Preface.

**PART I BACKGROUND MATERIAL.**

**1 Sets and Functions.**

1.1 Sets in General.

1.2 Sets of Numbers.

1.3 Functions.

**2 Real Numbers.**

2.1 Review of the Order Relations.

2.2 Completeness of Real Numbers.

2.3 Sequences of Real Numbers.

2.4 Subsequences.

2.5 Series of Real Numbers.

2.6 Intervals and Connected Sets.

**3 Vector Functions.**

3.1 Vector Spaces: The Basics.

3.2 Bilinear Functions.

3.3 Multilinear Functions.

3.4 Inner Products.

3.5 Orthogonal Projections.

3.6 Spectral Theorem.

**PART II DIFFERENTIATION.**

**4 Normed Vector Spaces.**

4.1 Preliminaries.

4.2 Convergence in Normed Spaces.

4.3 Norms of Linear and Multilinear Transformations.

4.4 Continuity in Normed Spaces.

4.5 Topology of Normed Spaces.

**5 Derivatives.**

5.1 Functions of a Real Variable.

5.2 Differentiable Functions.

5.3 Existence of Derivatives.

5.4 Partial Derivatives.

5.5 Rules of Differentiation.

5.6 Differentiation of Products.

**6 Diffeomorphisms and Manifolds.**

6.1 The Inverse Function Theorem.

6.2 Graphs.

6.3 Manifolds in Parametric Representations.

6.4 Manifolds in Implicit Representations.

6.5 Differentiation on Manifolds.

**7 Higher-Order Derivatives.**

7.1 Definitions.

7.2 Change of Order in Differentiation.

7.3 Sequences of Polynomials.

7.4 Local Extremal Values.

**PART III INTEGRATION.**

**8 Multiple Integrals.**

8.1 Jordan Sets and Volume.

8.2 Integrals.

8.3 Images of Jordan Sets.

8.4 Change of Variables.

**9 Integration on Manifolds.**

9.1 Euclidean Volumes.

9.2 Integration on Manifolds.

9.3 Oriented Manifolds.

9.4 Integrals of Vector Fields.

9.5 Integrals of Tensor Fields.

9.6 Integration on Graphs.

**10 Stokes’ Theorem.**

10.1 Basic Stokes’ Theorem.

10.2 Flows.

10.3 Flux and Change of Volume in a Flow.

10.4 Exterior Derivatives.

10.5 Regular and Almost Regular Sets.

10.6 Stokes’ Theorem on Manifolds.

**PART IV APPENDICES.**

**Appendix A: Construction of the Real Numbers.**

A.1 Field and Order Axioms in Q.

A.2 Equivalence Classes of Cauchy Sequences in Q.

A.3 Completeness of R.

**Appendix B: Dimension of a Vector Space.**

B.1 Bases and Linearly Independent Subsets.

**Appendix C: Determinants.**

C.1 Permutations.

C.2 Determinants of Square Matrices.

C.3 Determinant Functions.

C.4 Determinant of a Linear Transformation.

C.5 Determinants on Cartesian Products.

C.6 Determinants in Euclidean Spaces.

C.7 Trace of an Operator.

**Appendix D: Partitions of Unity.**

D.1 Partitions of Unity.

Index.