Modulets indhold, forløb og pædagogik
Purpose:
The purpose of this course is to provide the students with
knowledge of, and to support the students in their understanding
of, mathematical theories and methods of general applicability
within the analysis of linear systems on an application level. In
addition the course supports the students in their understanding of
complex function theory and vector analysis.
Læringsmål
Viden
- demonstrate an understanding of concepts, theories and methods
used within the area of complex function theory,
including:analytical functions and their derivatives
- Cauchy-Riemann equations
- curve integrals
- graphical representations of standard complex mappings; Möbius
(and its special cases), trigonometric, polynomial, logarithm, and
exponential.
- demonstrate an understanding of concepts, theories and methods
used within the area of series theory and Fourier transformation,
including:
- sampling continuous time functionsDiscrete Fourier
Transformation and Fast Fourier Transformation
- demonstrate an understanding of concepts, theories and methods
used within the area of vector analysis, including:
- skalarfelter og vektorfelter.
- rumlige integraler, herunder kurveintegraler, fladeintegraler
og volumenintegralerflux og cirkulation.
- Greens sætning, Stokeses sætning, Gausses sætning og
Helmholtzes sætning
Færdigheder
- apply the presented concepts, theories and methods used within
the area of complex function theory to:
- determine function properties; continuity and analyticity
complex functions
- apply Cauchy-Riemann equation to functions to determine if a
function is analytical
- Möbius transform and its special cases, including dilation,
translation, rotation, and inversion.
- Cauchy’s integral theorem and integral formula
- design of Möbius transform based on mapping points
- curve integrals, closed curve integrals, finding critical
points for functions
- apply Cauchy’s integral theorem og formula to analytical
functions
- apply the presented concepts, theories and methods used within
the areas of series theory and Fourier transformation to:
-
- Convergence tests Series and sequences
- Series analysis with special focus at convergence test (e.g. by
Comparison Test, by Ratio Test or by Root Test)
- specification and analysis of Power Series with special focus
at Convergence and calculation of the Radius of Convergence R by
Cauchy-Hadamards formula
- Power Series development by Taylor and Maclaurin
approximation
- development of Fourier Series for periodic functions
- development of Fourier Series for even and odd functions – and
for arbitrary periods (2L)
- development of Fourier Integrals
- development of the Fourier Transformation for real and
complex functions
- calculate amplitude specters and phase specters for Fourier
Series and for Fourier Transforms.
- Power Series, coefficients and the center
- Radius of Convergence R – Cauchy-Hadamards formular
- Taylor and Maclaurin power series
- Fourier Series
- Fourier Integrals
- Amplitude and phase specters by the Fourier transform
- Apply the presented concepts, theories and methods used within
the area of vector analysis to:
- Parametriske beskrivelser af kurser og flader
- Konservative felter og solenoidale felter
- Begrebet potentialefunktion
- Anvendelse af Jacobianten i forbindelse med
variabelsubstitutioner
- Fremstille parametriske repræsentationer af kurver og flader ud
fra verbale, formelle eller grafiske beskrivelser
- Skitsere givne kurser og flader
- Evaluere kurveintegraler, dobbeltintegraler, fladeintegraler og
volumenintegraler
- Foretage variabelskift under anvendelse af Jacobiant
- Bestemme divergens, gradient og rotation for givne skalar- og
vektorfelter
- Evaluere rumlige integraler under anvendelse og Gausses sætning
og Stokeses sætning
- Bestemme en potentialfunktion for et givent konservativt felt
samt kontrollere løsningen
- Evaluere vejuafhængige kurveintegraler ved at finde
stamfunktion
- Parameterization of curve integrals and path-independent curve
integrals
Kompetencer
- The Fourier transform
- Amplitude and phase specters by the Fourier transform
- Fremstille parametriske repræsentationer af kurver og flader ud
fra verbale, formelle eller grafiske beskrivelser
Undervisningsform
Forelæsninger, opgaveregning, workshops, selvstudie.
Eksamen
Prøver
Prøvens navn | Beregningsteknik indenfor elektronikområdet 1 |
Prøveform | Skriftlig eller mundtlig |
ECTS | 5 |
Bedømmelsesform | 7-trins-skala |
Censur | Intern prøve |
Vurderingskriterier | Vurderingskriterierne er angivet i Universitetets
eksamensordning |