stuvwxyz 1234567890 !@#$%^&*( Calculus III )-=_+,./; ’[]?:"{ }\| Tunc Geveci Copyright © 2011 by Tunc Geveci All rights reserved No part of this publication may be reprinted, reproduced, transmitted, or utilized in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information retrieval system without the written permission of University Readers, Inc First published in the United States of America in 2011 by Cognella, a division of University Readers, Inc Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe 15 14 13 12 11 12345 Printed in the United States of America ISBN: 978-1-935551-45-4 Contents 11 Vectors 11.1 Cartesian Coordinates in 3D and Surfaces 11.2 Vectors in Two and Three Dimensions 11.3 The Dot Product 11.4 The Cross Product 1 18 27 12 Functions of Several Variables 12.1 Tangent Vectors and Velocity 12.2 Acceleration and Curvature 12.3 Real-Valued Functions of Several Variables 12.4 Partial Derivatives 12.5 Linear Approximations and the Differential 12.6 The Chain Rule 12.7 Directional Derivatives and the Gradient 12.8 Local Maxima and Minima 12.9 Absolute Extrema and Lagrange Multipliers 35 35 46 61 65 74 85 94 107 121 13 Multiple Integrals 13.1 Double Integrals over Rectangles 13.2 Double Integrals over Non-Rectangular Regions 13.3 Double Integrals in Polar Coordinates 13.4 Applications of Double Integrals 13.5 Triple Integrals 13.6 Triple Integrals in Cylindrical and Spherical Coordinates 13.7 Change of Variables in Multiple Integrals 131 131 138 144 152 157 167 179 14 Vector Analysis 14.1 Vector Fields, Divergence and Curl 14.2 Line Integrals 14.3 Line Integrals of Conservative Vector Fields 14.4 Parametrized Surfaces and Tangent Planes 14.5 Surface Integrals 14.6 Green’s Theorem 14.7 Stokes’ Theorem 14.8 Gauss’ Theorem 187 187 194 210 220 239 259 274 278 K Answers to Some Problems 285 L Basic Differentiation and Integration formulas 309 iii Preface This is the third volume of my calculus series, Calculus I, Calculus II and Calculus III This series is designed for the usual three semester calculus sequence that the majority of science and engineering majors in the United States are required to take Some majors may be required to take only the first two parts of the sequence Calculus I covers the usual topics of the first semester: Limits, continuity, the derivative, the integral and special functions such exponential functions, logarithms, and inverse trigonometric functions Calculus II covers the material of the second semester: Further techniques and applications of the integral, improper integrals, linear and separable first-order differential equations, infinite series, parametrized curves and polar coordinates Calculus III covers topics in multivariable calculus: Vectors, vector-valued functions, directional derivatives, local linear approximations, multiple integrals, line integrals, surface integrals, and the theorems of Green, Gauss and Stokes An important feature of my book is its focus on the fundamental concepts, essential functions and formulas of calculus Students should not lose sight of the basic concepts and tools of calculus by being bombarded with functions and differentiation or antidifferentiation formulas that are not significant I have written the examples and designed the exercises accordingly I believe that "less is more" That approach enables one to demonstrate to the students the beauty and utility of calculus, without cluttering it with ugly expressions Another important feature of my book is the use of visualization as an integral part of the exposition I believe that the most significant contribution of technology to the teaching of a basic course such as calculus has been the effortless production of graphics of good quality Numerical experiments are also helpful in explaining the basic ideas of calculus, and I have included such data Remarks on some icons: I have indicated the end of a proof by ¥, the end of an example by Ô and the end of a remark by ♦ Supplements: An instructors’ solution manual that contains the solutions of all the problems is available as a PDF file that can be sent to an instructor who has adopted the book The student who purchases the book can access the students’ solutions manual that contains the solutions of odd numbered problems via www.cognella.com Acknowledgments: ScientificWorkPlace enabled me to type the text and the mathematical formulas easily in a seamless manner Adobe Acrobat Pro has enabled me to convert the LaTeX files to pdf files Mathematica has enabled me to import high quality graphics to my documents I am grateful to the producers and marketers of such software without which I would not have had the patience to write and rewrite the material in these volumes I would also like to acknowledge my gratitude to two wonderful mathematicians who have influenced me most by demonstrating the beauty of Mathematics and teaching me to write clearly and precisely: Errett Bishop and Stefan Warschawski v vi PREFACE Last, but not the least, I am grateful to Simla for her encouragement and patience while I spent hours in front a computer screen Tunc Geveci (tgeveci@math.sdsu.edu) San Diego, January 2011 Chapter 11 Vectors In this chapter we will introduce the concept of a vector and discuss the relevant algebraic operations We will also discuss the dot product that is related to angles between vectors and the cross product that produces a vector that is orthogonal to a pair of vectors These concepts and operations will be needed when we develop the calculus of functions of several variables 11.1 Cartesian Coordinates in 3D and Surfaces Cartesian Coordinates in Three Dimensions Our starting point is the familiar Cartesian coordinate plane Let’s designate the axes as the x and y axes Picture the xy-plane as a plane in the three-dimensional space The third axis is placed so that it is perpendicular to the xy-plane and its origin coincides with the origin of the xy-plane The positive direction is determined by the right-hand rule Let us label the third axis as the z-axis z 2 -4 y -2 -2 -4 -2 x -4 Figure We will associate an ordered triple (x, y, z) with each point P in space as follows: If P is the points at which the three axes intersect, we will call P the origin and denote it by O The ordered triple (0, 0, 0) is associated with O Let P be a point other than the origin Consider the line that passes through P and is perpendicular to the xy-plane Let Q be the intersection CHAPTER 11 VECTORS of that line with the xy-plane We will associate with Q the triple (x, y, 0), where x and y are determined as Cartesian coordinates in the xy-plane Consider the plane that passes through P and is parallel to the xy-plane If z is the point at which that plane intersects the third axis, we will associate the ordered triple (x, y, z) with the point P We are speaking of an “ordered triple”, since the order of the numbers x,y and z matters We will identify P with the triple (x, y, z), and refer to “the point (x, y, z)”, just as we identify a point in the Cartesian coordinate plane with the corresponding order pair of numbers Thus, we have described the Cartesian coordinate system in the three-dimensional space The system is also referred to as a rectangular coordinate system The set of all ordered triples (x, y, z) of real real numbers will be denoted by R3 Thus, R3 can be identified with the three-dimensional space that is equipped with the Cartesian coordinate system, just as the set of all ordered pairs (x, y) if real numbers can be denoted as R2 and identified with the set of points in the Cartesian coordinate plane The xy-plane consists points of the form (x, y, 0), the xz-plane consists of points of the form (x, 0, z), and the yz-plane consists of points of the form (0, y, z) We will refer to these planes as the coordinate planes The first octant consists of points (x, y, z) such that x ≥ 0, y ≥ and z ≥ Definition The (Euclidean) distance between P1 = (x1 , y1 , z1 ) and P2 = (x2 , y2 , z2 ) is dist (P1 , P2 ) = q (x2 − x1 )2 + (y2 − y1 )2 + (z2 − z1 )2 Surfaces Definition Let F (x, y, z) be an expression in the variables x, y, z and let C be a constant The set of points (x, y, z) ∈ R3 such that F (x, y, z) = C is a surface in R3 We say that the surface is the graph of the equation F (x, y, z) = C Example Let a, b, c, d be given constants The graph of the equation ax + by + cz = d is a plane For example, the equation x+y+z = describes a plane that intersects the coordinate axes at the points (1, 0, 0), (0, 1, 0) and (0, 0, 1) z x y Figure The equation z = describes a plane that is parallel to the the xy-plane 11.1 CARTESIAN COORDINATES IN 3D AND SURFACES Figure Definition Given a point P0 = (x0 , y0 , z0 ) and a positive number r, the set of points P = (x, y, z) such that (x − x0 )2 + (y − y0 )2 + (z − z0 )2 = r2 is the sphere of radius r centered at P0 Indeed, P is such a point if dist (P, P0 ) = q (x − x0 )2 + (y − y0 )2 + (z − z0 )2 = r For example, the sphere of radius centered at (3, 3, 1) is the graph of the equation 2 (x − 3) + (y − 3) + (z − 1) = Figure displays that sphere Figure Example Figure shows the surface that is the graph of the equation z = x2 + y This surface is referred to as a paraboloid CHAPTER 11 VECTORS z -2 0 x y -2 Figure Since z = x2 + y ≥ for each (x, y) ∈ R2 , the surface is above the xy-plane If (x, y) = (0, 0), then z = so that (0, 0, 0) is on the surface If c > 0, the intersection of the surface with the plane z = c is a circle whose projection onto the xy-plane is the circle x2 + y = c2 that is centered at (0, 0) and has radius c As c increases, these concentric circles expand The intersection of the surface with a plane of the form x = c is a parabola whose projection onto the yz-plane is the graph of the equation z = c2 + y Similarly, the intersection of the surface with a plane of the form y = c is a parabola whose projection onto the xz-plane is the graph of the equation z = x + c2 Ô Example Figure shows the surface that is described by the equation x2 − y + z = Such a surface is referred to as a hyperboloid of one sheet z 5 -5 0 x y -5 -5 Figure Since x2 + z = + y , the intersection of the surface with a plane of the form y = c is a circle that projects onto the xz-plane as the circle x2 + y = + c2 √ that is centered at (0, 0) and has radius + c2 296 APPENDIX K ANSWERS TO SOME PROBLEMS b) f (1.8, 0.8) ∼ = 0.420 009 = L (1.8, 0.8) ∼ c) According to a calculator f (1.8, 0.8) ∼ = 0.418 224 The absolute error is |f (1.8, 0.8) − L (1.8, 0.8)| ∼ = 1.8 × 10−3 a) L (x, y, z) = √ 14 + √ (x − 1) + √ (y − 2) + √ (z − 3) 14 14 14 b) f (0.9, 2.2, 2.9) ∼ = 741 66 = L (0.9, 2.2, 2.9) ∼ c) According to a calculator f (0.9, 2.2, 2.9) ∼ = f (0.9, 2.2, 2.9) The absolute error is |f (0.9, 2.2, 2.9) − L (0.9, 2.2, 2.9)| ∼ = × 10−3 a) xz xy yz dx + √ dy + √ dz df = √ xyz xyz xyz b) f (0.9, 2.9, 3.1) − f (1, 3, 3) ∼ = df (1, 3, 3, −0.1, −0.1, 0.1) = −0.15 Therefore, f (0.9, 2.9, 3.1) ∼ = f (1, 3, 3) − 0.15 = − 0.15 = 85 c) According to a calculator f (0.9, 2.9, 3.1) ∼ = 844 47 The absolute error is |f (0.9, 2.9, 3.1) − 85| ∼ = 5.5 × 10−3 Answers of Some Problems of Section 12.6 a) dz d f (x (t) , y (t)) = dt dt b) We have ∂z dx ∂z dy + ∂x dt ∂y dt à ! à ! x y p = cos (t) + p (−2 sin (t)) x2 + y x2 + y = sin (t) cos (t) = −q sin2 (t) + cos2 (t) f (x (t) , y (t)) = f (sin (t) , cos (t)) = Therefore, d d f (x (t) , y (t)) = dt dt q sin2 (t) + cos2 (t) q sin (t) cos (t) sin2 (t) + cos2 (t) = − q sin2 (t) + cos2 (t) 297 a) ∂z dz ∂x ∂ f (x (u, v)) = = = u u dx u ! ả u u =− − 3/2 2 x u + v2 (u + v ) ∂z dz ∂x ∂ f (x (u, v)) = = = ∂v ∂v dx v ! ả v v = − 3/2 2 x u + v2 (u + v ) and b) We have Therefore and a) µ ¶ ³p ´ f (x (u, v)) = f √ = ln u2 + v2 u2 + v2 ả u f (x (u, v)) = ln √ =− , ∂u ∂u u + v2 u2 + v ả v ∂ f (x (u, v)) = ln √ =− ∂v ∂v u + v2 u2 + v ∂z ∂ f (x (u, v) , y (u, v)) = ∂u ∂u ∂z ∂ f (x (u, v) , y (u, v)) = ∂v ∂v dz ∂x dz y + dx u dy u ả ả x y cos (v) + sin (v) = 0, = − x + y2 x2 + y = dz ∂x dz ∂y + dx ∂v dy ∂v ¶ µ ¶ µ x y (−u sin (v)) + (u cos (v)) = = − x + y2 x2 + y = b) We have f (x (u, v) , y (u, v)) = f (u cos (v) , u sin (v)) = arcsin (sin (v)) = v Therefore, ∂ ∂ f (x (u, v) , y (u, v)) = and f (x (u, v) , y (u, v)) = ∂u ∂v b) c) u (x, 0) = sin (x) , u (x, 2) = cos (x) , u (x, 4) = − cos (x) sin(x) y cos(x) y 1.0 0.5 -10 -5 -0.5 10 x -1.0 -10 -5 -1 -5 -1 -cos(x) y -10 10 x 10 x 298 APPENDIX K ANSWERS TO SOME PROBLEMS 11 a) x ∂z y ∂z =− , = ∂x z ∂y z b) ¯ ¯ ∂z ¯¯ ∂z ¯¯ = − , = − = = ∂x ¯x=3,y=6,z=6 ∂y ¯x=3,y=3,z=6 Therefore, the tangent plane is the graph of the equation z =6− (x − 3) + (y − 6) z y x Answers of Some Problems of Section 12.7 a) ∇f (x, y) = 8xi + 18yj b) √ 24 Du f (3, 4) = a) ∇f (x, y) = 2xex b) −y i − 2yex −y j √ Du f (2, 1) = − 10e3 a) ∇f (x, y, z) = 2xi − 2yj + 4zk b) Du f (1, −1, 2) = √ a) v = ∇f (2, 3) = − i − 3/2 j 133/2 13 299 The corresponding rate of increase of f is ||v|| = 13 b) w = −∇f (2, 3) = i + 3/2 j 3/2 13 13 The corresponding rate of decrease of f is 1/13 a) b) ¯ ³ ³ π ´´ dσ ³ π ´ ³ √ ´ ³ √ ´ √ ¯ d f (σ (t))¯¯ = ∇f σ · = 3e2 i − 2e2 j · −i + 3j = −4 3e dt dt t=π/6 √ Ddσ/dt f (σ (π/6)) = −2 3e2 11 a) Let f (x, y) = 2x2 + 3y ∇f (x, y) = 4xi + 6yj ∇f (2, 3) = 8i + 18j is orthogonal to the curve f (x, y) = 35 at (2, 3) b) The tangent line is the graph of the equation (x − 2) + 18 (y − 3) = 13 a) Let f (x, y) = e25−x −y2 ∇f (3, 4) = −6i − 8j is orthogonal to the curve f (x, y) = at (3, 4) b) The tangent line is the graph of the equation −6 (x − 3) − (y − 4) = 15 a) Let f (x, y, z) = x2 − y + z ∇f (2, 2, 1) = 4i − 4j + 2k is orthogonal to the surface at (2, 2, 1) b) The plane that is tangent to the surface at (2, 2, 1) is the graph of the equation (x − 2) − (y − 2) + (z − 1) = 17 Let f (x, y, z) = x − sin (y) cos (z) ∇f (1, π/2, 0) = i is orthogonal to the surface at (1, π/2, 0) b) The plane that is tangent to the surface at (1, π/2, 0) is the graph of the equation x − = ⇔ x = 300 APPENDIX K ANSWERS TO SOME PROBLEMS Answers of Some Problems of Section 12.8 1.(−1/5, −3/5) is the only critical point The function has a saddle point at (−1/5, −3/5) Any point on the line y = −x is a critical point The function attains its absolute maximum or minimum on the line y = −x (−18/5, −11/15).is the only critical point The function has a local (and absolute) minimum at (−18/5, −11/15) The critical points are (0, 0), (1, −1) and (−1, 1) The function has a saddle pioint at (0, 0), local maxima at (1, −1) and (−1, 1) Answers of Some Problems of Section 12.9 The maximum value of f on the circle x2 + y = is f √ ¢ √ ¡ √ value is f − 2, − = −2 √ ¡√ √ ¢ 2, = 2 and the minimum The minimum value of f (x, y) subject to 4x2 + y = is -−2, the maximum value is The minimum value of f in D is 0, and its maximum value in D is Answers of Some Problems of Section 13.1 81 ln (2) 1 12 e − e4 − e6 + e2 3 π 12 Answers of Some Problems of Section 13.2 The integral is 32 y z x -2 -1 2 y x The integral is 64/3 y y 1 -1 x The integral is 76 35 The integral is e−1 x 301 The integral is 11 a) z 0 x y y b) The integral is √ x Answers of Some Problems of Section 13.3 a) y x b) The integral is 32 a) y 3 -3 b) The integral is 9π a) x 302 APPENDIX K ANSWERS TO SOME PROBLEMS y 1 x b) The integral is 3π 64 a) y 1 x -1 b) The area of D is π/4 The volume of D is 4π 3/2 12 11 a) b) The integral is 2/3 Answers of Some Problems of Section 13.5 The volume of the region is 81π Z Z Z x2 dxdydz = D Z Z Z D Z Z Z D zex+y dxdydz = (e − 1)2 zdxdydz = Z Z dxdy = R π 303 Z Z Z xydxdxydz = D 28 Answers of Some Problems of Section 13.6 ³√ √ ´ 2, 2, ! √ 3 ,4 − , 2 à r= √ r= √ 2, θ = 3π , z=4 π 2, θ = − , z = Z Z Z p x2 + y dxdydz = 384π D 11 Z Z Z D 13 The volume of the region is ¡ ¢ ez dxdydz = π e6 − − e ´ 4π ³ − 33/2 ! Ã√ √ √ , , 2 15 17 ³ √ √ ´ − 6, 6, 19 ρ= 21 23 25 27 √ π 3π , θ=− 2, φ = √ 3π π ρ = 2, φ = , θ = Z Z Z 15 π zdxdydz = 16 D Z Z Z 1562 π x2 dxdydz = 15 D Z Z Z 29 The volume of the region D is dxdydz = D √ π ³√ ´ 3−1 π 304 APPENDIX K ANSWERS TO SOME PROBLEMS Answers of Some Problems of Section 14.1 a) ∇ · F (x, y, z) = yz b) ∇ × F (x, y, z) = −x2 i + 3xyi − xzk a) ∇ · F (x, y, z) = b) a) ∇ × F (x, y, z) = (−x cos (xy) + x sin (xz)) i + y cos (xy) j − z sin (xz) k ∇ · F (x, y, z) = 2y + 2z b) ∇ × F (x, y, z) = Answers of Some Problems of Section 14.2 Z y ds = C Z ´ ³ 3/2 145 − 54 xey ds = C Z (y−3)(z−4) C (x − 2) e Z C Z Z C 13 a) ¢ π x2 i + y j · dσ = − Z (−2xyi + (y + 1) j) · Tds = F·d σ = C b) 15 Z C −xydx + Z C Z C Z C 17 √ ¢ 14 ¡ e −1 ds = 12 (cos (y) i + sin (z) j + xk) · dσ = C 11 ¡ e e5 − 2 −xydx + dy = ln x2 + 28 dy x2 + µ 17 3x2 dx − 2y dy = − ¶ + 255 − sin (x) dx + cos (x) dy = −6 305 Answers of Some Problems of Section 14.3 b) f (x, y) = x2 − 3yx + g (y) = x2 − 3yx + 2y − 8y + K is a potential for F (K is an arbitrary constant) b) f (x, y) = ex sin (y) + K is a is a potential for F (K is an arbitrary constant) a) f (x, y) = is a potential for F b) Z a) C 2 x y F · dσ = f (x, y, z) = xyz + z is a potential for F b) Z C Z y2 dx + 2y arctan (x) dy = + x2 C where F · dσ = 77 Z df, C f (x, y) = y arctan (x) Z C y2 dx + 2y arctan (x) dy = π + x2 Answers of Some Problems of Section 14.4 a) b) a) b) ´ 3√3 ,1 = j+ k N 2 ³π √ à √ ! µ ¶ 3 3 3 z− =0 y− + 2 2 ³ √ ´ ³ π´ √ N 2, = 2i − + j + 12k ´ ³ √ ´ ³ √ √ ´ √ ³ x − − + (y − 4) + 12 z − 2 = 306 APPENDIX K ANSWERS TO SOME PROBLEMS a) b) √ ³ π´ 3 N 2, i− j + k =− 4 à √ µ √ ! ¶ √ ´ 3 1³ − x− − y− + z − = 4 2 a) b) √ ³ π´ k = −e2 i + ej + N 1, 2 ! ả 1 3 −e (x − 1) + e y − e + e = z− 2 2 Answers of Some Problems of Section 14.5 The area of the surface is 48π The area of the surface is 8π The area of the surface is ả Ă Â 1 2p 1 e (e + 1) + arcsinh e2 − 2π − arcsinh (1) 2 2 Z Z y dS = 8π Z Z xdS = −4π S S 13 Z 15 S xi + yj + zk · dS = 12π (x2 + y + z ) Z Z (xi+yj + zk) · dS =16π S Answers of Some Problems of Section 14.6 a) Z C y dx − x dy = −3 b) −3 3 Z Z D Z Z D ¡ ¢ x + y dxdy ¡ ¢ x + y dxdy = −24π 307 a) Let D be the rectangular region with vertices (0, 0) , (5, 0) , (5, π) and (0, π) Then Z Z Z cos (y) dx + x2 sin (y) dy = (2x + 1) sin (y) dxdy C b) D Z Z (2x + 1) sin (y) dxdy = 60 D a) Let D be the triangular region with vertices (0, 0) , (2, 6) and (2, 0) Z Z Z F · dσ = − 2xdxdy C D b) − Z Z D Z C1 Z C 2xdxdy = −16 F · dσ = F · nds = 24π Answers of Some Problems of Section 14.7 16π 243 π Answers of Some Problems of Section 14.8 3 192 Appendix L Basic Differentiation and Integration formulas Basic Differentiation Formulas d r x = rxr−1 dx d sin (x) = cos (x) dx d cos (x) = − sin(x) dx d sinh(x) = cosh(x) dx d cosh (x) = sinh(x) dx d tan(x) = dx cos2 (x) d x a = ln (a) ax dx d loga (x) = dx x ln (a) d arcsin (x) = √ dx − x2 10 d arccos (x) = − √ dx − x2 11 d arctan(x) = dx + x2 Basic Antidifferentiation Formulas C denotes an arbitrary constant Z Z Z Z R xr dx = xr+1 + C (r 6= −1) r+1 dx = ln (|x|) + C x sin (x) dx = − cos (x) + C cos (x) dx = sin (x) + C 10 sinh(x)dx = cosh(x) + C 309 R Z Z Z R cosh(x)dx = sinh(x) + C ex dx = ex + C ax dx = ax + C (a > 0) ln (a) dx = arctan (x) + C + x2 √ dx = arcsin (x) + C − x2 Index Absolute extrema, 122 Acceleration, 46 Arc length, 52 Level curves, 62 Level surface, 64 Limit, 66 Line integral of a vector field, 200 Line integrals, 196 differential form notation, 204 fundamental theorem of line integrals, 212 independence of path, 213 line integrals of conservative fields, 212 Line integrals of scalar functions, 196 Linear approximations, 75 Binormal, 49 Cartesian Coordinates, Chain rule, 85 Circulation of a vector field, 268 Conservative vector fields, 215 Continuity, 66 Continuity equation, 283 Curvature, 53 radius of curvature, 54 Cylindrical coordinates, 169 Differential, 80 Directional derivatives, 94 Distance traveled, 52 Divergence Theorem, 281 Double integrals, 133, 140 double integrals in polar coordinates, 146 Fubini’s Theorem, 136 Riemann sums, 134 Doubly connected region, 265 Flux across a curve, 270 Flux integrals, 249 differential form notation, 255 Gauss’ Theorem, 280 Gradient, 97 chain rule and the gradient, 101 gradient and level curves, 101 gradient and level surfaces, 104 Green’s Theorem, 261 Implicit differentiation, 90 Incompressible flow, 271 Irrotational flow, 271 Jacobian, 183 Lagrange multipliers, 124 Mass, 154 mass density, 154 Maxima and minima, 107 discriminant, 112 Second derivative test, 112 Mobius band, 230 Moments, 155 center of mass, 155 Moving frame, 48 Normal distribution, 158 Orientation of a curve, 197 Parametrized curves, 35 derivative of a vector-valued function, 40 tangent line, 42 tangent vectors, 40 unit tangent, 42 Parametrized surfaces, 222 normal vectors, 227 orientation, 230 tangent planes, 227 Partial derivatives, 65 higher-order partial derivaives, 71 Planes normal vactor, 31 Potential, 215 Potential function existence of a potential, 271 Principal normal, 49 Probability, 157 310 INDEX joint density function, 157 Projection, 23 Random variables, 157 Real-valued functions graphs, 61 Real-valued functions of several variables, 61 Simple closed curve, 261 Simply connected region, 261 Spherical coordinates, 172 Stokes’ Theorem, 276 Surface area, 241 Surface integrals of scalar functions, 246 Surfaces of revolution, 226 Tangent plane tangent plane to a graph, 75 Transformations, 182 Triangle Inequality, 19 Triple Integrals change of variables, 181 triple integrals in Cartesian coordinates, 159 Triple integrals triple integrals in cylindrical coordinates, 170 triple integrals in spherical coordinates, 174 Vectors, addition, angle between vectors, 20 component, 24 cross product, 27 direction cosines, 23 dot product, 18 length, linear combination, 12 normalization, 12 orthogonal vectors, 22 orthogonality, 20 scalar multiplication, scalar triple product, 30 standard basis vectors, 13 subtraction, 11 unit vector, 12 Velocity, 45, 46 Work, 200 311 ... Integration formulas 309 iii Preface This is the third volume of my calculus series, Calculus I, Calculus II and Calculus III This series is designed for the usual three semester calculus sequence that...stuvwxyz 1234567890 !@#$%^&*( Calculus III )-=_+,./; ’[]?:"{ }| Tunc Geveci Copyright © 2011 by Tunc Geveci All rights reserved No part of this publication may... differential equations, infinite series, parametrized curves and polar coordinates Calculus III covers topics in multivariable calculus: Vectors, vector-valued functions, directional derivatives, local