Calculus Illustrated. Volume 4: Calculus in Higher Dimensions

Preface
Functions in multidimensional spaces
	Multiple variables, multiple dimensions
	Euclidean spaces and Cartesian systems of dimensions 1, 2, 3,...
	Geometry of distances
	Where vectors come from
	Vectors in Rn
	Algebra of vectors
	Convex, affine, and linear combinations of vectors
	The magnitude of a vector
	Parametric curves
	The angles between vectors; the dot product
	Projections and decompositions of vectors
	Sequences and topology in Rn
	The coordinatewise treatment of sequences
	Partitions of the Euclidean space
	Discrete forms
Parametric curves
	Parametric curves
	Limits
	Continuity
	Location - velocity - acceleration
	The change and the rate of change: the difference and the difference quotient
	The instantaneous rate of change: derivative
	Computing derivatives
	Properties of difference quotients and derivatives
	Compositions and the Chain Rule
	What the derivative says about the difference quotient: the Mean Value Theorem
	Sums and integrals
	The Fundamental Theorem of Calculus
	Algebraic properties of sums and integrals
	The rate of change of the rate of change
	Reversing differentiation: antiderivatives
	The speed
	The curvature
	The arc-length parametrization
	Re-parametrization
	Lengths of curves
	Arc-length integrals: weight
	The helix
Functions of several variables
	Overview of functions
	Linear functions: lines in R2 and planes in R3
	An example of a non-linear function
	Graphs
	Limits
	Continuity
	The difference and the partial difference quotients
	The average and the instantaneous rates of change
	Linear approximations and differentiability
	Partial differentiation and optimization
	The second difference quotient with respect to a repeated variable
	The second difference and the difference quotient with respect to mixed variables
	The second partial derivatives
The gradient
	Overview of differentiation
	Gradients vs. vector fields
	The change and the rate of change of a function of several variables
	The gradient
	Algebraic properties of the difference quotients and the gradients
	Compositions and the Chain Rule
	Differentiation under multiplication and division
	The gradient is perpendicular to the level curves
	Monotonicity of functions of several variables
	Differentiation and anti-differentiation
	When is anti-differentiation possible?
	When is a vector field a gradient?
The integral
	Volumes and the Riemann sums
	Properties of the Riemann sums
	The Riemann integral over rectangles
	The weight as the 3d Riemann sum
	The weight as the 3d Riemann integral
	Lengths, areas, volumes, and beyond
	Outside the sandbox
	Triple integrals
	The n-dimensional case
	The center of mass
Vector fields
	What are vector fields?
	Motion under forces: a discrete model
	The algebra and geometry of vector fields
	Summation along a curve: flow and work
	Line integrals: work
	Sums along closed curves reveal exactness
	Path-independence of integrals
	How a ball is spun by the stream
	The Fundamental Theorem of Discrete Calculus of degree 2
	Green\'s Theorem: the Fundamental Theorem of Calculus for vector fields in dimension 2
Exercises
	Exercises: Basic calculus
	Exercises: Algebra and geometry
	Exercises: Parametric curves
	Exercises: Functions of several variables
	Exercises: Integrals
	Exercises: Vector fields
	Examples
Index