Derivatives and Integrals of Multivariable Functions

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Springer Science & Business Media, Aug 22, 2003 - Mathematics - 319 pages
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This text is appropriate for a one-semester course in what is usually called ad vanced calculus of several variables. The approach taken here extends elementary results about derivatives and integrals of single-variable functions to functions in several-variable Euclidean space. The elementary material in the single- and several-variable case leads naturally to significant advanced theorems about func tions of multiple variables. In the first three chapters, differentiability and derivatives are defined; prop erties of derivatives reducible to the scalar, real-valued case are discussed; and two results from the vector case, important to the theoretical development of curves and surfaces, are presented. The next three chapters proceed analogously through the development of integration theory. Integrals and integrability are de fined; properties of integrals of scalar functions are discussed; and results about scalar integrals of vector functions are presented. The development of these lat ter theorems, the vector-field theorems, brings together a number of results from other chapters and emphasizes the physical applications of the theory.
 

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Contents

Differentiability of Multivariable Functions
ix
12 Derivatives and Partial Derivatives
6
13 The Chain Rule
11
1 4 Higher Derivatives
19
Derivatives of Scalar Functions
29
22 The Mean Value Theorem
33
23 Extreme Values and the Derivative
38
24 Extreme Values and the Second Derivative
43
43 Domains of Integrability
112
44 Integrability and Sets of Zero Volume
119
Integrals of Scalar Functions
131
52 Properties of Integrals
140
53 Change of Variable
144
54 Generalized Integrals
157
55 Line Integrals
165
56 Surface Integrals
182

25 Implicit Scalar Functions
48
26 Curves Surfaces Tangents and Normals
56
Derivatives of Vector Functions
69
32 The Inverse Function Theorem
75
33 The Implicit Function Theorem
81
34 Lagranges Method
92
Integrability of Multivariable Functions
101
42 Integrability in a Box
105
Vector Integrals and the VectorField Theorems
197
62 PathIndependence
206
The Theorems of Green and Stokes
214
64 Gausss Theorem
231
Solutions to Exercises
243
References
309
Index
311
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