| TATA43 |
Calculus in several variables, 8 ECTS credits.
/Flervariabelanalys/
For:
FyN
I
Ii
Mat
MED
Y
Yi
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Prel. scheduled
hours: 76
Rec. self-study hours: 137
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Area of Education: Science
Main field of studies: Mathematics, Applied Mathematics
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Advancement level
(G1, G2, A): G1
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Aim:
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
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Prerequisites: (valid for students admitted to programmes within which the course is offered)
Linear algebra and Calculus
Note: Admission requirements for non-programme students usually also include admission requirements for the programme and threshhold requirements for progression within the programme, or corresponding.
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Supplementary courses:
Vector analysis, Complex analysis, Fourier analysis
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Organisation:
Lectures and lessons
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Course contents:
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.
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Course literature:
Persson, A, Böiers, L-C: Analys i flera variabler, Studentlitteratur, Lund 2005.
Problemsamling utgiven av matematiska institutionen.
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Examination: |
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Written examination
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8 ECTS
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Course language is Swedish.
Department offering the course: MAI.
Director of Studies: Jesper Thorén
Examiner: Lars Alexandersson (Y,Yi,MED,Mat,FyN), Göran Bergqvist (I,Ii)
Link to the course homepage at the department
Course Syllabus in Swedish
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