Vector Calc Open Math.pdf

(3307 KB) Pobierz
ÅÙÐØ Ú Ö Ð Ò Î ØÓÖ
David A. SANTOS
dsantos@ccp.edu
January 15, 2009 Version
Ð ÙÐÙ×
ree to photocopy and distribute
Mathesis iuvenes tentare rerum quaelibet ardua semitasque non usitatas pandere docet.
ii
Copyright c 2007 David Anthony SANTOS. Permission is granted to
copy, distribute and/or modify this document under the terms of the
GNU Free Documentation License, Version 1.2 or any later version
published by the Free Software Foundation; with no Invariant Sec-
tions, no Front-Cover Texts, and no Back-Cover Texts. A copy of the
license is included in the section entitled “GNU Free Documentation
License”.
Contents
Preface
1 Vectors and Parametric Curves
1.1 Points and Vectors on the Plane . . . . . . . .
1.2 Scalar Product on the Plane . . . . . . . . . .
1.3 Linear Independence . . . . . . . . . . . . . .
1.4 Geometric Transformations in two dimensions
1.5 Determinants in two dimensions . . . . . . .
1.6 Parametric Curves on the Plane . . . . . . . .
1.7 Vectors in Space . . . . . . . . . . . . . . . .
1.8 Cross Product . . . . . . . . . . . . . . . . . .
1.9 Matrices in three dimensions . . . . . . . . .
1.10 Determinants in three dimensions . . . . . . .
1.11 Some Solid Geometry . . . . . . . . . . . . . .
1.12 Cavalieri, and the Pappus-Guldin Rules . . . .
1.13 Dihedral Angles and Platonic Solids . . . . . .
1.14 Spherical Trigonometry . . . . . . . . . . . .
1.15 Canonical Surfaces . . . . . . . . . . . . . . .
1.16 Parametric Curves in Space . . . . . . . . . .
1.17 Multidimensional Vectors . . . . . . . . . . .
2 Differentiation
2.1 Some Topology . . . . . . . . . . . .
2.2 Multivariable Functions . . . . . . .
2.3 Limits . . . . . . . . . . . . . . . . .
2.4 Definition of the Derivative . . . . . .
2.5 The Jacobi Matrix . . . . . . . . . .
2.6 Gradients and Directional Derivatives
2.7 Levi-Civitta and Einstein . . . . . . .
2.8 Extrema . . . . . . . . . . . . . . . .
2.9 Lagrange Multipliers . . . . . . . . .
3 Integration
3.1 Differential Forms . . . . .
3.2 Zero-Manifolds . . . . . . .
3.3 One-Manifolds . . . . . . .
3.4 Closed and Exact Forms . .
3.5 Two-Manifolds . . . . . . .
3.6 Change of Variables . . . .
3.7 Change to Polar Coordinates
3.8 Three-Manifolds . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
v
1
1
11
15
17
25
31
41
50
58
62
64
68
70
74
79
87
90
97
97
99
100
104
107
116
122
124
129
133
. 133
. 136
. 137
. 141
. 146
. 155
. 162
. 168
iii
iv
3.9 Change of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 172
3.10 Surface Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
3.11 Green’s, Stokes’, and Gauss’ Theorems . . . . . . . . . . . . . . . . 179
A Answers and Hints
186
Answers and Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
GNU Free Documentation License
1. APPLICABILITY AND DEFINITIONS . . . . . .
2. VERBATIM COPYING . . . . . . . . . . . . . .
3. COPYING IN QUANTITY . . . . . . . . . . . .
4. MODIFICATIONS . . . . . . . . . . . . . . . .
5. COMBINING DOCUMENTS . . . . . . . . . . .
6. COLLECTIONS OF DOCUMENTS . . . . . . .
7. AGGREGATION WITH INDEPENDENT WORKS
8. TRANSLATION . . . . . . . . . . . . . . . . .
9. TERMINATION . . . . . . . . . . . . . . . . .
10. FUTURE REVISIONS OF THIS LICENSE . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
250
. 250
. 250
. 250
. 251
. 251
. 252
. 252
. 252
. 252
. 252
Preface
These notes started during the Spring of 2003. They are meant to be a gentle
introduction to multivariable and vector calculus.
Throughout these notes I use Maple™ version 10 commands in order to illus-
trate some points of the theory.
I would appreciate any comments, suggestions, corrections, etc., which can
be addressed to the email below.
David A. SANTOS
dsantos@ccp.edu
v

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika:

Zgłoś jeśli naruszono regulamin