Calculus in Several Variables, 6 credits

Flervariabelanalys, 6 hp

TATA69

Main field of study

Mathematics Applied Mathematics

Course level

First cycle

Course type

Programme course

Examiner

Göran Bergqvist (I,Ii) Hans Lundmark (DPU,EMM,M)

Director of studies or equivalent

Jesper Thorén

Education components

Preliminary scheduled hours: 64 h
Recommended self-study hours: 96 h
ECV = Elective / Compulsory / Voluntary
Course offered for Semester Period Timetable module Language Campus ECV
6CDPU Design and Product Development, M Sc in Engineering 3 (Autumn 2017) 1 4 Swedish Linköping C
6CEMM Energy-Environment-Management M Sc in Engineering 3 (Autumn 2017) 1 4 Swedish Linköping C
6CMMM Mechanical Engineering, M Sc in Engineering 3 (Autumn 2017) 1 4 Swedish Linköping C

Main field of study

Mathematics, Applied Mathematics

Course level

First cycle

Advancement level

G1X

Course offered for

  • Design and Product Development, M Sc in Engineering
  • Energy-Environment-Management M Sc in Engineering
  • Mechanical Engineering, M Sc in Engineering
  • Industrial Engineering and Management, M Sc in Engineering
  • Industrial Engineering and Management - International, M 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

TEN1Written examination6 creditsU, 3, 4, 5

Grades

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

Department

Matematiska institutionen

Director of Studies or equivalent

Jesper Thorén

Examiner

Göran Bergqvist (I,Ii) Hans Lundmark (DPU,EMM,M)

Course website and other links

Education components

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

Course literature

Additional literature

Books

  • Persson, A, Böiers, L-C, (2005) Analys i flera variabler Studentlitteratur, Lund

Other

  • Problemsamling utgiven av matematiska institutionen
Code Name Scope Grading scale
TEN1 Written examination 6 credits U, 3, 4, 5

Regulations (apply to LiU in its entirety)

The university is a government agency whose operations are regulated by legislation and ordinances, which include the Higher Education Act and the Higher Education Ordinance. In addition to legislation and ordinances, operations are subject to several policy documents. The Linköping University rule book collects currently valid decisions of a regulatory nature taken by the university board, the vice-chancellor and faculty/department boards.

LiU’s rule book for education at first-cycle and second-cycle levels is available at http://styrdokument.liu.se/Regelsamling/Innehall/Utbildning_pa_grund-_och_avancerad_niva. 

Additional literature

Books

Persson, A, Böiers, L-C, (2005) Analys i flera variabler Studentlitteratur, Lund

Other

Problemsamling utgiven av matematiska institutionen

Note: The course matrix might contain more information in Swedish.

I = Introduce, U = Teach, A = Utilize
I U A Modules Comment
1. DISCIPLINARY KNOWLEDGE AND REASONING
1.1 Knowledge of underlying mathematics and science (G1X level)
X
X
X
TEN1

                            
1.2 Fundamental engineering knowledge (G1X level)

                            
1.3 Further knowledge, methods, and tools in one or several subjects in engineering or natural science (G2X level)

                            
1.4 Advanced knowledge, methods, and tools in one or several subjects in engineering or natural sciences (A1X level)

                            
1.5 Insight into current research and development work

                            
2. PERSONAL AND PROFESSIONAL SKILLS AND ATTRIBUTES
2.1 Analytical reasoning and problem solving
X
X
TEN1

                            
2.2 Experimentation, investigation, and knowledge discovery
X

                            
2.3 System thinking

                            
2.4 Attitudes, thought, and learning
X
X
TEN1

                            
2.5 Ethics, equity, and other responsibilities

                            
3. INTERPERSONAL SKILLS: TEAMWORK AND COMMUNICATION
3.1 Teamwork

                            
3.2 Communications
X

                            
3.3 Communication in foreign languages

                            
4. CONCEIVING, DESIGNING, IMPLEMENTING AND OPERATING SYSTEMS IN THE ENTERPRISE, SOCIETAL AND ENVIRONMENTAL CONTEXT
4.1 External, societal, and environmental context

                            
4.2 Enterprise and business context

                            
4.3 Conceiving, system engineering and management

                            
4.4 Designing

                            
4.5 Implementing

                            
4.6 Operating

                            
5. PLANNING, EXECUTION AND PRESENTATION OF RESEARCH DEVELOPMENT PROJECTS WITH RESPECT TO SCIENTIFIC AND SOCIETAL NEEDS AND REQUIREMENTS
5.1 Societal conditions, including economic, social, and ecological aspects of sustainable development for knowledge development

                            
5.2 Economic conditions for knowledge development

                            
5.3 Identification of needs, structuring and planning of research or development projects

                            
5.4 Execution of research or development projects

                            
5.5 Presentation and evaluation of research or development projects

                            

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