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