Course syllabus
Flerdimensionell analys med vektoranalys
Calculus in Several Variables
FMA435, 7,5 credits, G1 (First Cycle)
Valid for: 2014/15
Decided by: Education Board B
Date of Decision: 2014-04-08
General Information
Main field: Technology.
Compulsory for: Pi1
Language of instruction: The course will be given in Swedish
Aim
The course aims at giving a basic treatment of calculus in
several variables, including three-dimensional vector analysis.
Particular emphasis is put on the role which calculus in several
variables plays in applications in different subjects of
technology, in order to give the future engineer a good foundation
for further studies in mathematics as well as other subjects. The
aim is furthermore to develop the student's ability in problem
solving and to assimilate mathematical text.
Learning outcomes
Knowledge and understanding
For a passing grade the student must
- be able to compute with and handle elementary functions
of several variables within the framework of the course, together
with their derivatives and integrals, with confidence.
- be familiar with different representations of curves, surfaces
and volumes in two and three dimensions, and be able to use them in
computations.
- be able to carry out (specified) changes of variables in
partial differential equations, and by this means to
solve such equations.
- be familiar with the theory of optimization, local as well as
global, and be able to find the solution in simple cases.
- be able to demonstrate an ability to independently choose
methods to evaluate double and triple integrals, and be able to
carry out the solution essentially correct.
- be able to demonstrate an ability to independently choose
methods to evaluate curve and surface integrals, and be able to
carry out the solution essentially correct.
- be familiar with the important theorems of vector analysis,
and have an understanding of their physical interpretation.
- be able to demonstrate a good ability to carry out algebraic
calculations within the context of the course.
- be able to give a general account of, and to illustrate the
meaning of, such mathematical concepts in calculus in several
variables that are used to construct and study mathematical models
in the applications.
- be able to account for the contents of some central
definitions, theorems and proofs.
Competences and skills
For a passing grade the student must
- in the context of problem solving, be able to demonstrate an
ability to independently choose and use mathematical concepts and
methods within calculus in several variables.
- in the context of problem solving be able to integrate concepts
from different parts of the course.
- be able to demonstrate an ability to construct and analyse
some simple mathematical models in calculus in several
variables.
- be able to demonstrate an ability to explain mathematical
arguments in a structured and logically clear way.
- have a basic ability to use Maple for visualisation and formula
manipulation, and be aware of its possibilities and
limitations.
Contents
Part 1. Calculus in several variables
- Generalities on functions of several variables.
function surfaces, level surfaces, surfaces in parameter form,
curvilinear coordinates.
- Partial derivatives. Differentiability, tangent planes, error
propagation. The chain rule. Applications to partial differential
equations. Gradient, directional derivative, level curves. Study of
stationary points. Curves, tangent, arc length. Surfaces, normal
direction, tangent plane. Functional (Jacobi) matrix and
determinant, linearisation. Implicit functions.
- Optimization on compact and non-compact domains. Optimization
with constraints.
- Double and triple integrals. Iterated integration. Change of
variables. Integration using level curves. Improper integrals.
Applications: volume, moment of inertia, centre of gravity.
- Curve integrals. Green's formula with applications. Potential
and exact differential.
- Computer work. Visualization and formula manipulation using
Maple.
Part 2. Threedimensional vector analysis
Surface integrals. Flux integrals. Divergence and rotation.
Gauss' and Stokes' theorems. Potential and exact
differential. The continuity equation.
Examination details
Grading scale: TH
Assessment: In the first subcourse a written test comprising theory and problem solving. (This test is identical to the one given for the course FMA430). In the second subcourse a written test. The final grade is the grade obtained in the first test. Computer work.
Parts
Code: 0108. Name: Calculus in Several Variables.
Credits: 6. Grading scale: TH. Assessment: Written test comprising theory and problem solving. The test is identical to the one given for the course FMA430. Contents: See part 1 above. (Coinciding with the contents of the course FMA430.)
Code: 0208. Name: Three-dimensional Vector Analysis.
Credits: 1,5. Grading scale: UG. Assessment: Written test in the middle of the study period. Retakes in appropriate examination periods. Contents: See part 2 above.
Code: 0308. Name: Computer Work.
Credits: 0. Grading scale: UG.
Admission
Required prior knowledge: FMAA01 or FMAA05, Calculus in One Variable, FMA420 Linear Algebra.
The number of participants is limited to: No
The course overlaps following course/s: FMA025, FMA430, FMAF15
Reading list
- Jonas Månsson och Patrik Nordbeck: FLERDIMENSIONELL ANALYS, Tredje upplagan. Studentlitteratur, 2013, ISBN: 9789144080833.
- ÖVNINGAR I FLERDIMENSIONELL ANALYS. Studentlitteratur, 2013, ISBN: 9789144092508.
Contact and other information
Course coordinator: Studierektor Anders Holst, Studierektor@math.lth.se
Course homepage: http://www.maths.lth.se/course/flerdimvec/