Concepts in Calculus III UNIVERSITY PRESS OF FLORIDA Florida A&M University, Tallahassee Florida Atlantic University, Boca Raton Florida Gulf Coast University, Ft. Myers Florida International University, Miami Florida State University, Tallahassee New College of Florida, Sarasota University of Central Florida, Orlando University of Florida, Gainesville University of North Florida, Jacksonville University of South Florida, Tampa University of West Florida, Pensacola Orange Grove Texts Plus Concepts in Calculus III Multivariable Calculus Sergei Shabanov University of Florida Department of Mathematics University Press of Florida Gainesville • Tallahassee • Tampa • Boca Raton Pensacola • Orlando • Miami • Jacksonville • Ft. Myers • Sarasota Copyright 2012 by the University of Florida Board of Trustees on behalf of the University of Florida Department of Mathematics This work is licensed under a modified Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 Unported License. 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ISBN 978-1-61610-162-6 Orange Grove Texts Plus is an imprint of the University Press of Florida, which is the scholarly publishing agency for the State University System of Florida, comprising Florida A&M University, Florida Atlantic University, Florida Gulf Coast University, Florida International University, Florida State University, New College of Florida, University of Central Florida, University of Florida, University of North Florida, University of South Florida, and University of West Florida. University Press of Florida 15 Northwest 15th Street Gainesville, FL 32611-2079 http://www.upf.com Contents Chapter 11. Vectors and the Space Geometry 1 71 . Rectangular Coordinates in Space 1 72. Vectors in Space 12 73. The Dot Product 25 74. The Cross Product 38 75. The Triple Product 51 76. Planes in Space 65 77. Lines in Space 73 78. Quadric Surfaces 82 Chapter 12. Vector Functions 97 79 . Curves in Space and Vector Functions 97 80. Differentiation of Vector Functions 111 81. Integration of Vector Functions 120 82. Arc Length of a Curve 128 83. Curvature of a Space Curve 136 84. Applications to Mechanics and Geometry 147 Chapter 13. Differentiation of Multivariable Functions 163 85 . Functions of Several Variables 163 86. Limits and Continuity 173 87. A General Strategy for Studying Limits 183 88. Partial Derivatives 196 89. Higher-Order Partial Derivatives 202 90. Linearization of Multivariable Functions 211 91. Chain Rules and Implicit Differentiation 221 92. The Differential and Taylor Polynomials 231 93. Directional Derivative and the Gradient 245 94. Maximum and Minimum Values 257 95. Maximum and Minimum Values (Continued) 268 96. Lagrange Multipliers 278 Chapter and section numbering continues from the previous volume in the series, Concepts in Calculus II. vi CONTENTS Chapter 14. Multiple Integrals 293 97 . Double Integrals 293 98. Properties of the Double Integral 301 99. Iterated Integrals 310 100. Double Integrals Over General Regions 315 101. Double Integrals in Polar Coordinates 330 102. Change of Variables in Double Integrals 341 103. Triple Integrals 356 104. Triple Integrals in Cylindrical and Spherical Coordinates 369 105. Change of Variables in Triple Integrals 382 106. Improper Multiple Integrals 392 107. Line Integrals 403 108. Surface Integrals 408 109. Moments of Inertia and Center of Mass 423 Chapter 15. Vector Calculus 437 110 . Line Integrals of a Vector Field 437 111. Fundamental Theorem for Line Integrals 446 112. Green’s Theorem 458 113. Flux of a Vector Field 470 114. Stokes’ Theorem 481 115. Gauss-Ostrogradsky (Divergence) Theorem 490 Acknowledgments 501 CHAPTER 11 Vectors and the Space Geometry Our space may be viewed as a collection of points. Every geometri- cal figure, such as a sphere, plane, or line, is a special subset of points in space. The main purpose of an algebraic description of various objects in space is to develop a systematic representation of these objects by numbers. Interestingly enough, our experience shows that so far real numbers and basic rules of their algebra appear to be sufficient to de- scribe all fundamental laws of nature, model everyday phenomena, and even predict many of them. The evolution of the Universe, forces bind- ing particles in atomic nuclei, and atomic nuclei and electrons forming atoms and molecules, star and planet formation, chemistry, DNA struc- tures, and so on, all can be formulated as relations between quantities that are measured and expressed as real numbers. Perhaps, this is the most intriguing property of the Universe, which makes mathemat- ics the main tool of our understanding of the Universe. The deeper our understanding of nature becomes, the more sophisticated are the mathematical concepts required to formulate the laws of nature. But they remain based on real numbers. In this course, basic mathematical concepts needed to describe various phenomena in a three-dimensional Euclidean space are studied. The very fact that the space in which we live is a three-dimensional Euclidean space should not be viewed as an absolute truth. All one can say is that this mathematical model of the physical space is sufficient to describe a rather large set of physical phenomena in everyday life. As a matter of fact, this model fails to describe phenomena on a large scale (e.g., our galaxy). It might also fail at tiny scales, but this has yet to be verified by experiments. 71. Rectangular Coordinates in Space The elementary object in space is a point. So the discussion should begin with the question: How can one describe a point in space by real numbers? The following procedure can be adopted. Select a particular point in space called the origin and usually denoted O. Set up three mutually perpendicular lines through the origin. A real number is associated with every point on each line in the following way. The origin corresponds to 0. Distances to points on one side of the line 1 2 11. VECTORS AND THE SPACE GEOMETRY from the origin are denoted by positive real numbers, while distances to points on the other half of the line are denoted by negative numbers (the absolute value of a negative number is the distance). The half-lines with the grid of positive numbers will be indicated by arrows pointing from the origin to distinguish the half-lines with the grid of negative numbers. The described system of lines with the grid of real numbers on them is called a rectangular coordinate system at the origin O.The lines with the constructed grid of real numbers are called coordinate axes. 71.1. Points in Space as Ordered Triples of Real Numbers. The position of any point in space can be uniquely specified as an ordered triple of real numbers relative to a given rectangular coordinate system. Consider a rectangular box whose two opposite vertices (the endpoints of the largest diagonal) are the origin and a point P , while its sides that are adjacent at the origin lie on the axes of the coordinate system. For every point P , there is only one such rectangular box. It is uniquely determined by its three sides adjacent at the origin. Let the number x denote the position of one such side that lies on the first axis; the numbers y and z do so for the second and third sides, respectively. Note that, depending on the position of P ,thenumbersx, y,and z may be negative, positive, or even 0. In other words, any point in space is associated with a unique ordered triple of real numbers (x, y, z) determined relative to a rectangular coordinate system. This ordered triple of numbers is called rectangular coordinates of a point. To reflect the order in (x, y, z), the axes of the coordinate system will be denoted as x, y,andz axes. Thus, to find a point in space with rectangular coordinates (1, 2, −3), one has to construct a rectangular box with a vertex at the origin such that its sides adjacent at the origin occupy the intervals [0, 1], [0, 2], and [−3, 0] along the x, y,andz axes, respectively. The point in question is the vertex opposite to the origin. 71.2. A Point as an Intersection of Coordinate Planes. The plane con- taining the x and y axes is called the xy plane. For all points in this plane, the z coordinate is 0. The condition that a point lies in the xy plane can therefore be stated as z =0. Thexz and yz planes can be defined similarly. The condition that a point lies in the xz or yz plane reads y =0orx = 0, respectively. The origin (0, 0, 0) can be viewed as the intersection of three coordinate planes x =0,y =0,andz =0. Consider all points in space whose z coordinate is fixed to a particular value z = z 0 (e.g., z = 1). They form a plane parallel to the xy plane that lies |z 0 | units of length above it if z 0 > 0 or below it if z 0 < 0. 71. RECTANGULAR COORDINATES IN SPACE 3 Figure 11.1. Left:AnypointP in space can be viewed as the intersection of three coordinate planes x = x 0 , y = y 0 , and z = z 0 ; hence, P can be given an algebraic description as an ordered triple of numbers P =(x 0 ,y 0 ,z 0 ). Right: Translation of the coordinate system. The origin is moved toapoint(x 0 ,y 0 ,z 0 ) relative to the old coordinate system while the coordinate axes remain parallel to the axes of the old system. This is achieved by translating the origin first along the x axis by the distance x 0 (as shown in the figure), then along the y axis by the distance y 0 , and finally along the z axis by the distance z 0 . As a result, a point P that had coordinates (x, y, z) in the old system will have the coordi- nates x  = x − x 0 , y  = y − y 0 ,andz  = z − z 0 in the new coordinate system. ApointP with coordinates (x 0 ,y 0 ,z 0 ) can therefore be viewed as an intersection of three coordinate planes x = x 0 , y = y 0 ,andz = z 0 as shown in Figure 11.1. The faces of the rectangle introduced to specify the position of P relative to a rectangular coordinate system lie in the coordinate planes. The coordinate planes are perpendicular to the cor- responding coordinate axes: the plane x = x 0 is perpendicular to the x axis, and so on. 71.3. Changing the Coordinate System. Since the origin and directions of the axes of a coordinate system can be chosen arbitrarily, the co- ordinates of a point depend on this choice. Suppose a point P has coordinates (x, y, z). Consider a new coordinate system whose axes are 4 11. VECTORS AND THE SPACE GEOMETRY parallel to the corresponding axes of the old coordinate system, but whose origin is shifted to the point O  with coordinates (x 0 , 0, 0). It is straightforward to see that the point P would have the coordinates (x −x 0 ,y,z) relative to the new coordinate system (Figure 11.1, right panel). Similarly, if the origin is shifted to a point O  with coordinates (x 0 ,y 0 ,z 0 ), while the axes remain parallel to the corresponding axes of the old coordinate system, then the coordinates of P are transformed as (11.1) (x, y, z) −→ (x −x 0 ,y− y 0 ,z−z 0 ) . One can change the orientation of the coordinate axes by rotating them about the origin. The coordinates of the same point in space are different in the original and rotated rectangular coordinate systems. Algebraic relations between old and new coordinates can be established. A simple case, when a coordinate system is rotated about one of its axes, is discussed in Study Problem 11.2. It is important to realize that no physical or geometrical quantity should depend on the choice of a coordinate system. For example, the length of a straight line segment must be the same in any coordinate system, while the coordinates of its endpoints depend on the choice of the coordinate system. When studying a practical problem, a coordi- nate system can be chosen in any way convenient to describe objects in space. Algebraic rules for real numbers (coordinates) can then be used to compute physical and geometrical characteristics of the objects. The numerical values of these characteristics do not depend on the choice of the coordinate system. 71.4. Distance Between Two Points. Consider two points in space, P 1 and P 2 . Let their coordinates relative to some rectangular coordinate system be (x 1 ,y 1 ,z 1 )and(x 2 ,y 2 ,z 2 ), respectively. How can one calcu- late the distance between these points, or the length of a straight line segment with endpoints P 1 and P 2 ?ThepointP 1 is the intersection point of three coordinate planes x = x 1 , y = y 1 ,andz = z 1 .Thepoint P 2 is the intersection point of three coordinate planes x = x 2 , y = y 2 , and z = z 2 . These six planes contain faces of the rectangular box whose largest diagonal is the straight line segment between the points P 1 and P 2 . The question therefore is how to find the length of this diagonal. Consider three sides of this rectangular box that are adjacent, say, at the vertex P 1 . The side parallel to the x axis lies between the coordinate planes x = x 1 and x = x 2 and is perpendicular to them. So the length of this side is |x 2 − x 1 |. The absolute value is necessary as the difference x 2 − x 1 may be negative, depending on the values of x 1 and x 2 , whereas the distance must be nonnegative. Similar arguments [...]... added in any order Take the first vector, then move the second vector parallel to itself so that its initial point coincides with the terminal point of the first vector The third vector is moved parallel so that its initial point coincides with the terminal point of the second vector, and so on Finally, make a vector whose initial point coincides with the initial point of the first vector and whose terminal... coordinate system Let the origin be positioned at the initial point of the motion and let the coordinate axes be directed along the three mutually perpendicular lines parallel to which the point has moved In this coordinate system, the final point has the coordinates (3, 6, 6) The distance between this point and the origin (0, 0, 0) is √ D = 32 + 62 + 62 = 9(1 + 4 + 4) = 9 Rotations in Space The origin... parallelogram rule Given the first vector in the sum, all other vectors are transported parallel so that the initial point of the next vector in the sum coincides with the terminal point of the previous one The sum is the vector that originates from the initial point of the first vector and terminates at the terminal point of the last vector It does not depend on the order of vectors in the sum that the addition... it is obtained from ˆ a by dividing the latter by its length a , that is, a = sa, where s = 1/ a Right: A unit vector in a plane can always be viewed as an oriented segment whose initial point is at the origin of a coordinate system and whose terminal point lies on the circle of unit radius centered at the origin ˆ If θ is the polar angle in the plane, then a = cos θ, sin θ, 0 72 VECTORS IN SPACE... 2 − 4y = (y − 2)2 − 4 In the xy plane, the inequality describes a disk of radius 3 whose center 71 RECTANGULAR COORDINATES IN SPACE 11 is the point (1, 2, 0) As the algebraic condition imposes no restriction on the z coordinate of points in the set, in any plane z = z0 parallel to the xy plane, the x and y coordinates satisfy the same inequality, and hence the corresponding points also form a disk... 72 VECTORS IN SPACE 15 parallel transport such that their initial points coincide with the origin, their final points coincide too and hence have the same coordinates By virtue of the correspondence between vectors and points in space, this definition reflects the fact that two same points should have the same position vectors −→ − Example 11.3 Find the components of a vector P1 P2 if the coordinates of... cos θ cos φ + r sin θ sin φ = x cos φ + y sin φ , y = r sin θ = r sin(θ − φ) = r sin θ cos φ − r cos θ sin φ = y cos φ − x sin φ These equations define the transformation (x, y) → (x , y ) of the old coordinates to the new ones The inverse transformation (x , y ) → (x, y) can be found by solving the equations for (x, y) A simpler way is to note that if (x , y ) are viewed as “old” coordinates and (x,... terminal point coincides with the terminal point of the last vector in the sum To visualize this process, imagine a man walking along the first vector, then going parallel to the second vector, then parallel to the third vector, and so 20 11 VECTORS AND THE SPACE GEOMETRY on The endpoint of his walk is independent of the order in which he chooses the vectors Algebraic Addition of Vectors Definition 11.7... that has endpoints A(1, 2, 3) and B(−1, 5, 1) and is directed from A to B (ii) The vector that has endpoints A(1, 2, 3) and B(−1, 5, 1) and is directed from B to A (iii) The vector that has the initial point A(1, 2, 3) and the final point C that is the midpoint of the line segment AB, where B = (−1, 5, 1) (iv) The position vector of a point P obtained from the point A(−1, 2, −1) by transporting the latter... displacement vector BB coincides with the ship’s velocity u because B travels the distance u parallel to −→ − u This suggests a simple geometrical rule for finding AB as shown in − → Figure 11.6 Take the vector AB = v, place the vector u so that its initial point coincides with B, and make the oriented segment with the initial point of v and the final point of u in this diagram The resulting vector is the displacement . coordinate axes by rotating them about the origin. The coordinates of the same point in space are different in the original and rotated rectangular coordinate. the origin. 71.2. A Point as an Intersection of Coordinate Planes. The plane con- taining the x and y axes is called the xy plane. For all points in this plane,