The book is suitable for a one-year course at the advanced undergraduate level.by omitting certain chapters,a one semester course can be based on it.for instance,if the students already have a good knowledge of partial differentiation and the elementary topology of e',then substantial parts of chapters 4,5,7,and 8 can be covered in a semester.some knowledge of linear algebra is presumed.however,results from linear algebra are reviewed as needed (in some cases without proof.

Chapter 1 Euclidean spaces

 1.l The real number system

 1.2 Euclidean En

 1.3 Elementary geometry of En

 1.4 Basic topological notions in En

 *1.5 Convex sets

Chapter 2 Elementary topology of En

 2.1 Functions

 2.2 Limits and continuity of transformations

 2.3 Sequences in En

 2.4 Bo|zano-Weierstrass theorem

 2.5 Relative neighborhoods, continuous transformations

 2.6 Topological spaces

 2.7 Connectedness

 2.8 Compactness

 2.9 Metric spaces

 2.10 Spaces of continuous functions

 *2.11 Noneuclidean norms on En

Chapter 3 Differentiation of real-valued functions

 3.1 Directional and partial derivatives

 3.2 Linear functions

 3.3 Differentiable functions

 3.4 Functions of class C（q）

 3.5 Relative extrema

 *3.6 Convex and concave functions

Chapter 4 Vector-valued functions of several variables

 4.1 Linear transformations

 4.2 Affine transformations

 4.3 Differentiable transformations

 4.4 Composition

 4.5 The inverse function theorem

 4.6 The implicit function theorem

 4.7 Manifolds

 4.8 The multiplier rule

Chapter 5 Integration

 5.1 Intervals

 5.2 Measure

 5.3 Integrals over En

 5.4 Integrals over bounded sets

 5.5 Iterated integrals

 5.6 Integrals of continuous functions

 5.7 Change of measure under affine transformations

 5.8 Transformation of integrals

 5.9 Coordinate systems in En

 5.10 Measurable sets and functions; further properties

 5.11 Integrals: general definition, convergence theorems

 5.12 Differentiation under the integral sign

 5.13 LP-spaces

Chapter 6 Curves and line integrals

 6.1 Derivatives

 6.2 Curves in En

 6.3 Differential l-forms

 6.4 Line integrals

 *6.5 Gradient method

 *6.6 Integrating factors; thermal systems

Chapter 7 Exterior algebra and differential calculus

 7.1 Covectors and differential forms of degree 2

 7.2 Alternating multilinear functions

 7.3 Multicovectors

 7.4 Differential forms

 7.5 Multivectors

 7.6 Induced linear transformations

 7.7 Transformation law for differential forms

 7.8 The adjoint and codifferential

 *7.9 Special results for n=3

 *7.10 Integrating factors （continued）

Chapter 8 Integration on manifolds

 8.1 Regular transformations

 8.2 Coordinate systems on manifolds

 8.3 Measure and integration on manifolds

 8.4 The divergence theorem

 *8.5 Fluid flow

 8.6 Orientations

 8.7 Integrals oft-forms

 8.8 Stokes's formula

 8.9 Regular transformations on submanifolds

 8.10 Closed and exact differential forms

 8.11 Motion of a particle

 8.12 Motion of several particles

Appendix 1 Axioms for a vector space

Appendix 2 Mean value theorem; Taylor's theorem

Appendix 3 Review of Riemann integration

Appendix 4 Monotone functions

References

Answers to problems

Index