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Calculus in Several Variables, 8 credits
Flervariabelanalys, 8 hp
TATA43
Main field of study
Mathematics Applied MathematicsCourse level
First cycleCourse type
Programme courseExaminer
Vladimir TkatjevDirector of studies or equivalent
Jesper ThorénEducation components
Preliminary scheduled hours: 76 hRecommended self-study hours: 137 h
ECV = Elective / Compulsory / Voluntary
Main field of study
Mathematics, Applied MathematicsCourse level
First cycleAdvancement level
G1XCourse offered for
- Biomedical Engineering, M Sc in Engineering
- Applied Physics and Electrical Engineering, M Sc in Engineering
- Physics, Bachelor´s Programme
- Mathematics, Bachelor´s Programme
- Applied Physics and Electrical Engineering - 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 CalculusIntended 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, (local) extremal point, and multiple integral
- cite, explain and use central theorems such as the max-min existence theorem, differentiability implies existence of partial derivatives, the chain rule, Taylor's formula, the characterization of stationary points, the theorem on local maxima and minima with constraints, 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 and global maxima and minima, with or without constraints
- explain the behavior of an implicitly given function, for example by Taylor expansion through 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 and global extrema. Extremal problems with constraints by means of linearly dependent gradients. 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 | 8 credits | U, 3, 4, 5 |
Grades
Four-grade scale, LiU, U, 3, 4, 5Other information
Supplementary courses: Vector analysis, Complex analysis, Fourier analysis
Department
Matematiska institutionenDirector of Studies or equivalent
Jesper ThorénExaminer
Vladimir TkatjevCourse website and other links
http://www.mai.liu.se/kurser/index-amne-tm.htmlEducation components
Preliminary scheduled hours: 76 hRecommended self-study hours: 137 h
Course literature
Additional literature
Books
- Persson, A, Böiers, L-C, (2005) Analys i flera variabler Studentlitteratur, Lund
Compendia
Other
- Problemsamling utgiven av matematiska institutionen
| Code | Name | Scope | Grading scale |
|---|---|---|---|
| TEN1 | Written examination | 8 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
Compendia
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
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X
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X
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TEN1
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| 1.2 Fundamental engineering knowledge (G1X level) |
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| 1.3 Further knowledge, methods, and tools in one or several subjects in engineering or natural science (G2X level) |
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| 1.4 Advanced knowledge, methods, and tools in one or several subjects in engineering or natural sciences (A1X level) |
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| 1.5 Insight into current research and development work |
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| 2. PERSONAL AND PROFESSIONAL SKILLS AND ATTRIBUTES | ||||||
| 2.1 Analytical reasoning and problem solving |
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X
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X
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TEN1
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| 2.2 Experimentation, investigation, and knowledge discovery |
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X
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| 2.3 System thinking |
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| 2.4 Attitudes, thought, and learning |
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X
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X
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TEN1
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| 2.5 Ethics, equity, and other responsibilities |
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| 3. INTERPERSONAL SKILLS: TEAMWORK AND COMMUNICATION | ||||||
| 3.1 Teamwork |
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| 3.2 Communications |
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X
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| 3.3 Communication in foreign languages |
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| 4. CONCEIVING, DESIGNING, IMPLEMENTING AND OPERATING SYSTEMS IN THE ENTERPRISE, SOCIETAL AND ENVIRONMENTAL CONTEXT | ||||||
| 4.1 External, societal, and environmental context |
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| 4.2 Enterprise and business context |
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| 4.3 Conceiving, system engineering and management |
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| 4.4 Designing |
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| 4.5 Implementing |
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| 4.6 Operating |
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| 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 |
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| 5.2 Economic conditions for knowledge development |
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| 5.3 Identification of needs, structuring and planning of research or development projects |
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| 5.4 Execution of research or development projects |
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| 5.5 Presentation and evaluation of research or development projects |
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