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