Multivariable and Vector Calculus (De Gruyter Textbook)

Foreword to the first edition
Foreword to the second edition
Notes to the reader
Contents
1 Introduction/background
	1.1 What is multivariable and vector calculus?
	1.2 Vectors, lines, and planes in R^3
		1.2.1 Vectors
		1.2.2 Planes in R^3
		1.2.3 Lines in R^3
		1.2.4 Projections
	1.3 Basic surfaces in R^3
		1.3.1 Quadratic surfaces
	1.4 Polar, cylindrical, and spherical coordinates
		1.4.1 Polar coordinates in R^2
		1.4.2 Cylindrical and spherical coordinates in R^3
2 Vector functions
	2.1 Limits, derivatives, and integrals for vector functions
	2.2 Parametric curves in R^2 and R^3
	2.3 Particle motion in R^2 and R^3
		2.3.1 Tangent vectors
		2.3.2 Normal vectors
		2.3.3 Acceleration
	2.4 Arc length
		2.4.1 Arc length between fixed points \\alpha and \\omega
		2.4.2 Arc length as a function of time: s(t)
		2.4.3 When it all goes wrong: a nonrectifiable curve
	2.5 Acceleration decomposition
	2.6 A twist: motion in R^3 and the binormal vector
3 Multivariable derivatives—differentiation in R^n
	3.1 Limits in R^n
		3.1.1 Definitions and the basics
		3.1.2 0/0 indeterminate form
		3.1.3 Something that does not work
	3.2 Continuity in R^n
		3.2.1 Definition and examples
		3.2.2 Types of discontinuities
		3.2.3 Piecewise continuity
	3.3 The derivative in R^n
		3.3.1 Partial derivatives
		3.3.2 Higher-order partial derivatives
		3.3.3 Tangent planes and unique tangent planes
		3.3.4 Existence of the tangent plane
		3.3.5 Multivariable derivatives
		3.3.6 Linear approximations
	3.4 The chain rule in R^n
		3.4.1 The basic chain rule
		3.4.2 Several interesting extensions
		3.4.3 Implicit partial differentiation
	3.5 Directional derivatives
4 Implications of multivariable derivatives
	4.1 Level curves, level surfaces
	4.2 The gradient \\nabla F for the surface F(x, y, z) = 0
	4.3 Maximums and minimums for continuous functions on closed and bounded domains
	4.4 Local extrema
	4.5 Lagrange multipliers
5 Multiple integrals-integration in R^n
	5.1 Riemann integration versus iterated integrals
		5.1.1 Single-variable Riemann integration
		5.1.2 Multivariable Riemann integration
		5.1.3 Iterated integrals
		5.1.4 The Fubini theorem and the relationship between Riemann and iterated integrals
		5.1.5 When it all goes wrong: functions that are not Riemann integrable
	5.2 Double integrals: integration over domains in R^2
		5.2.1 Integration using rectangular coordinates
		5.2.2 Polar integration
		5.2.3 \"What does dA or dx dy become?
		5.2.4 What does it all mean? What do double integrals represent?
	5.3 Triple integrals: integration over domains in R^3
		5.3.1 Integration using rectangular coordinates
		5.3.2 Integration using cylindrical and spherical coordinates
6 Vector fields and vector calculus
	6.1 Line integrals: integration along curves in R^2 or R^3
		6.1.1 Direct evaluation of line integrals
		6.1.2 Path dependence; path independence
		6.1.3 Flow crossing a curve
	6.2 Surface integrals: integration over surfaces in R^3
	6.3 Differential operators
		6.3.1 Definitions
		6.3.2 Why is div(u) actually divergence?
	6.4 The theorems of Gauss, Green, and Stokes
		6.4.1 The divergence theorem
		6.4.2 Green\'s identities
		6.4.3 Stokes\' theorem
	6.5 The power of the divergence theorem: the Laplace and Poisson equations, and the Neumann problem
	6.6 All together now: a unified theorem
Bibliography
Index