Modern Multidimensional Calculus

CONTENTS
Chapter 1. Function and Variables
1-1 Elementary functions
1-2 Coordinate variables
1-3 Derivative and integrals
1-4 Polynomial approximations
Chapter 2. Differentials
2-1 Differentials: geometric picture
2-2 Differentials: algebraic theory
2-3 Manifold
2-4 Nigher-order derivatives
2-5 Line integrals
2-6 Comments on basic concepts
Chapter 3. Vectors in the Plane
3-1 Vectors
3-2 Differential geometry of plane curves
3-3 Area
3-4 Plane motion
Chapter 4. Topics in Linear Algebra
4-1 Matrix algebra
4-2 Linear transformations
4-3 Determinants
4-4 Elementary operations
4-5 Vectors in three dimensions
4-6 Orthonormal bases
4-7 Rotations
4-8 Quadratic and bilinear forms
4-9 Analytic geometry: lines and planes
Chapter 5: Partial Derivatives
5-1 Functions on n-tuples
5-2 Partial derivatives of functions
5-3 Partial derivatives of variables
5-4 Differentials
5-5 Geometric representations
5-6 Foundations of the calculus of variables
Chapter 6. Applications
6-1 Geometry in polar coordinates
6-2 Implicit relations, one-dimensional loci
6-3 Constrained maxima and minima, one dimension
6-4 Constrained maximan and minima, several dimensions
6-5 Chain rules
6-6 Inversion
6-7 Implicit relation, multidimensional loci
Chapter 7. Vector Differential Calculus
7-1 Introduction
7-2 Gradients
7-3 Divergence and curl
7-4 Coordinate-free definitions
7-5 Curvilinear coordinates
Chapter 8. Iterated Integrals
8-1 Twofold iterated integrals
8-2 Applications
8-3 Quadric surfaces
8-4 Threefold iterated integrals
Chapter 9. Multiple Integrals
9-1 Oriented manifold
9-2 Exterior products
9-3 Multiple and iterated integrals
9-4 Change of variable
9-5 Mass
9-6 Probability
9-7 Moments
Chapter 10. Line and Surface Integrals
10-1 Line integrals, recapitulation
10-2 Surface integrals
10-3 Surface area
10-4 Stokes type theorems
10-5 Vector integral calculus
10-6 Integrals independent of the path
ANSWERS
INDEX