Classroom-tested in a Princeton University honors course, this text offers a unified introduction to advanced calculus. Starting with an abstract treatment of vector spaces and linear transforms, the authors present a corresponding theory of integration, concluding with a series of applications to analytic functions of complex variable. 1959 edition"-

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