Calculus Illustrated. Volume 4: Calculus in Higher Dimensions

Description

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