Exercises with Maple
Rolf Brigola, Georg-Simon-Ohm-University of Applied Sciences, Nürnberg, Germany
Exercises for First Semester Students in Engineering
Exercise 1
Solve Practice Problems in the Maple Tutorial
Exercise 2
How far away is the horizon?
Exercise 3
Solve a Optimization Problem
Exercise 4
Make your first Fourier expansion, observe the Gibbs phenomenon,
look at Fejer's trigonometric approximation
Exercise 5
Calculate the curve known as the Conchoide of Nikomedes
Exercise 6
Solve a Combinatorial Problem
Exercise 7
Calculate Amplitude and Phase of a Harmonic Oscillation
Exercise 8
Calculate Distances on the Earth's Surface
Exercise 9
The Fibonacci Sequence and the Golden Section
Exercise 10
Solve Exercise 6 with the Fibonacci Numbers
Exercise 11
Calculate the Convergence Radius of a Power Series
Exercise 12
Calculate the Cauchy Product of Two Power Series
Exercise 13
On Different Definitions of a Mean Value
Exercise 14
Make some calculations with complex numbers
Exercise 15
Continue with complex calculations
Exercise 16
And again some simple calculations with complex numbers
Exercise 17
Calculate and plot images of circles and lines under the mapping w=1/z
Exercise 18
Design a Butterworth Filter, First Steps
A Solution for Exercise 2
A Solution for Exercise 5
A Solution for Exercise 6
A Solution for Exercise 7
A Solution for Exercise 8
A Solution for Exercise 9
Exercises for Second Semester Students in Engineering
Linear Algebra
Exercise 1 Orthogonal Projection, Volume of a Parallelepiped, Equation of a Line
Exercise 2 General Solution of a System of Linear Equations
Exercise 3 Solution of a System of Linear Equations depending on Parameters
Exercise 4 Compute the Determinant of a Product of Simple Matrices
Exercise 5 Compute a Orthogonal Projection into a Subspace
Exercise 6 Compute the Matrix, which describes a Linear Mapping, and its Diagonalization
Exercise 7 Compute the Matrix of a Rotation
Exercise 8 Compute a Polynomial Approximation for the Sine Function with Minimal RMS-Error
Exercise 9 Compute a Trigonometric Approximation with Minimal RMS-Error for the Sawtooth Function
Exercise 10 A First Experience with Regularization of Ill-Posed Linear Problems
Exercise 11 Write a procedure to calculate a rotation of a vector around a given axis
Exercise 12 Write a procedure for a numerical solution of linear convolution problems. Test it with and without regularization
Exercise 13 Program the Gram-Schmidt Orthogonalization Procedure for spatial
vectors and function families as well
A Solution for Exercise 1
A Solution for Exercise 2
A Solution for Exercise 3
A Solution for Exercise 4
A Solution for Exercise 5
A Solution for Exercise 6
A Solution for Exercise 7
A Solution for Exercise 8
A Solution for Exercise 9
A Solution for Exercise 10
A Solution for Exercise 11
Another Solution for Exercise 11
A Solution for Exercise 12
A Solution for Exercise 13
Ordinary Differential Equations, Time-Invariant Linear Systems
Exercise 1 Transform a Explicit Higher Order Differential Equation into a First Order System
Exercise 2 Solve a First Order System using the Jordan Form of the System Matrix
Exercise 3 On Transfer Functions, Causal Impulse Response, Frequency Characteristics
and Realization of Time-Invariant Linear Systems
Exercise 4 Write a Procedure for the Solution of a Time-Invariant Linear First Order System
Exercise 5 Use the Laplace Transform to Compute exponential(At) for a Given Matrix A
Exercise 6 Solve a Differential Equation with Non-Constant Coefficients
Exercise 7 Write a procedure for solving linear initial value problems P(D)u=Q(D)f with generalized inputs f
A Solution for Exercise 1
A Solution for Exercise 2
A Solution for Exercise 3
A Solution for Exercise 4
A Solution for Exercise 5
A Solution for Exercise 6
A Solution for Exercise 7
Exercises for Second and Third Semester Students in Engineering
The examples are covered in detail in the author's book
Fourier-Analysis und Distributionen,
Eine Einführung mit Anwendungen.
Fourier Series
Exercise 1 Trigonometric Approximation 1
Exercise 2 Trigonometric Approximation 2
Exercise 3 Gibbs phenomenon and smoothing
Exercise 4 Scaling and translation in time
Exercise 5 Modulation of the amplitude
Exercise 6 Differentiation and Integration of Fourier Series
Exercise 7 Smoothness properties versus decay of spectral values
Exercise 8 Periodic Convolution, Aspects of System Theory
Exercise 9 The Laplace equation on a disc with given boundary values
Exercise 10 Limits of special series
Exercise Solutions 1-10 and Further Examples for Fourier Series
Discrete Signals and Discrete Fourier Transform
Exercise 11 Trigonometric Interpolation
Exercise 12 DFT spectrum, associated frequencies and alias effects
Exercise 13 Make first experiences with sound signal processing.
For that exercise please download the sound data files described in the worksheet
Exercise 13 continued On Shannon's Sampling Theorem and Bandwidth Conditions
A Solution for Exercise 12
A Solution for Exercise 13
A Solution for Exercise 13 continued
Distributions, Fourier Transform, LTI-Systems and Realization by Circuits
Exercise 14 Generalized Fourier series, periodic Dirac-impulses and derivatives
Exercise 15 On Eigenfunctions, Causal Impulse Response and Realization of LTI-Systems
Exercise 16 Construct a Butterworth Lowpass Filter for a Given Specification
Exercise 17 Examples where the Convolution Theorem does not hold
Exercise 18 Write a procedure for solving linear initial value problems P(D)u=Q(D)f with generalized inputs f
A Solution for Exercise 14
A Solution for Exercise 15
A Solution for Exercise 16
A Solution for Exercise 18
More on Discrete Transforms and Related Material
Exercise 19 Design a discrete Butterworth Lowpass Filter using the Bilinear Transform
Exercise 20 Basic Properties of Chebyshev Polynomials;
Approximation and Interpolation with Discrete Cosine Transforms, Gibbs Phenomenon, Alias Effects
Exercise 21 Program the Clenshaw-Curtis Quadrature
Exercise 22 Design analog and discrete Chebyshev Lowpass Filters
Exercise 23 Approximate the Fourier Transform of a Triangle with a DFT, Without and With Zero Padding
Exercise 24 Compare Interpolation With Equidistant Nodes and with Chebyshev Nodes in the Famous Runge Example
Exercise 25 Construct a Causal FIR Filter From the Ideal Lowpass Using Blackman and Hamming Windows
Exercise 26 On time frequency analysis: Construct a series of DFT's for a frequency modulated signal (.mw Maple 14 Worksheet)
Exercise 27 Construct an Analog and a Discrete Notch Filter Using the Bilinear Transform
A Solution for Exercise 19
A Solution for Exercise 20
A Solution for Exercise 21
A Solution for Exercise 22
A Solution for Exercise 23
A Solution for Exercise 24
A Solution for Exercise 25
A Solution for Exercise 26 (this is a .mw Maple 14 Worksheet)
A Solution for Exercise 27
Distributions and First Partial Differential Equations
Exercise 1 The Coulomb Potential for a Given Charge Density in Space
Exercise 2 A First Contact with the Finite Element Method:
Dirichlet's Boundary Value Problem for a Rectangular Membrane
A Solution for Exercise 1
A Solution for Exercise 2