Multivariable Calculus

Description

Preface I Preliminaries Rn Vector arithmetic Linear transformations The matrix of a linear transformation Matrix multiplication The geometry of the dot product Determinants Exercises for Chapter 1 II Vector-valued functions of one variable Paths and curves Parametrizations Velocity, acceleration, speed, arclength Integrals with respect to arclength The geometry of curves: tangent and normal vectors The cross product The geometry of space curves: Frenet vectors Curvature and torsion The Frenet-Serret formulas The classification of space curves Exercises for Chapter 2 III Real-valued functions Real-valued functions: preliminaries Graphs and level sets More surfaces in R3 The equation of a plane in R3 Open sets Continuity Some properties of continuous functions The Cauchy-Schwarz and triangle inequalities Limits Exercises for Chapter 3 Real-valued functions: differentiation The first-order approximation Conditions for differentiability The mean value theorem The C1 test The Little Chain Rule Directional derivatives f as normal vector Higher-order partial derivatives Smooth functions Max/min: critical points Classifying nondegenerate critical points The second-order approximation Sums and differences of squares Max/min: Lagrange multipliers Exercises for Chapter 4 Real-valued functions: integration Volume and iterated integrals The double integral Interpretations of the double integral Parametrizations of surfaces Polar coordinates (r, ) in R2 Cylindrical coordinates (r, , z) in R3 Spherical coordinates (, , ) in R3 Integrals with respect to surface area Triple integrals and beyond Exercises for Chapter 5 IV Vector-valued functions Differentiability and the chain rule Continuity revisited Differentiability revisited The chain rule: a conceptual approach The chain rule: a computational approach Exercises for Chapter 6 Change of variables Change of variables for double integrals A word about substitution Examples: linear changes of variables, symmetry Change of variables for n-fold integrals Exercises for Chapter 7 V Integrals of vector fields Vector fields Examples of vector fields Exercises for Chapter 8 Line integrals Definitions and examples Another word about substitution Conservative fields Green's theorem The vector field W The converse of the mixed partials theorem Exercises for Chapter 9 Surface integrals What the surface integral measures The definition of the surface integral Stokes's theorem Curl fields Gauss's theorem The inverse square field A more substitution-friendly notation for surface integrals Independence of parametrization Exercises for Chapter 10 Working with differential forms Integrals of differential forms Derivatives of differential forms A look back at the theorems of multivariable calculus Exercises for Chapter 11 Answers to selected exercises Index