrals by means of iterated integration and using various changes of variables, notably linear, plane polar and spherical
• investigate convergence of improper multiple integrals and calculate their values
• verify that results and partial results are correct or reasonable

Course content

The space R^n. Fundamental notions from topology. Functions from R^n to R^p. Function graphs, level curves and level surfaces. Limit and continuity. Partial derivatives. Differentiability and differential. The chain rule. Gradient, normal, tangent and tangent plane. Directional derivative. Taylor's formula. Local extrema. Implicitly defined functions and implicit differentiation. Multiple integrals. Iterated integration. Change of variables. Area, volume, mass and center of mass. Improper multiple integrals.

Teaching and working methods

Lectures and lessons

Examination

TEN1

Written examination

U, 3, 4, 5

6 credits

Grades

Four-grade scale, LiU, U, 3, 4, 5

Other information

Supplementary courses: Vector analysis

Department

Matematiska institutionen

Director of Studies or equivalent

Jesper Thorén

Examiner

Vitalij Tjatyrko

Course website and other links

http://www.mai.liu.se/kurser/index-amne-tm.html

Education components

Preliminary scheduled hours: 64 h
Recommended self-study hours: 96 h

Course literature

Persson, A, Böiers, L-C: Analys i flera variabler, Studentlitteratur, Lund 2005. Problemsamling utgiven av matematiska institutionen.