Teaching And Exercises

Working with MAPLE (for Analysis I/II and Linear Algebra I/II)

Lecture notes

• Part I: Introduction to MAPLE
• Principles of Working with MAPLE
• Part II: Analysis with MAPLE
1. Factorial, binomial coefficient, finite sums, finite products, absolute value, square root, solve equations and inequalities
2. Real Sequences, limit values-, convergence, divergence, graphical presentations, recursive sequences, sequences definition by cases and upper/lower bounds
3. Infinite sums (series) and infinite products, convergence, divergence and limit values
4. One-dimensional real-valued functions, derivatives, roots, extreme values, symmetries, monotonies, intersections, continuity, limit values, connections of two functions, graphical presentation and special functions
5. Complex numbers, neutral elements, inverse elements, imaginary unit, arithmetic operations, real part, imaginary part, complex conjugation, absolute value, argument,polar coordinates and graphical presentations
6. Differentiation of one-dimensional real-valued functions
7. Integration of one-dimensional real-valued functions, antiderivatives, improper integral, indefinite integrals, special antiderivatives and graphical presentations
8. Real-valued power series, radius of convergence, Cauchy-Hadamard formula, Taylor series and Lagrange remainder
9. Curves, differentiable curves, regular curves, singular points, velocity of a curve, tangent vectors, tangent unit vectors, angle of intersection, arc length, graphical presentations and projection of a curve
10. Multidimensional real-valued functions, derivatives, roots, extreme values, saddle points, gradient, hessian, curves of intersection, singularities, limit values, connections of two functions and graphical presentations
11. Differentiation of multidimensional real-valued functions, partial differentiation, vector field, gradient, gradient field, divergence, curl, laplacian, jacobian and hessian
12. Ordinary differential equations, linear differential equations, differential equations of higher order, fundamental systems, wronskian, systems of linear differential equations and graphical presentations
• Part III: Linear algebra with MAPLE
1. Vectors, arithmetic operations, length, standardization, orthogonality, angle, inner product, cross product, triple product, transposition, linear independency, basis and dimension
2. Points, lines, plains, plain with three points, plain with a point and two vectors, plain with a point and a normal vector, points of intersection, lines of intersection, distances, prependicular and angle of intersection
3. matrices, arithmetic operations, inverse of a matrix, transposition, power of a matrix, matrix exponential, rank, dimension, row and column operations, special representations of matrices, trace, condition, positive and negative (semi-) definite, orthogonal, unitary and similiar matrices
4. Linear system of equations, solutions of linear equations and linear equations with dependence of parameters
5. Determinant, computation of determinants and determinants with dependence of parameters
6. Theory of eigenvalues, eigenvalues, eigenvectors, eigenspaces, characteristic polynomials, minimal polynomials, diagonalizable matrices and triangular matrices
7. Nilpotent matrices, degree of a nilpotent matrix, Jordan normal form and Jordan-Chevalley decomposition
8. Polynomials, gcd, lcm, polynomial long division, factorization, numerically solving of polynomial equations, roots, norm, irreducibility, squarefree polynomials and special polynomials

Literature

• [1]: Maplesoft: Maple User Manual. A division of Waterloo Maple Inc. 1996-2008, Canada
• [2]: Thomas Westermann: Mathematische Probleme lösen mit Maple. Springer, Berlin, 3. Auflage, 2008, ISBN-13 978-3540777205
• [3]: Dorothea Bahns, Christoph Schweigert: Softwarepraktikum - Analysis und Lineare Algebra: Ein MAPLE-Arbeitsbuch mit vielen Beispielen und Lösungen. Vieweg und Teubner, 1. Auflage, 2007, ISBN-13 978-3834803702
• [4]: Andre Heck: Introduction to Maple. Springer, Berlin, 3. Auflage, 2003, ISBN-13 978-0387002309
• [5]: Wilhelm Forst, Dieter Hoffmann: Funktionentheorie erkunden mit Maple. Springer, Berlin, 1. Auflage, 2002, ISBN-13 978-3540425434
• [6]: Detlef Wille: Repetitorium der Linearen Algebra (Teil 1). Binomi Verlag, 4. Auflage, 2001, ISBN 3-923 923-40-6
• [7]: Michael Holz, Detlef Wille: Repetitorium der Linearen Algebra (Teil 2). Binomi Verlag, 1. Auflage, 2002, ISBN 3-923 923-42-2
• [8]: Hans-Jürgen Dobner, Bernd Engelmann: Analysis 1 Grundlagen und Differenzialrechnung. Fachbuchverlag Leipzig, 1. Auflage, 2002, ISBN 3-446-22120-4
• [9]: Hans-Jürgen Dobner, Bernd Engelmann: Analysis 2 Integralrechnung und mehrdimensionale Analysis. Fachbuchverlag Leipzig, 1. Auflage, 2003, ISBN 3-446-22240-5
• [10]: Otto Forster: Analysis 1 Differential- und Integralrechnung. Vieweg, 6. Auflage, 2001, ISBN 3-528-57224-8
• [11]: Otto Forster: Analysis 2 Differentialrechnung im IR^n, gewöhnliche Differentialgleichungen. Vieweg, 5. Auflage, 2002, ISBN 3-528-37231-1

Corrections and annotations

• 21.04.2009: I ask all participants, who have not attend on Monday 20.04.2009 in the lecture and want to attend next week, to activate your (by security reasons meanwhile closed) account in the computer lab U5-142. For the activation of your account the computer lab needs a so-called account application, which you can download under Application Form in german and in english.

I wish all participants of this lecture a lot of success!