Advanced Calculus (Dover Books on Mathematics)

✍ Scribed by H.K Nickerson, D.C. Spencer, N.E. Steenrod

• Publisher: Dover Publications
• Year: 2011
• Tongue: English
• Leaves: 449
• Edition: Reprint
• Category: Library

✦ Synopsis

"This book is a radical departure from all previous concepts of advanced calculus," declared the Bulletin of the American Mathematics Society, "and the nature of this departure merits serious study of the book by everyone interested in undergraduate education in mathematics."
Classroom-tested in a Princeton University honors course, it offers students a unified introduction to advanced calculus. Starting with an abstract treatment of vector spaces and linear transforms, the authors introduce a single basic derivative in an invariant form. All other derivatives — gradient, divergent, curl, and exterior — are obtained from it by specialization. The corresponding theory of integration is likewise unified, and the various multiple integral theorems of advanced calculus appear as special cases of a general Stokes formula. The text concludes by applying these concepts to analytic functions of complex variables.

✦ Table of Contents

I. The Algebra of Vector Spaces
1. Axioms
2. Redundancy
3. Cartesian spaces
4. Exercises
5. Associativity and commutativity
6. Notations
7. Linear subspaces
8. Exercises
9. Independent sets of vectors
10. Bases and dimension
11. Exercises
12. Parallels and affine subspaces
13. Exercises

II. Linear Transformations of Vector Spaces
1. Introduction
2. Properties of linear transformations
3. Exercises
4. Existence of linear transformations
5. Matrices
6. Exercises
7. Isomorphisms of vector spaces
8. The space of linear transformations
9. Endomorphisms
10. Exercises
11. Quotient; direct sum
12. Exact sequences

III. The Scalar Product
1. Introduction
2. Existence of scalar products
3. Length and angle
4. Exercises
5. Orthonormal bases
6. Isometries
7. Exercises

IV. Vector Products in R3
1. Introduction
2. The triple product
3. Existence of a vector product
4. Properties of the vector product
5. Analytic geometry
6. Exercises
V. Endomorphisms
1. Introduction
2. The determinant
3. Exercises
4. Proper vectors
5. The adjoint
6. Exercises
7. Symmetric endomorphisms
8. Skew-symmetric endomorphisms
9. Exercises

VI. Vector-Valued Functions of a Scalar
1. Limits and continuity
2. The derivative
3. Arclength
4. Acceleration
5. Steady flows
6. Linear differential equations
7. General differential equations
8. Planetary motion
9. Exercises

VII. Scalar-Valued Functions of a Vector
1. The derivative
2. Rate of change along a curve
3. Gradient; directional derivative
4. Level surfaces
5. Exercises
6. Reconstructing a function from its gradient
7. Line integrals
8. The fundamental theorem of calculus
9. Exercises

VIII. Vector-Valued Functions of a Vector
1. The derivative
2. Taylor’s expansion
3. Exercises
4. Divergence and curl
5. The divergence and curl of a flow
6. Harmonic fields
7. Exercises

IX. Tensor Products and the Standard Algebras
1. Introduction
2. The tensor product
3. Exercises
4. Graded vector spaces
5. Graded algebras
6. The graded tensor algebra
7. The commutative algebra
8. Exercises
9. The exterior algebra of a finite dimensional vector space
10. Exercises

X. Topology and Analysis
1. Topological spaces
2. Hausdorff spaces
3. Some theorems in analysis
4. The inverse and implicit function theorems
5. Exercises

XI. Differential Calculus of Forms
1. Differentiability classes
2. Associated structures
3. Maps; change of coordinates
4. The exterior derivative
5. Riemannian metric
6. Exercises

XII. Integral Calculus of Forms
1. Introduction
2. Standard simplexes
3. Singular differentiable chains; singular homology
4. Integrals of forms over chains
5. Exercises
6. Cohomology; de Rham theorem
7. Exercises
8. Green’s formulas
9. Potential theory on euclidean domains
10. Harmonic forms and cohomology
11. Exercises

XIII. Complex Structure
1. Introduction
2. Complex vector spaces
3. Relations between real and complex vector spaces
4. Exercises
5. Complex differential calculus of forms
6. Holomorphic maps and holomorphic functions
7. Poincaré Lemma
8. Exercises
9. Hermitian and Kähler metrics
10. Complex Green’s formulas
11. Exercises

✦ Subjects

topology,vector,analysis