6 CHAPTER 6 VECTOR CALCULU We now easil compute this last integral using the parametriation given b rt a cos t i a sin t j, t. Thus C F dr C F dr Frt rt dt a sin ta sin t a cos ta cos t a cos t a sin t dt dt M We end this section b using Green s Theorem to discuss a result that was stated in the preceding section. KETCH OF PROOF OF THEOREM 6.3.6 We re assuming that F P i Q j is a vector field on an open simpl-connected region D, that P and Q have continuous first-order partial derivatives, and that P Q throughout D If C is an simple closed path in D and R is the region that C encloses, then Green s Theorem gives Q C F dr C P d Q d R P da da R A curve that is not simple crosses itself at one or more points and can be broken up into a number of simple curves. We have shown that the line integrals of F around these simple curves are all and, adding these integrals, we see that C F dr for an closed curve C. Therefore C F dr is independent of path in D b Theorem 6.3.3. It follows that F is a conservative vector field. M 6.4 EXERCIE 4 Evaluate the line integral b two methods: (a) directl and (b) using Green s Theorem.. C d d, C is the circle with center the origin and radius. C d d, C is the rectangle with vertices,, 3,, 3,, and, 3. C d 3 d, C is the triangle with vertices,, (, ), and (, ) 4. C d d, C consists of the line segments from, to, and from, to, and the parabola from, to, 5 Use Green s Theorem to evaluate the line integral along the given positivel oriented curve. 5. C d d, C is the triangle with vertices,,,, and, 4 6. C cos d sin d, C is the rectangle with vertices,, 5,, 5,, and, C ( e s ) d cos d 7., C is the boundar of the region enclosed b the parabolas and 8. C e d 4 d, C is the boundar of the region between the circles and 4 9., C is the circle C 3 d 3 d 4., C is the ellipse C sin d cos d 4 Use Green s Theorem to evaluate C F dr. (Check the orientation of the curve before appling the theorem.). F, s 3, s, C consists of the arc of the curve sin from, to, and the line segment from, to,
ECTION 6.5 CURL AND DIVERGENCE 6. F, cos, sin, C is the triangle from, to, 6 to, to, 3. F, e, e, C is the circle 5 oriented clockwise 4. F, ln, tan, C is the circle 3 oriented counterclockwise 5 6 Verif Green s Theorem b using a computer algebra sstem to evaluate both the line integral and the double integral. 5. P, e, Q, e, C consists of the line segment from, to, followed b the arc of the parabola from, to, 6. P, 3 5, Q, 3 8, C is the ellipse 4 4 7. Use Green s Theorem to find the work done b the force F, i j in moving a particle from the origin along the -ais to,, then along the line segment to,, and then back to the origin along the -ais. 8. A particle starts at the point,, moves along the -ais to,, and then along the semicircle s4 to the starting point. Use Green s Theorem to find the work done on this particle b the force field F,, 3 3. 9. Use one of the formulas in (5) to find the area under one arch of the ccloid t sin t, cos t. ;. If a circle C with radius rolls along the outside of the circle 6, a fied point P on C traces out a curve called an epiccloid, with parametric equations 5 cos t cos 5t, 5 sin t sin 5t. Graph the epiccloid and use (5) to find the area it encloses.. (a) If C is the line segment connecting the point, to the point,, show that (b) If the vertices of a polgon, in counterclockwise order, are,,,,..., n, n, show that the area of the polgon is A 3 3 A C d d n n n n n n (c) Find the area of the pentagon with vertices,,,,, 3,,, and,.. Let D be a region bounded b a simple closed path C in the -plane. Use Green s Theorem to prove that the coordinates of the centroid, of D are A C d where A is the area of D. 3. Use Eercise to find the centroid of a quarter-circular region of radius a. 4. Use Eercise to find the centroid of the triangle with vertices,, a,, and a, b, where a and b. 5. A plane lamina with constant densit, occupies a region in the -plane bounded b a simple closed path C. how that its moments of inertia about the aes are I 3 3 d C 6. Use Eercise 5 to find the moment of inertia of a circular disk of radius a with constant densit about a diameter. (Compare with Eample 4 in ection 5.5.) 7. If F is the vector field of Eample 5, show that C F dr for ever simple closed path that does not pass through or enclose the origin. 8. Complete the proof of the special case of Green s Theorem b proving Equation 3. 9. Use Green s Theorem to prove the change of variables formula for a double integral (Formula 5.9.9) for the case where f, : R d d A C d I 3 3 d C, u, du dv v Here R is the region in the -plane that corresponds to the region in the uv-plane under the transformation given b tu, v, hu, v. [Hint: Note that the left side is AR and appl the first part of Equation 5. Convert the line integral over R to a line integral over and appl Green s Theorem in the uv-plane.] 6.5 CURL AND DIVERGENCE In this section we define two operations that can be performed on vector fields and that pla a basic role in the applications of vector calculus to fluid flow and electricit and magnetism. Each operation resembles differentiation, but one produces a vector field whereas the other produces a scalar field.
78 CHAPTER 6 VECTOR CALCULU Converting to polar coordinates, we obtain A 3 s 4r r dr d ( 8 ) 3 4r 3 ] 3 d 3 6 (37s37 ) rs 4r dr M The question remains whether our definition of surface area (6) is consistent with the surface area formula from single-variable calculus (8..4). We consider the surface obtained b rotating the curve f, a b, about the -ais, where f and f is continuous. From Equations 3 we know that parametric equations of are f cos f sin a b To compute the surface area of we need the tangent vectors r i f cos j f sin k Thus r f sin j f cos k i j k r r f cos f sin f sin f cos f f i f cos j f sin k and so r r s f f f cos f sin because f. Therefore the area of is A D s f f f s f r r da b f s f d a b a f s f d d This is precisel the formula that was used to define the area of a surface of revolution in single-variable calculus (8..4). 6.6 EXERCIE Determine whether the points P and Q lie on the given surface.. ru, v u 3v, 5u v, u v P7,, 4, Q5,, 5 5. 6. rs, t s, t, t s rs, t s sin t, s, s cos t. 3 6 Identif the surface with the given vector equation. 3. ru, v u v, u v, u v P3,, 5, Q, 3, 4 ru, v u v i 3 v j 4u 5v k 4. ru, v sin u i 3 cos u j v k, v ; 7 Use a computer to graph the parametric surface. Get a printout and indicate on it which grid curves have u constant and which have v constant. 7. ru, v u, v 3, u v, u, v 8. ru, v u v, u, v, u, v 9. ru, v u cos v, u sin v, u 5, u, v
ECTION 6.6 PARAMETRIC URFACE AND THEIR AREA 79. ru, v cos u sin v, sin u sin v, cos v ln tanv, u,. v 6.. sin v, cos u sin 4v, sin u sin 4v, u, v. u sin u cos v, u cos u cos v, 3 8 Match the equations with the graphs labeled I VI and give reasons for our answers. Determine which families of grid curves have u constant and which have v constant. 3. 4. ru, v u cos v i u sin v j sin u k, 5. ru, v sin v i cos u sin v j sin u sin v k 6. u3 cos v cos 4u, u3 cos v sin 4u, 3u u sin v 7. cos 3 u cos 3 v, sin 3 u cos 3 v, 8. u cos v, u sin v, I ru, v u cos v i u sin v j v k II u sin v u sin3 v u 9 6 Find a parametric representation for the surface. 9. The plane that passes through the point,, 3 and contains the vectors i j k and i j k. The lower half of the ellipsoid 4. The part of the hperboloid that lies to the right of the -plane. The part of the elliptic paraboloid 4 that lies in front of the plane 3. The part of the sphere 4 that lies above the cone s 4. The part of the sphere 6 that lies between the planes and 5. The part of the clinder 6 that lies between the planes and 5 6. The part of the plane 3 that lies inside the clinder 7 8 Use a computer algebra sstem to produce a graph that looks like the given one. 7. 8. 3 III IV _3 _3 5 _ ; 9. Find parametric equations for the surface obtained b rotating the curve e, 3, about the -ais and use them to graph the surface. V VI ; 3. Find parametric equations for the surface obtained b rotating the curve 4 4,, about the -ais and use them to graph the surface. ; 3. (a) What happens to the spiral tube in Eample (see Figure 5) if we replace cos u b sin u and sin u b cos u? (b) What happens if we replace cos u b cos u and sin u b sin u? ; 3. The surface with parametric equations cos r cos sin r cos r sin where r and, is called a Möbius strip. Graph this surface with several viewpoints. What is unusual about it?
8 CHAPTER 6 VECTOR CALCULU 33 36 Find an equation of the tangent plane to the given parametric surface at the specified point. If ou have software that graphs parametric surfaces, use a computer to graph the surface and the tangent plane. 33. u v, 3u, u v;, 3, 34. u, v, uv; u, v 35. ru, v u i u sin v j u cos v k; 36. ru, v uv i u sin v j v cos u k; u, v u, v 37 47 Find the area of the surface. 37. The part of the plane 3 6 that lies in the first octant 38. The part of the plane 5 that lies inside the clinder 9 39. The surface 3 3 3,, 4. The part of the plane with vector equation ru, v v, u v, 3 5u v that is given b u, v 4. The part of the surface that lies within the clinder 4. The part of the surface 3 that lies above the triangle with vertices,,,, and, 43. The part of the hperbolic paraboloid that lies between the clinders and 4 44. The part of the paraboloid that lies inside the clinder 9 45. The part of the surface 4 that lies between the planes,,, and 46. The helicoid (or spiral ramp) with vector equation ru, v u cos v i u sin v j v k, u, v 47. The surface with parametric equations u, uv, v, u, v 48 49 Find the area of the surface correct to four decimal places b epressing the area in terms of a single integral and using our calculator to estimate the integral. 48. The part of the surface cos that lies inside the clinder 49. The part of the surface e that lies above the disk 4 5. (a) Use the Midpoint Rule for double integrals (see ection 5.) with si squares to estimate the area of the surface, 6, 4. (b) Use a computer algebra sstem to approimate the surface area in part (a) to four decimal places. Compare with the answer to part (a). 5. Find the area of the surface with vector equation ru, v cos 3 u cos 3 v, sin 3 u cos 3 v, sin 3 v, u, v. tate our answer correct to four decimal places. 53. Find the eact area of the surface 3 4, 4,. 54. (a) et up, but do not evaluate, a double integral for the area of the surface with parametric equations au cos v, bu sin v, u, u, v. (b) Eliminate the parameters to show that the surface is an elliptic paraboloid and set up another double integral for the surface area. ; (c) Use the parametric equations in part (a) with a and b 3 to graph the surface. (d) For the case a, b 3, use a computer algebra sstem to find the surface area correct to four decimal places. 55. (a) how that the parametric equations a sin u cos v, b sin u sin v, c cos u, u, v, represent an ellipsoid. ; (b) Use the parametric equations in part (a) to graph the ellipsoid for the case a, b, c 3. (c) et up, but do not evaluate, a double integral for the surface area of the ellipsoid in part (b). 56. (a) how that the parametric equations a cosh u cos v, b cosh u sin v, c sinh u, represent a hperboloid of one sheet. ; (b) Use the parametric equations in part (a) to graph the hperboloid for the case a, b, c 3. (c) et up, but do not evaluate, a double integral for the surface area of the part of the hperboloid in part (b) that lies between the planes 3 and 3. 57. Find the area of the part of the sphere 4 that lies inside the paraboloid. 58. The figure shows the surface created when the clinder intersects the clinder. Find the area of this surface. 5. Find, to four decimal places, the area of the part of the surface that lies above the square. Illustrate b graphing this part of the surface.
ECTION 6.7 URFACE INTEGRAL 9 6.7 EXERCIE. Let be the boundar surface of the bo enclosed b the planes,,, 4,, and 6. Approimate e. d b using a Riemann sum as in Definition, taking the patches ij to be the rectangles that are the faces of the bo and the points P ij * to be the centers of the rectangles.. A surface consists of the clinder,, together with its top and bottom disks. uppose ou know that f is a continuous function with f,, f,, 3 f,, 4 Estimate the value of f,, d b using a Riemann sum, taking the patches ij to be four quarter-clinders and the top and bottom disks. 3. Let H be the hemisphere 5,, and suppose f is a continuous function with f 3, 4, 5 7, f 3, 4, 5 8, f 3, 4, 5 9, and f 3, 4, 5. B dividing H into four patches, estimate the value of H f,, d. 4. uppose that f,, t(s ), where t is a function of one variable such that t 5. Evaluate, where is the sphere f,, d 4. 5 8 Evaluate the surface integral. 5. d, is the part of the plane 3 that lies above the rectangle, 3, 6. d, is the triangular region with vertices (,, ), (,, ), and (,, ) 7. d, is the part of the plane that lies in the first octant 8. d, is the surface 3 3 3,, 9. d, is the surface with parametric equations u, u sin v, u cos v, u, v. s d, is the helicoid with vector equation ru, v u cos v i u sin v j v k, u, v. d, is the part of the cone that lies between the planes and 3. d, is the surface,, 3. d, is the part of the paraboloid that lies inside the clinder 4 4. d, is the part of the sphere 4 that lies inside the clinder and above the -plane 5. d, is the hemisphere 4, 6. d, is the boundar of the region enclosed b the clinder 9 and the planes and 5 7. d, is the part of the clinder that lies between the planes and 3 in the first octant 8. d, is the part of the clinder 9 between the planes and, together with its top and bottom disks 9 3 Evaluate the surface integral F d for the given vector field F and the oriented surface. In other words, find the flu of F across. For closed surfaces, use the positive (outward) orientation. 9. F,, i j k, is the part of the paraboloid 4 that lies above the square,, and has upward orientation. F,, i j k, is the helicoid of Eercise with upward orientation. F,, e i e j k, is the part of the plane in the first octant and has downward orientation. F,, i j 4 k, is the part of the cone s beneath the plane with downward orientation 3. F,, i j k, is the part of the sphere 4 in the first octant, with orientation toward the origin 4. F,, i j k, is the hemisphere 5,, oriented in the direction of the positive -ais 5. F,, j k, consists of the paraboloid,, and the disk, 6. F,, i 4 j k, is the surface e,,, with upward orientation
9 CHAPTER 6 VECTOR CALCULU 7. F,, i j 3 k, is the cube with vertices,, 8. F,, i j 5 k, is the boundar of the region enclosed b the clinder and the planes and 9. F,, i j k, is the boundar of the solid half-clinder s, 3. F,, i j k, is the surface of the tetrahedron with vertices,,,,,,,,, and,, 3. Evaluate d correct to four decimal places, where is the surface,,. 3. Find the eact value of d, where is the surface in Eercise 3. 33. Find the value of d correct to four decimal places, where is the part of the paraboloid 3 that lies above the -plane. 34. Find the flu of F,, sin i j e 5 k across the part of the clinder 4 4 that lies above the -plane and between the planes and with upward orientation. Illustrate b using a computer algebra sstem to draw the clinder and the vector field on the same screen. 35. Find a formula for F d similar to Formula for the case where is given b h, and n is the unit normal that points toward the left. 36. Find a formula for F d similar to Formula for the case where is given b k, and n is the unit normal that points forward (that is, toward the viewer when the aes are drawn in the usual wa). 37. Find the center of mass of the hemisphere a,, if it has constant densit. 38. Find the mass of a thin funnel in the shape of a cone s, 4, if its densit function is,,. 39. (a) Give an integral epression for the moment of inertia about the -ais of a thin sheet in the shape of a surface if the densit function is. (b) Find the moment of inertia about the -ais of the funnel in Eercise 38. 4. Let be the part of the sphere 5 that lies above the plane 4. If has constant densit k, find (a) the center of mass and (b) the moment of inertia about the -ais. 4. A fluid has densit 87 kgm 3 and flows with velocit v i j k, where,, and are measured in meters and the components of v in meters per second. Find the rate of flow outward through the clinder 4,. 4. eawater has densit 5 kgm 3 and flows in a velocit field v i j, where,, and are measured in meters and the components of v in meters per second. Find the rate of flow outward through the hemisphere 9,. 43. Use Gauss s Law to find the charge contained in the solid hemisphere a,, if the electric field is E,, i j k 44. Use Gauss s Law to find the charge enclosed b the cube with vertices,, if the electric field is E,, i j k 45. The temperature at the point,, in a substance with conductivit K 6.5 is u,,. Find the rate of heat flow inward across the clindrical surface 6, 4. 46. The temperature at a point in a ball with conductivit K is inversel proportional to the distance from the center of the ball. Find the rate of heat flow across a sphere of radius a with center at the center of the ball. 47. Let F be an inverse square field, that is, Fr cr r 3 for some constant c, where r i j k. how that the flu of F across a sphere with center the origin is independent of the radius of. I 6.8 TOKE THEOREM tokes Theorem can be regarded as a higher-dimensional version of Green s Theorem. Whereas Green s Theorem relates a double integral over a plane region D to a line integral around its plane boundar curve, tokes Theorem relates a surface integral over a surface to a line integral around the boundar curve of (which is a space curve). Figure shows
ECTION 6.8 TOKE THEOREM 97 6.8 EXERCIE. A hemisphere H and a portion P of a paraboloid are shown. uppose F is a vector field on 3 whose components have continuous partial derivatives. Eplain wh. F,, i j 3 k, C is the curve of intersection of the plane 5 and the clinder 9 H curl F d P curl F d. (a) Use tokes Theorem to evaluate C F dr, where F,, i j k H 4 P 4 and C is the curve of intersection of the plane and the clinder 9 oriented counterclockwise as viewed from above. ; (b) Graph both the plane and the clinder with domains chosen so that ou can see the curve C and the surface that ou used in part (a). ; (c) Find parametric equations for C and use them to graph C. 6 Use tokes Theorem to evaluate curl F d.. F,, cos i e sin j e k, is the hemisphere 9,, oriented upward. (a) Use tokes Theorem to evaluate C F dr, where F,, i 3 3 j k and C is the curve of intersection of the hperbolic paraboloid and the clinder oriented counterclockwise as viewed from above. ; (b) Graph both the hperbolic paraboloid and the clinder with domains chosen so that ou can see the curve C and the surface that ou used in part (a). ; (c) Find parametric equations for C and use them to graph C. 3. F,, i j k, is the part of the paraboloid that lies inside the clinder 4, oriented upward 4. F,, 3 i sin j k, is the part of the cone that lies between the planes and 3, oriented in the direction of the positive -ais 5. F,, i j k, consists of the top and the four sides (but not the bottom) of the cube with vertices,,, oriented outward [Hint: Use Equation 3.] 6. F,, e cos i j k, is the hemisphere s, oriented in the direction of the positive -ais [Hint: Use Equation 3.] 7 Use tokes Theorem to evaluate C F dr. In each case C is oriented counterclockwise as viewed from above. 7. F,, i j k, C is the triangle with vertices (,, ), (,, ), and (,, ) 8. F,, e i e j e k, C is the boundar of the part of the plane in the first octant 9. F,, i j e k, C is the circle 6, 5 3 5 Verif that tokes Theorem is true for the given vector field F and surface. 3. F,, i j k, is the part of the paraboloid that lies below the plane, oriented upward 4. F,, i j k, is the part of the plane that lies in the first octant, oriented upward 5. F,, i j k, is the hemisphere,, oriented in the direction of the positive -ais 6. Let C be a simple closed smooth curve that lies in the plane. how that the line integral C d d 3 d depends onl on the area of the region enclosed b C and not on the shape of C or its location in the plane. 7. A particle moves along line segments from the origin to the points,,,,,,,,, and back to the origin under the influence of the force field F,, i j 4 k Find the work done.
98 CHAPTER 6 VECTOR CALCULU 8. Evaluate C sin d cos d 3 d where C is the curve rt sin t, cos t, sin t, t. [Hint: Observe that C lies on the surface.] 9. If is a sphere and F satisfies the hpotheses of tokes Theorem, show that. curl F d. uppose and C satisf the hpotheses of tokes Theorem and f, t have continuous second-order partial derivatives. Use Eercises 4 and 6 in ection 6.5 to show the following. (a) C f t dr f t d (b) C f f dr (c) C f t tf dr WRITING PROJECT N The photograph shows a stained-glass window at Cambridge Universit in honor of George Green. Courtes of the Masters and Fellows of Gonville and Caius College, Universit of Cambridge, England www.stewartcalculus.com The Internet is another source of information for this project. Click on Histor of Mathematics. Follow the links to the t. Andrew s site and that of the British ociet for the Histor of Mathematics. THREE MEN AND TWO THEOREM Although two of the most important theorems in vector calculus are named after George Green and George tokes, a third man, William Thomson (also known as Lord Kelvin), plaed a large role in the formulation, dissemination, and application of both of these results. All three men were interested in how the two theorems could help to eplain and predict phsical phenomena in electricit and magnetism and fluid flow. The basic facts of the stor are given in the margin notes on pages 56 and 93. Write a report on the historical origins of Green s Theorem and tokes Theorem. Eplain the similarities and relationship between the theorems. Discuss the roles that Green, Thomson, and tokes plaed in discovering these theorems and making them widel known. how how both theorems arose from the investigation of electricit and magnetism and were later used to stud a variet of phsical problems. The dictionar edited b Gillispie [] is a good source for both biographical and scientific information. The book b Hutchinson [5] gives an account of tokes life and the book b Thompson [8] is a biograph of Lord Kelvin. The articles b Grattan-Guinness [3] and Gra [4] and the book b Cannell [] give background on the etraordinar life and works of Green. Additional historical and mathematical information is found in the books b Kat [6] and Kline [7].. D. M. Cannell, George Green, Mathematician and Phsicist 793 84: The Background to His Life and Work (Philadelphia: ociet for Industrial and Applied Mathematics, ).. C. C. Gillispie, ed., Dictionar of cientific Biograph (New York: cribner s, 974). ee the article on Green b P. J. Wallis in Volume XV and the articles on Thomson b Jed Buchwald and on tokes b E. M. Parkinson in Volume XIII. 3. I. Grattan-Guinness, Wh did George Green write his essa of 88 on electricit and magnetism? Amer. Math. Monthl, Vol. (995), pp. 387 396. 4. J. Gra, There was a joll miller. The New cientist, Vol. 39 (993), pp. 4 7. 5. G. E. Hutchinson, The Enchanted Voage and Other tudies (Westport, CT : Greenwood Press, 978). 6. Victor Kat, A Histor of Mathematics: An Introduction (New York: HarperCollins, 993), pp. 678 68. 7. Morris Kline, Mathematical Thought from Ancient to Modern Times (New York: Oford Universit Press, 97), pp. 683 685. 8. lvanus P. Thompson, The Life of Lord Kelvin (New York: Chelsea, 976).
ECTION 6.9 THE DIVERGENCE THEOREM 3 P P FIGURE 4 The vector field F= i+ j the origin. [This is a special case of Gauss s Law (Equation 6.7.) for a single charge. The relationship between and is 4.] Another application of the Divergence Theorem occurs in fluid flow. Let v,, be the velocit field of a fluid with constant densit. Then F v is the rate of flow per unit area. If P,, is a point in the fluid and B a is a ball with center P and ver small radius a, then div FP div FP for all points in B a since div F is continuous. We approimate the flu over the boundar sphere as follows: This approimation becomes better as a l and suggests that 8 a F d Ba div F dv Ba div FP dv div FP VB a div FP lim F d a l VB a a Equation 8 sas that div FP is the net rate of outward flu per unit volume at P. (This is the reason for the name divergence.) If div FP, the net flow is outward near P and P is called a source. If div FP, the net flow is inward near P and P is called a sink. For the vector field in Figure 4, it appears that the vectors that end near P are shorter than the vectors that start near P. Thus the net flow is outward near P, so div FP and P is a source. Near P, on the other hand, the incoming arrows are longer than the outgoing arrows. Here the net flow is inward, so div FP and P is a sink. We can use the formula for F to confirm this impression. ince F i j, we have div F, which is positive when. o the points above the line are sources and those below are sinks. a 6.9 EXERCIE 4 Verif that the Divergence Theorem is true for the vector field F on the region E.. F,, 3 i j k, E is the cube bounded b the planes,,,,, and. F,, i j k, E is the solid bounded b the paraboloid 4 and the -plane 3. F,, i j k, E is the solid clinder, 4. F,, i j k, E is the unit ball 5 5 Use the Divergence Theorem to calculate the surface integral F d; that is, calculate the flu of F across. 5. F,, e sin i e cos j k, is the surface of the bo bounded b the planes,,,,, and 6. F,, 3 i 3 j 4 k, is the surface of the bo with vertices,, 3 7. F,, 3 i e j 3 k, is the surface of the solid bounded b the clinder and the planes and 8. F,, 3 i j k, is the surface of the solid bounded b the hperboloid and the planes and 9. F,, sin i cos j cos k, is the ellipsoid a b c. F,, i j k, is the surface of the tetrahedron bounded b the planes,,, and. F,, cos i e j sin k, is the surface of the solid bounded b the paraboloid and the plane 4. F,, 4 i 3 j 4 k, is the surface of the solid bounded b the clinder and the planes and 3. F,, 4 3 i 4 3 j 3 4 k, is the sphere with radius R and center the origin
4 CHAPTER 6 VECTOR CALCULU 4. F r r, where r i j k, consists of the hemisphere s and the disk in the -plane 5. F,, e tan i s3 j sin k, is the surface of the solid that lies above the -plane and below the surface 4 4,, 6. Use a computer algebra sstem to plot the vector field F,, sin cos i sin 3 cos 4 j sin 5 cos 6 k in the cube cut from the first octant b the planes,, and. Then compute the flu across the surface of the cube. 7. Use the Divergence Theorem to evaluate F d, where F,, i ( 3 3 tan ) j k and is the top half of the sphere. [Hint: Note that is not a closed surface. First compute integrals over and, where is the disk, oriented downward, and.] 8. Let F,, tan i 3 ln j k. Find the flu of F across the part of the paraboloid that lies above the plane and is oriented upward. 9. A vector field F is shown. Use the interpretation of divergence derived in this section to determine whether div F is positive or negative at P and at P. P _ P _ P P. (a) Are the points and sources or sinks for the vector field F shown in the figure? Give an eplanation based solel on the picture. (b) Given that F,,, use the definition of divergence to verif our answer to part (a). P _ P _ Plot the vector field and guess where div F and where div F. Then calculate div F to check our guess.. F,,. F,, 3. Verif that div E for the electric field E Q. 4. Use the Divergence Theorem to evaluate d where is the sphere. 5 3 Prove each identit, assuming that and E satisf the conditions of the Divergence Theorem and the scalar functions and components of the vector fields have continuous second-order partial derivatives. 5. a n d, where a is a constant vector 6. F d, where F,, i j k 7. 8. 9. 3. VE 3 curl F d D n f d E f dv f t n d E f t tf n d E 3. uppose and E satisf the conditions of the Divergence Theorem and f is a scalar function with continuous partial derivatives. Prove that f t f t dv f n d E These surface and triple integrals of vector functions are vectors defined b integrating each component function. [Hint: tart b appling the Divergence Theorem to F f c, where c is an arbitrar constant vector.] 3. A solid occupies a region E with surface and is immersed in a liquid with constant densit. We set up a coordinate sstem so that the -plane coincides with the surface of the liquid and positive values of are measured downward into the liquid. Then the pressure at depth is p t, where t is the acceleration due to gravit (see ection 6.5). The total buoant force on the solid due to the pressure distribution is given b the surface integral F f t t f dv pn d f dv 3 where n is the outer unit normal. Use the result of Eercise 3 to show that F Wk, where W is the weight of the liquid displaced b the solid. (Note that F is directed upward because is directed downward.) The result is Archimedes principle: The buoant force on an object equals the weight of the displaced liquid.
A6 APPENDIX I ANWER TO ODD-NUMBERED EXERCIE EXERCIE 6.3 N PAGE 53. 4 3. f, 3 8 K 5. f, e sin K 7. f, e sin K 9. f, ln 3 K. (b) 6 3. (a) f, (b) 5. (a) f,, (b) 77 7. (a) f,, cos (b) 9.. 3 3. No 5. Conservative 9. (a) Yes (b) Yes (c) Yes 3. (a) Yes (b) Yes (c) No EXERCIE 6.4 N PAGE 6 8. 3. 3 5. 7. 3 9.. 3 3. 65 9 5. 8e 48e 7. 9.. (c) 3. 4a3, 4a3 if the region is the portion of the disk a in the first quadrant EXERCIE 6.5 N PAGE 68. (a) i 3 j k (b) 3. (a) i j k (b) (s) 5. (a) (b) s 7. (a),, (b) 9. (a) Negative (b) curl F. (a) Zero (b) curl F points in the negative -direction 3. f,, 3 K 5. f,, K 7. Not conservative 9. No EXERCIE 6.6 PAGE 78 N. P: no; Q: es 3. Plane through, 3, containing vectors,, 4,,, 5 5. Hperbolic paraboloid 7. constant 4 3 4. 3. IV 5. II 7. III 9.. 3. u v, u v, 3 u v,, s sin cos, sin sin, cos, 4, 5., 4 cos, 4 sin, 5, 9., e cos, e sin, 3, constant u constant [or,, s4, ] 3. (a) Direction reverses (b) Number of coils doubles 33. 3 3 3 35. 37. 3s4 4 39. 53 5 7 4. 3(s ) 43. 6(7s7 5s5) 45. s 7 4 [ln( s) ln s7] 47. 4 49. 3.9783 5. (a) 4.55 (b) 4.476 45 53. 8 s4 5 6 ln[(s5 3s7)(3s5 s7)] 55. (b) _ 9. _ u constant.5 u constant constant (c) s36 sin4 u cos v 9 sin 4 u sin v 4 cos u sin u du dv 57. 59. a 4 _ EXERCIE 6.7 N PAGE 9. 49.9 3. 5. 7s4 7. s34 9. 5s548 4. 364s3 3. 6(39s7 ) 5. 7. 9 6
APPENDIX I ANWER TO ODD-NUMBERED EXERCIE A7 73 9.. 3. 4 8 6 3 5. 7. 48 9. 3..64 33. 3.4895 35. F d D Ph Q Rh da, where D projection of on -plane 37.,, a 39. (a) I,, d (b) 439s5 8 4. kgs 43. 3a 3 45. EXERCIE 6.8 N PAGE 97 3. 5. 7. 9.. (a) 8 (b) (c) 3 cos t, 3 sin t, 3cos t sin t, t 7. 3 8 3 EXERCIE 6.9 N PAGE 3 3 5. 7. 9 9.. 33 3. 5. 34s6 8 arcsin(s33) 7. 9. Negative at P, positive at P. div F in quadrants I, II; div F in quadrants III, IV CHAPTER 6 REVIEW N PAGE 6 True-False Qui. False 3. True 5. False 7. True Eercises. (a) Negative (b) Positive 4 3. 6s 5. 5 7. 9. 3 4e. f, e e 3. 7. 5. 6 (7 5s5) 7. 6(39s7 ) 9. 643 33. 37. 4 39. 8 5 5 4 _ 8 48 CHAPTER 7 EXERCIE 7. N PAGE 7. c e 3 c e 3. c cos 4 c sin 4 5. c e 3 c e 3 7. c c e 9. e c cos 3 c sin 3. c e (s3)t c e (s3)t 3. P e t [c cos( t) c sin( t)] 5. All solutions approach either or as l. 7. e 3 e 9. e / e. 3 cos 4 sin 4 3. e cos 3 sin 5. 7. e 3 e 3 e 3 cos( ) 4 sin( ) e 3 9. No solution 3. e cos 3 e 33. (b), n a positive integer; C sinnl EXERCIE 7. N PAGE 4. 3. 5. 7. 9. c e c e 3 7 4 c c e 4 cos 4 sin 4 e c cos c sin e 3 cos sin e 3 6 e ( ). 3 The solutions are all asmptotic to p cos 3 sin as l. Ecept for p, all _3 p 8 solutions approach either or as l. 3. 5. 7. 9.. 3. 5. 7. f _3 3 n L _3 _ g sin 3 p Ae B C D cos E F G sin p A B Ce 9 p e A B C cos 3 D E F sin 3 c cos( ) c sin( ) 3 cos c e c e e c sin c cos sin lnsec tan c ln e e c e ln e e e [c c ln tan ] EXERCIE 7.3 N PAGE 3..35 cos(s5 t) 3. 5. 49 5 e 6t 6 5 e t kg