Discovering Mathematics with Maple von R J/Kaashoek Stroeker

his book grew out of the wish to let students of econometrics get acquainted T with the powerful techniques of computer algebra at an early stage in their curriculum. As no textbook available at the time met our requirements as to content and presentation, we had no other choice than to write our own course material. The try-out on a group of 80 first year students was not without success, and after adding some necessary modifications, the same material was presented to a new group of students of similar size the year after. Some more adjustments were made, and the final result now lies before you. Working with computer algebra packages like Derive, Mathematica, and Maple over many years convinced us of the favourable prospects of computer algebra as a means of improving the student's understanding of the difficult concepts on which mathematical techniques are often based. Moreover, advanced mathematical ed­ ucation, be it for mathematics itself or for mathematical statistics, operations research and other branches of applied mathematics, can greatly profit from the large amount of non-trivial mathematical knowledge that is stored in a computer algebra system. Admittedly, the fact remains that many a tough mathematical problem, such as solving a complicated non-linear system or obtaining a finite ex­ pression for a multiple parameter integral, can not easily be handled by computer algebra either, if at all.

Inhaltsangabe1 A Tour of Maple V.- 1.1 Introduction.- 1.2 Maple's User Interface.- 1.3 Just a Maple Session.- 1.3.1 Numerical Calculations.- 1.3.2 Symbolic Calculations.- 1.4 Worksheets and Assignments.- 1.4.1 Worksheet la. Numbers.- 1.4.2 Assignments la. Numbers.- 1.4.3 Worksheet lb. Variables and Names.- 1.4.4 Assignments lb. Variables and Names.- 1.5 Exercises.- 1.6 Survey of Used Maple Expressions.- 2 Functions and Sequences.- 2.1 Introduction.- 2.2 Selected Concepts from Analysis.- 2.3 A Tricky Probability Function.- 2.3.1 Mathematical Functions.- 2.3.2 Investigating a Probability Function.- 2.3.3 Investigating a Strange Sequence.- 2.4 Worksheets and Assignments.- 2.4.1 Worksheet 2a. Functions.- 2.4.2 Assignments 2a. Functions.- 2.4.3 Worksheet 2b. Sequences.- 2.4.4 Assignments 2b. Sequences.- 2.5 Exercises.- 2.6 Survey of Used Maple Expressions.- 3 Matrices and Vectors.- 3.1 Introduction.- 3.2 Selected Concepts from Linear Algebra.- 3.3 A Least Squares Filter.- 3.3.1 Matrices and Vectors.- 3.3.2 Removing Noise from a Data Set.- 3.4 Worksheets and Assignments.- 3.4.1 Worksheet 3a. Matrices.- 3.4.2 Assignments 3a. Matrices.- 3.4.3 Worksheet 3b. Vectors.- 3.4.4 Assignments 3b. Vectors.- 3.5 Exercises.- 3.6 Survey of Used Maple Expressions.- 4 Counting and Summation.- 4.1 Introduction.- 4.2 Selected Concepts from Discrete Mathematics.- 4.3 Binomial Coefficients.- 4.3.1 Maple's binomial Function.- 4.3.2 Two Binomial Identities.- 4.3.3 The Binomial Series.- 4.3.4 The De Moivre-Laplace Limit Theorem.- 4.4 Worksheets and Assignments.- 4.4.1 Worksheet 4a. Symbolic Summation.- 4.4.2 Assignments 4a. Symbolic Summation.- 4.4.3 Worksheet 4b. Counting and Random Numbers.- 4.4.4 Assignments 4b. Counting and Random Numbers.- 4.5 Exercises.- 4.6 Survey of Used Maple Expressions.- 5 Derivative and Integral.- 5.1 Introduction.- 5.2 Further Concepts from Analysis.- 5.3 Riemann's Non-differentiable Function.- 5.3.1 Investigating the Riemann Function.- 5.3.2 Finding a Primitive Function.- 5.4 Worksheets and Assignments.- 5.4.1 Worksheet 5a. Differentiation.- 5.4.2 Assignment 5a. Differentiation.- 5.4.3 Worksheet 5b. Integration.- 5.4.4 Assignments 5b. Integration.- 5.5 Exercises.- 5.6 Survey of Used Maple Expressions.- 6 Vector Spaces and Linear Mappings.- 6.1 Introduction.- 6.2 Further Concepts from Linear Algebra.- 6.3 Diagonalization and the SVD.- 6.3.1 A Simple Orthogonal Projection.- 6.3.2 Singular Value Decomposition.- 6.4 Worksheets and Assignments.- 6.4.1 Worksheet 6a. Linear Mappings.- 6.4.2 Assignments 6a. Linear Mappings.- 6.4.3 Worksheet 6b. Eigenvectors and Eigenspaces.- 6.4.4 Assignments 6b. Eigenvectors and Eigenspaces.- 6.5 Exercises.- 6.6 Survey of Used Maple Expressions.- A Exercises in Experimentation.- A.1 Upper Bound for a Parameterized Function.- A.2 Matrix Differentiation.- A.3 Chaotic Sequences.- B Hints and Answers.- B.1 Exercises 1.5.- B.2 Exercises 2.5.- B.3 Exercises 3.5.- B.4 Exercises 4.5.- B.5 Exercises 5.5.- B.6 Exercises 6.5.- C Quick Guide to Selected Maple Commands.- C.1 Linear Algebra.- C.1.1 Elementary Matrix Algebra.- C.1.2 Elementary Vector Algebra.- C.1.4 Eigenvalues and Eigenvectors.- C.2 Mathematical Analysis.- C.2.1 Writing Your Own Procedures.- C.2.2 Plotting the Graph of a Function.- C.2.3 Computing Limits.- C.2.4 Differential Calculus.- C.2.5 Integration.- C.2.6 Solving Equations.- C.2.7 Solving Differential/Difference Equations.- C.3 Statistics.- C.3.1 Probabilities.- C.3.2 Generating Random Numbers.- C.3.3 Analysis of Statistical Data.- C.4 Linear Programming.- C.5 Mathematical Finance.