目錄
Preface and Acknowledgments
Notations
1 Basic Features of Euclidean Space, Rn
1.1 Real numbers
1.1.1 Convergence of sequences of real numbers
1.2 Rn as a vector space
1.3 Rn as an inner product space
1.3.1 The inner product and norm in Rn
1.3.2 Orthogonality
1.3.3 The cross product in R3
1.4 Rn as a metric space
1.5 Convergence of sequences in Rn
1.6 Compactness
1.7 Equivalent norms (*)
1.8 Solved problems for Chapter 1
2 Functions on Euclidean Spaces
2.1 Functions from Rn to Rm
2.2 Limits of functions
2.3 Continuous functions
2.4 Linear transformations
2.5 Continuous functions on compact sets
2.6 Connectedness and convexity
2.6.1 Connectedness
2.6.2 Path-connectedness
2.6.3 Convex sets
2.7 Solved problems for Chapter 2
3 Differential Calculus in Several Variables
3.1 Differentiable functions
3.2 Partial and directional derivatives, tangent space
3.3 Homogeneous functions and Euler's equation
3.4 The mean value theorem
3.5 Higher order derivatives
3.5.1 The second derivative
3.6 Taylor's theorem
3.6.1 Taylor's theorem in one variable
3.6.2 Taylor's theorem in several variables
3.7 Maxima and minima in several variables
3.7.1 Local extrema for functions in several variables
3.7.2 Degenerate critical points
3.8 The inverse and implicit function theorems
3.8.1 The Inverse Function theorem
3.8.2 The Implicit Function theorem
3.9 Constrained extrema, Lagrange multipliers
3.9.1 Applications to economics
3.10 Functional dependence
3.11 Morse's leInma (*)
3.12 Solved problems for Chapter 3
4 Integral Calculus in Several Variables
4.1 The integral in Rn
4.1.1 Darboux sums. Integrability condition
4.1.2 The integral over a bounded set
4.2 Properties of multiple integrals
4.3 Fubini's theorern
4.3.1 Center of mass, centroid, moment of inertia
4.4 Smooth Urysohn's lemma and partition of unity (*)
4.5 Sard's theorem (*)
4.6 Solved problems for Chapter 4
5 Change of Variables Formula, Improper Multiple Integrals
5.1 Change of variables formula
5.1.1 Change of variables; linear case
5.1.2 Change of variables; the general case
5.1.3 Applications, polar and spherical coordinates
5.2 Improper multiple integrals
5.3 Functions defined by integrals
5.3.1 Functions defined by improper integrals
5.3.2 Convolution of functions
5.4 The Weierstrass approximation theorem (*)
5.5 The Fourier transform (*)
5.5.1 The Schwartz space
5.5.2 The Fourier transform on Rn
5.6 Solved problems for Chapter 5
6 Line and Surface Integrals
6.1 Arc-length and Line integrals
6.1.1 Paths and curves
6.1.2 Line integrals
6.2 Conservative vector fields and Poincare's lemma
6.3 Surface area and surface integrals
6.3.1 Surface area
6.3.2 Surface integrals
6.4 Green's theorem and the divergence theorem in R2
6.4.1 The divergence theorem in R2
6.5 The divergence and curl
6.6 Stokes' theorem
6.7 The divergence theorem in R3
6.8 Differential forms (*)
6.9 Vector fields on spheres and Brouwer fixed point theorem (*)
6.9.1 Tangential vector fields on spheres
6.9.2 The Brouwer fixed point theorem
6.10 Solved problems for Chapter 6
7 Elements of Ordinary and Partial Differential Equations
7.1 Introduction
7.2 First order differential equations
7.2.1 Linear first order ODE
7.2.2 Equations with variables separated
7.2.3 Homogeneous equations
7.2.4 Exact equations
7.3 Picard's theorem (*)
7.4 Second order differential equations
7.4.1 Linear second order ODE with constant coefficients
7.4.2 Special types of second order ODE; reduction of order
7.5 Higher order ODE and systems of ODE
7.6 Some more advanced topics in ODE (*)
7.6.1 The method of Frobenius; second order equations with variable coefficients
7.6.2 The Hermite equation
7.7 Partial differential equations
7.8 Second order PDE in two variables
7.8.1 Classification and general solutions
7.8.2 Boundary value problems for the wave equation
7.8.3 Boundary value problems for Laplace's equation
7.8.4 Boundary value problems for the heat equation
7.8.5 A note on Fourier series
7.9 The Fourier transform method (*)
7.10 Solved problems for Chapter 7
8 An Introduction to the Calculus of Variations
8.1 Simple variational problems
8.1.1 Some classical problems
8.1.2 Sufficient conditions
8.2 Generalizations
8.2.1 Geodesics on a Riemannian surface
8.2.2 The principle of least action
8.3 Variational problems with constraints
8.4 Multiple integral variational problems
8.4.1 Variations of double integrals
8.4.2 The case of n variables
8.5 Solved problems for Chapter 8
Appendix A Countability and Decimal Expansions
Appendix B Calculus in One Variable
B.1 Differential calculus
B.2 Integral calculus
B.2.1 Complex-valued functions
B.3 Series
Appendix C Uniform Convergence
C.1 The Stone-Weierstrass theorem
Appendix D Linear Algebra
Bibliography
Index
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