Applied Mathematics: Body and Soul: Volume 1: Derivatives and Geometry in IR3

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Springer Science & Business Media, Apr 17, 2013 - Mathematics - 428 pages

Applied Mathematics: Body & Soul is a mathematics education reform project developed at Chalmers University of Technology and includes a series of volumes and software. The program is motivated by the computer revolution opening new possibilities of computational mathematical modeling in mathematics, science and engineering. It consists of a synthesis of Mathematical Analysis (Soul), Numerical Computation (Body) and Application. Volumes I-III present a modern version of Calculus and Linear Algebra, including constructive/numerical techniques and applications intended for undergraduate programs in engineering and science. Further volumes present topics such as Dynamical Systems, Fluid Dynamics, Solid Mechanics and Electro-Magnetics on an advanced undergraduate/graduate level.

Volume I (Derivatives and Geometry in R3) presents basics of Calculus starting with the construction of the natural, rational, real and complex numbers, and proceeding to analytic geometry in two and three space dimensions, Lipschitz continuous functions and derivatives, together with a variety of applications.

Volume II (Integrals and Geomtery in Rn) develops the Riemann integral as the solution to the problem of determining a function given its derivative, and proceeds to generalizations in the form of initial value problems for general systems of ordinary differential equations, including a variety of applications. Linear algebra including numerics is also presented.

Volume III (Calculus in Several Dimensions) presents Calculus in several variables including partial derivatives, multi-dimensional integrals, partial differential equations and finite element methods, together with a variety of applications modeled as systems of partial differential equations.

The authors are leading researchers in Computational Mathematics who have written various successful books.

Further information on Applied Mathematics: Body and Soul can be found at http://www.phi.chalmers.se/bodysoul/.

 

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Contents

Derivatives and Geometry in R
2
Solving Linear Algebraic Systems 651
3
Linear Algebra Tool Bag 685
16
Optimization 871
18
Applications Tool Bag 1103
19
The Mathematics Laboratory
21
Introduction to Modeling
25
A Very Short Calculus Course
33
Sequences and limits
165
The Square Root of Two
185
Real numbers
195
The Bisection Algorithm for fa 0
215
Do Mathematicians Quarrel?
221
The Function y a
241
Analytic Geometry in R 265
264
Analytic Geometry in R
313

Calculus Tool Bag I 585
34
The Integral 429
36
Natural Numbers and Integers
47
Mathematical Induction 63
62
Rational Numbers
71
Piecewise Linear Polynomials in R and
76
Pythagoras and Euclid
87
What is a Function?
103
Polynomial functions
119
Fourier Analysis Tool Bag 1185
120
Combinations of functions 141
140
Lipschitz Continuity
149
FEM for Boundary Value Problems in R and R
339
Complex Numbers
345
Analytic Functions 1107
352
The Derivative 355
354
Differentiation Rules
377
Newtons Method
393
Galileo Newton Hooke Malthus and Fourier 403
402
Trigonometric Functions 505
409
References
417
Differential Equations Tool
419
Lagrange and the Principle of Least Action 697
423
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