Basic Multivariable Calculus 1993

Basic Multivariable Calculus fills the need for a student-oriented text devoted exclusively to the third-semester course in multivariable calculus. In this text, the basic algebraic, analytic, and geometric concepts of multivariable and vector calculus are carefully explained, with an emphasis on developing the student's intuitive understanding and computational technique. A wealth of figures supports geometrical interpretation, while exercise sets, review sections, practice exams, and historical notes keep the students active in, and involved with, the mathematical ideas. All necessary linear algebra is developed within the text, and the material can be readily coordinated with computer laboratories. Basic Multivariable Calculus is the product of an extensive writing, revising, and class-testing collaboration by the authors of Calculus III (Springer-Verlag) and Vector Calculus (W.H. Freeman & Co.). Incorporating many features from these highly respected texts, it is both a synthesis of the authors' previous work and a new and original textbook. MULTIVARIABLE Preface Contents Algebra and Geometry of Euclidean Space Vectors The Standard Basis Vectors The Vector Joining Two Points Parametric Equation of a Line: Point-Direction Form Parametric Equation of a Line: Point-Point Form Inner Product, Length, and Distance ||j - i|| = 7(o -1)2+ (i-o)2 +(o-o)2 = 72. ♦ Angles and the Inner Product Perpendicular Vectors Cauchy-Schwarz Inequality P l|a|Pa- Orthogonal Projection Triangle Inequality Displacement and Velocity Real-World Problems vs. Made-Up Problems Exercises for Definition of the Cross Product The Cross Product Geometry of 2 x 2 Determinants Geometry of 3 x 3 Determinants Equation of a Plane in Space Distance from a Point to a Plane Cauchy-Schwarz Inequality H"iiH " WM Triangle Inequality Paths and Curves Velocity Vector Tangent Vector Tangent Line to a Path Exercises for Differentiation The Graph of a Function Level Curves Level Surfaces Plotting Surfaces Partial Differentiation Limits—Intuitive Approach Double and Single Limits in Two Variables The e,6 Definition of Limit Definition of Continuity Composition Continuity and Composition Tangent Plane to a Graph Differentiability /(xo.yo) + [ar, > The Derivative X^XO llx-Xoll »/(*.)= Differentiability and Continuity Condition for Differentiability The Chain Rule Curves and Tangents Curves on Graphs Tangents to Curves on Surfaces c'(t) = i/COi + A'Wj + kr(t)k. The Chain Rule for Two Intermediate and Two Independent Variables The Chain Rule for Three Intermediate and Two Independent Variables The Chain Rule—General Case Additional Derivative Rules The Chain Rule—General Case % Additional Derivative Rules The Gradient The Chain Rule and Gradients Gradients and Directional Derivatives Tangent Plane to a Surface Gradients and Tangent Planes .•*>x’« *.*3(V. c’A y J J ft « Wm A l) V Higher Derivatives and Extrema Equality of Mixed Partial Derivatives Equality of Mixed Partial Derivatives 0, A Second Order Taylor Formula z(y -yo) s ? + &2 g(s) = f(sx + (1 - s)x0, sy 4- (1 - s)yo) /(0,0) = l, ^f(0,0) = l, ^(0,0) = 0, ^(0.0) = -l, ^(0,0)= cos(0 + 2-0) = 1, §Z(0,0) =2cos(0 + 2-0) =2, &0,0) = 0, &o,o) = 0, Definition of Maxima and Minima Critical Points First Derivative Test 2y2 + y2 = 3y2 = 0, -2y2 + y2 = -y2 = 0, Absolute Maxima and Minima on Closed Intervals f'(x) = 2x - 4 = 0. Taylor's Formula Near a Critical Point ^(0,0) = ^(0,0) = 0. The Shape of Graphs of Quadratic Functions The Shape of Graphs of General Quadratic Functions Second Derivative Test 9/ ^=xcosy- x [2 - (x2 - y2)] = 0, y [-2 - (x2 - y2)] - 0. Exercises for Critical Pointiest for Constrained Extrema Method of Lagrange Multipliers 7(o,i) = /(o,-i) = -i, /(i,o) = 7(-i,0) = i, Finding the Absolute Maximum and Minimum of f(x,y') on a Region D Lagrange Multiplier Method in Space 4 Vector-Valued Functions Differentiation Rules 0 = ~ [c(t). c(t)] = c'(t). C(i) + C(t). C'(i) = 2c(t) • C'(i); Acceleration and Newton's Second Law Kepler's Law 5- 4[ci(t) + c2(t)] 4[ci(f) xc2(t)] Arc Length Arc Length Differential Arc Length in Rn Vector Fields F(x, y, z) = i — -g, r3 y, r3~z)- ♦ Flow Lines §4.3 Divergence ^-(s2y) + ci A x 4-—(-2/) = 1 4-(-1) = 0. ♦ dx dy Curl of a Vector Field Curl of a Gradient Divergence of a Curl The Laplace Operator Some Basic Identities of Vector Analysis div(/F) = A(/F1) + |;(/r2) + ^(/F3). 4- CW = j^72i+t’ + k^ = 2 c(f) = [cos^(t)]i + [sing-(t)]j Interlude: Where We Are Headed Multiple Integrals The Slice Method—Cava I ieri's Principle Double and Iterated Integrals b 1 Exercises for Definition of the Double Integral ILf' /fRf{x'y}dA’ Geometry of the Double Integral Existence of Integrals Properties of the Double Integral Reduction to Iterated Integrals IL The Double Integral over a Region d Continuity Implies Integrability Iterated Integrals for Elementary Regions If £> is a region of type 1? Mean Value Theorem for Single Integrals Mean Value Theorem for Double Integrals f fl (x + y2)dxdy L fi + y2}dydx l-^ILnx’v)dA-‘- The Triple Integral over a Box Reduction to Iterated Integrals Triple Integrals by Iterated Integration yyy # dxdydz y yyy exyydxdydz--B = [°’ x 1°’ x I0’ //LfiV - Double Integrals in Polar Coordinates J J M. The Gaussian Integral Cylindrical Coordinates Triple Integrals in Cylindrical Coordinates Spherical Coordinates Triple Integrals in Spherical Coordinates HL T The Jacobian Determinant Change of Variables Formula ffD^x,^dxdy = //D ^x^u,v^’y^u,v^ Jacobians Change of Variables in Triple Integrals W-= Average Value Coordinates of the Center of Mass Volume, Mass, Center of Mass, and Average Value for Regions in Space 17!,. = | Moments of Inertia /(x,y,z) = 1 = Gm//L Integrals Over Curves and Surfaces Line Integrals / F-ds= / tdt = 2tt2. ♦ . dz dy* dz* c =df1 + Si + 3k Line Integrals—Differential Form Notation Line Integrals of Gradient Fields Independence of Parametrization Line Integrals Along Geometric Curves T(0 = l|e'(0ll Work \r2 n/ Integrals of Scalar Functions Along Paths Parametrized Surface Tangent Plane n = Ou x - (-1,0,1) = -i + k. Area Element on a Graph a/ Surface Area = lfD l/WI \/l + LTM2 dudv = I" \f(u)171 + [/'(u)F dvdu = 2?r f |/(u)l71 + [/'(^)]2 du, J a Integral of a Scalar Function over a Surface ZZ'^gZZ,^ Surface Integral yy r • ds ~ yy * - x Orientation The Surface Integral for Graphs 1 1 _ 1 _ 1 _ 4 2 + 2 + 3 3 ”3' Independence of Parametrization Area of a Shadow Surface Integrals for Fluid Flow The Integral Theorems of Vector Analysis Orienting the Boundary Curve Green's Theorem Area of a Region Vector Form of Green's Theorem i k a d_ Q 0 _ 1 1 1 1 - 1 2 4 3 + 6 ~ 12' Gauss' Divergence Theorem in the Plane IL (& - dxdvIL(%+%)dxdy=IL di',tdxdy’ f[ Wl. . . / “x+ 0 in the first and third quadrants, Review Exercises (Chapter 4) 5.2 The Double Integral Over a Rectangle sin(l) 9 7- -e9)" 5(e4 ’ e} 16/9 76/3 25/6 2 e 5.3 The Double Integral Over Regions 5.5 Change of Variables 5.6 Applications of Multiple Integrals Review Exercises (Chapter 5) Chapter 6 Integrals Over Curves and Surfaces 6.1 Line Integrals 6.2 Parametrized Surfaces 6.3 Area of a Surface 6.4 Surface Integrals Review Exercises (Chapter 6) 7.3 Gauss' Theorem =yyy dxdydz 4- yyy 7.4 Path Independence and the Fundamental Theorems of Calculus Practice Exam 1 Practice Exam 2 Index