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All Textbook Solutions for Multivariable Calculus

Determine whether or not F is a conservative vector field. If it is, find a function f such that F = f. 8. F(x, y) = (2xy + y2) i + (x2 2xy3) j, y 0 Determine whether or not F is a conservative vector field. If it is, find a function f such that F = f. 9. F(x, y) = (y2 cos x + cos y) i + (2y sin x x sin y) j 10E The figure shows the vector field F(x, y) = 2xy, x2 and three curves that start at (1, 2) and end at (3, 2). (a) Explain why C F dr has the same value for all three curves. (b) What is this common value?12E (a) Find a function f such that F = f and (b) use part (a) to evaluate C F dr along the given curve C. 13. F(x, y) = x2y3 i + x3y2 j, C: r(t) = t3 2t, t3 + 2t, 0 t 1 (a) Find a function f such that F = f and (b) use part (a) to evaluate C F dr along the given curve C. 14. F(x, y) = (1 + xy)exy i + x2exy j, C: r(t) = cos t i + 2 sin t j, 0 t /2 (a) Find a function f such that F = f and (b) use part (a) to evaluate C F dr along the given curve C. 15. F(x, y, z) = yz i + xz j + (xy + 2z) k, C is the line segment from (1, 0, 2) to (4, 6, 3)16E 17E (a) Find a function f such that F = f and (b) use part (a) to evaluate C F dr along the given curve C. 18. F(x, y, z) = sin y i + (x cos y + cos z) j y sin z k, C: r(t) = sin t i + t j + 2t k, 0 t /2 Show that the line integral is independent of path and evaluate the integral. 19. C 2xey dx + (2y x2ey) dy, C is any path from (1, 0) to (2, 1)Show that the line integral is independent of path and evaluate the integral. 20. C sin y dx + (x cos y sin y) dy, C is any path from (2, 0) to (1, )21E 22E Find the work done by the force field F in moving an object from P to Q. 23. F(x, y) = x3 i + y3 j; P(1, 0), Q(2, 2)Find the work done by the force field F in moving an object from P to Q. 24. F(x, y) = (2x + y) i + x j; P(1, 1), Q(4, 3)Is the vector field shown in the figure conservative? Explain.Is the vector field shown in the figure conservative? Explain.Let F = f, where f(x, y) = sin(x 2y). Find curves C1 and C2 that are not closed and satisfy the equation. (a) C1F dr = 0 (b) C1 F dr = 1 29E Use Exercise 29 to show that the line integral C y dx + x dy + xyz dz is not independent of path.31E 32E 33E 34E Let F(x, y) = yi+xjx2+y2 (a) Show that P/y=Q/x. (b) Show that C F dr is not independent of path. [Hint: Compute C1 F dr and C2 F dr, where C1 and C2 are the upper and lower halves of the circle x2 + y2 = 1 from (1, 0) to (1, 0).] Does this contradict Theorem 6?Evaluate the line integral by two methods: (a) directly and (b) using Greens Theorem. 1. C y2 dx + x2y dy, C is the rectangle with vertices (0, 0), (5, 0), (5. 4), and (0, 4)Evaluate the line integral by two methods: (a) directly and (b) using Greens Theorem. 2. C y dx x dy, C is the circle with center the origin and radius 4 Evaluate the line integral by two methods: (a) directly and (b) using Greens Theorem. 3. C xy dx + x2y3 dy, C is the triangle with vertices (0, 0), (1, 0), and (1, 2)Evaluate the line integral by two methods: (a) directly and (b) using Greens Theorem. 4. C x2y2 dx + xy dy, C consists of the arc of the parabola y = x2 from (0, 0) to (1, 1) and the line segments from (1, 1) to (0, 1) and from (0, 1) to (0, 0)Use Greens Theorem to evaluate the line integral along the given positively oriented curve. 5. C yex dx + 2ex dy, C is the rectangle with vertices (0, 0), (3, 0), (3, 4), and (0, 4)Use Greens Theorem to evaluate the line integral along the given positively oriented curve. 6. C (x2 + y2) dx + (x2 y2) dy, C is the triangle with vertices (0, 0), (2, 1), and (0, 1)Use Greens Theorem to evaluate the line integral along the given positively oriented curve. 7. C (y + ex) dx + (2x + cos y2) dy, C is the boundary of the region enclosed by the parabolas y = x2 and x = y2 Use Greens Theorem to evaluate the line integral along the given positively oriented curve. 8. C y4 dx + 2xy3 dy, C is the ellipse x2 + 2y2 = 2 Use Greens Theorem to evaluate the line integral along the given positively oriented curve. 9. C y3 dx x3 dy, C is the circle x2 + y2 = 4 Use Greens Theorem to evaluate the line integral along the given positively oriented curve. 10. C (1 y3) dx + (x3 + ey2) dy, C is the boundary of the region between the circles x2 + y2 = 4 and x2 + y2 = 9 Use Greens Theorem to evaluate C F dr. (Check the orientation of the curve before applying the theorem.) 11. F(x, y) = y cos x xy sin x, xy + x cos x, C is the triangle from (0, 0) to (0, 4) to (2, 0) to (0, 0)Use Greens Theorem to evaluate C F dr. (Check the orientation of the curve before applying the theorem.) 12. F(x, y) = ex + y2, ey + x2, C consists of the arc of the curve y = cos x from (/2, 0) to (/2, 0) and the line segment from (/2, 0) to (/2, 0)Use Greens Theorem to evaluate C F dr. (Check the orientation of the curve before applying the theorem.) 13. F(x, y) = y cos y, x sin y, C is the circle (x 3)2 + (y + 4)2 = 4 oriented clockwise Use Greens Theorem to evaluate C F dr. (Check the orientation of the curve before applying the theorem.) 14. F(x,y)=x2+1,tan1x, C is the triangle from (0, 0) to (1, 1) to (0, 1) to (0, 0)17E A particle starts at the origin, moves along the x-axis to (5, 0), then along the quarter-circle x2 + y2 = 25, x 0, y 0 to the point (0, 5), and then down the y-axis back to the origin. Use Greens Theorem to find the work done on this particle by the force field F(x, y) = (sin x, sin y + xy2 + 13x3.Use one of the formulas in (5) to find the area under one arch of the cycloid x = t sin t, y = 1 cos t.If a circle C with radius 1 rolls along the outside of the circle x2 + y2 = 16, a fixed point P on C traces out a curve called an epicycloid, with parametric equations x = 5 cos t cos 5t, y = 5 sin t sin 5t. Graph the epicycloid and use (5) to find the area it encloses.(a) If C is the line segment connecting the point (x1, y1) to the point (x2, y2), show that C x dy y dx = x1y2 x2y1 (b) If the vertices of a polygon, in counterclockwise order, are (x1, y1), (x2, y2), , (xn, yn), show that the area of the polygon is A = 12[(x1, y2), (x2, y1) + (x2y3 x3y2) + + (xn1 yn xn yn1) + (xny1 x1yn)] (c) Find the area of the pentagon with vertices (0, 0), (2, 1), (1, 3), (0, 2), and (1, 1).Let D be a region bounded by a simple closed path C in the xy-plane. Use Greens Theorem to prove that the coordinates of the centroid (x, y) of D are x=12ACx2dy y=12ACy2dx where A is the area of D.Use Exercise 22 to find the centroid of a quarter-circular region of radius a.Use Exercise 22 to find the centroid of the triangle with vertices (0, 0), (a, 0), and (a, b), where a 0 and b 0.A plane lamina with constant density (x, y) = occupies region in the xy-plane bounded by a simple closed path C. Show that its moments of inertia about the axes are Ix=3Cy3dxIy=3Cx3dy 26E Use the method of Example 5 to calculate C F dr, where F(x,y)=2xyi+(y2x2)j(x2+y2)2 and C is any positively oriented simple closed curve that encloses the origin.Calculate C F dr, where F(x, y) = x2 + y, 3x y2 and C is the positively oriented boundary curve of a region D that has area 6.If F is the vector field of Example 5, show that C F dr = 0 for every simple closed path that does not pass through or enclose the origin.Complete the proof of the special case of Greens Theorem by proving Equation 3.Use Greens Theorem to prove the change of variables formula for a double integral (Formula 15.9.9) for the case where f(x, y) = 1: Rdxdy=S|(x,y)(u,v)|dudv Here R is the region in the xy-plane that corresponds to the region S in the uv-plane under the transformation given by x = g(u, v), y = h(u, v). [Hint: Note that the left side is A(R) and apply the first part of Equation 5. Convert the line integral over R to a line integral over S and apply Greens Theorem in the uv-plane.]Find (a) the curl and (b) the divergence of the vector field. 1. F(x, y, z) = xy2z2 i + x2yz2 j + x2y2z k Find (a) the curl and (b) the divergence of the vector field. 2. F(x, y, z) = x3yz2 j + y4z3 k Find (a) the curl and (b) the divergence of the vector field. 3. F(x, y, z) = xyez i + yzex k Find (a) the curl and (b) the divergence of the vector field. 4. F (x, y, z) = sin yz i + sin zx j + sin xy k Find (a) the curl and (b) the divergence of the vector field. 5. x, y, z) = i + j + k Find (a) the curl and (b) the divergence of the vector field. 6. F(x, y, z) = ln(2y + 3z) i + ln(x + 3z) j + ln(x + 2y) k Find (a) the curl and (b) the divergence of the vector field. 7. F(x, y, z) = ex sin y, ey sin z, ez sin x Find (a) the curl and (b) the divergence of the vector field. 8. F(x, y, z) = arctan(xy), arctan(yz), arctan(zx)The vector field F is shown in the xy-plane and looks the same in all other horizontal planes. (In other words, F is independent of z and its z-component is 0.) (a) Is div F positive, negative, or zero? Explain. (b) Determine whether curl F = 0. If not, in which direction doescurl F point? 9.The vector field F is shown in the xy-plane and looks the same in all other horizontal planes. (In other words, F is independent of z and its z-component is 0.) (a) Is div F positive, negative, or zero? Explain. (b) Determine whether curl F = 0. If not, in which direction doescurl F point? 10.The vector field F is shown in the xy-plane and looks the same in all other horizontal planes. (In other words, F is independent of z and its z-component is 0.) (a) Is div F positive, negative, or zero? Explain. (b) Determine whether curl F = 0. If not, in which direction doescurl F point? 11.Let f be a scalar field and F a vector field. State whether each expression is meaningful. If not, explain why. If so, state whether it is a scalar field or a vector field. (a) curl f (b) grad f (c) div F (d) curl(grad f) (e) grad F (f) grad(div F) (g) div(grad f) (h) grad(div f) (i) curl(curl F) (j) div(div F) (k) (grad f) (div F) (l) div(curl(grad f))13E Determine whether or not the vector field is conservative. If it is conservative, find a function f such that F = f. 14. F(x, y, z) = xyz4 i + x2z4 j + 4x2yz3 k Determine whether or not the vector field is conservative. If it is conservative, find a function f such that F = f. 15. F(x, y, z) = z cos y i + xz sin y j + x cos y k Determine whether or not the vector field is conservative. If it is conservative, find a function f such that F = f. 16. F(x, y, z) = i + sin z j + y cos z k Determine whether or not the vector field is conservative. If it is conservative, find a function f such that F = f. 17. F(x, y, z) = eyz i + xzeyz j + xyeyz k Determine whether or not the vector field is conservative. If it is conservative, find a function f such that F = f. 18. F(x, y, z) = ex sin yz i + zex cos yz j + yex cos yz k Is there a vector field G on 3 such that curl G = x sin y, cos y, z xy? Explain.Is there a vector field G on 3 such that curl G = x, y, z? Explain.Show that any vector field of the form F(x, y, z) = f(x) i + g(y) j + h(z) k where f, g, h are differentiable functions, is irrotational.Show that any vector field of the form F(x, y, z) = f(y, z) i + g(x, z) j + h(x, y) k is incompressible.Prove the identity, assuming that the appropriate partial derivatives exist and are continuous. If f is a scalar field and F, G are vector fields, then fF, F G, and F G are defined by (fF)(x, y, z) = f(x, y, z) F(x, y, z) (F G)(x, y, z) = F(x, y, z) G(x, y, z) (F G)(x, y, z) = F(x, y, z) G(x, y, z) 23. div(F + G) = div F + div G Prove the identity, assuming that the appropriate partial derivatives exist and are continuous. If f is a scalar field and F, G are vector fields, then fF, F G, and F G are defined by (fF)(x, y, z) = f(x, y, z) F(x, y, z) (F G)(x, y, z) = F(x, y, z) G(x, y, z) (F G)(x, y, z) = F(x, y, z) G(x, y, z) 24. curl(F + G) = curl F + curl G Prove the identity, assuming that the appropriate partial derivatives exist and are continuous. If f is a scalar field and F, G are vector fields, then fF, F G, and F G are defined by (fF)(x, y, z) = f(x, y, z) F(x, y, z) (F G)(x, y, z) = F(x, y, z) G(x, y, z) (F G)(x, y, z) = F(x, y, z) G(x, y, z) 25. div(fF) = f div F + F f Prove the identity, assuming that the appropriate partial derivatives exist and are continuous. If f is a scalar field and F, G are vector fields, then fF, F G, and F G are defined by (fF)(x, y, z) = f(x, y, z) F(x, y, z) (F G)(x, y, z) = F(x, y, z) G(x, y, z) (F G)(x, y, z) = F(x, y, z) G(x, y, z) 26. curl(fF) = f curl F + (f) F Prove the identity, assuming that the appropriate partial derivatives exist and are continuous. If f is a scalar field and F, G are vector fields, then fF, F G, and F G are defined by (fF)(x, y, z) = f(x, y, z) F(x, y, z) (F G)(x, y, z) = F(x, y, z) G(x, y, z) (F G)(x, y, z) = F(x, y, z) G(x, y, z) 27. div(F G) = G curl F F curl G Prove the identity, assuming that the appropriate partial derivatives exist and are continuous. If f is a scalar field and F, G are vector fields, then fF, F G, and F G are defined by (fF)(x, y, z) = f(x, y, z) F(x, y, z) (F G)(x, y, z) = F(x, y, z) G(x, y, z) (F G)(x, y, z) = F(x, y, z) G(x, y, z) 28. div(f g) = 0 Prove the identity, assuming that the appropriate partial derivatives exist and are continuous. If f is a scalar field and F, G are vector fields, then fF, F G, and F G are defined by (fF)(x, y, z) = f(x, y, z) F(x, y, z) (F G)(x, y, z) = F(x, y, z) G(x, y, z) (F G)(x, y, z) = F(x, y, z) G(x, y, z) 29. curl(curl F) = grad(div F) 2F Let r = x i + y j + z k and r = |r|. 30. Verify each identity. (a) r = 3 (b) (rr) = 4r (c) 2r3 = 12r Let r = x i + y j + z k and r = |r|. 31. Verify each identity. (a) r = r/r (b) r = 0 (c) (1/r) = r/r3 (d) ln r = r/r2 Let r = x i + y j + z k and r = |r|. 32. If F = r/rp, find div F. Is there a value of p for which div F = 0?Use Greens Theorem in the form of Equation 13 to prove Greens first identity: Df2gdA=Cf(g)ndsDfgdA where D and C satisfy the hypotheses of Greens Theorem and the appropriate partial derivatives of f and g exist and arc continuous. (The quantity g n = Dn g occurs in the line integral. This is the directional derivative in the direction of the normal vector n and is called the normal derivative of g.)34E 35E 36E This exercise demonstrates a connection between the curl vector and rotations. Let B be a rigid body rotating about the z-axis. The rotation can be described by the vector w = k, where is the angular speed of B, that is, the tangential speed of any point P in B divided by the distance d from the axis of rotation. Let r = x, y, z be the position vector of P. (a) By considering the angle in the figure, show that the velocity field of B is given by v = w r. (b) Show that v = y i + x j. (c) Show that curl v = 2w.Maxwells equations relating the electric field E and magnetic field H as they vary with time in a region containing no charge and no current can be stated as follows: divE=0divH=0curtE=1cHtcurlH=1cEt where c is the speed of light. Use these equations to prove the following: (a) (E)=1c22Et2 (b) (H)=1c22Ht2 (c) 2E=1c22Et2 [Hint: Use Exercise 29.] (d) 2H=1c22Ht2 We have seen that all vector fields of the form F = g satisfy the equation curt F = 0 and that all vector fields of the form F = curl G satisfy the equation div F = 0 (assuming continuity of the appropriate partial derivatives). This suggests the question: are there any equations that all functions of the form f = div G must satisfy? Shoes that the answer to this question is No by proving that every continuous function f on 3 is the divergence of sonic vector field. [Hint: Let G(x, y, z) = g(x, y, z), 0, 0, where g(x,y,z)=0xf(t,y,z)dt.]Determine whether the points P and Q lie on the given surface. 1. r(u, v) = u + v, u 2v, 3 + u v P(4, 5, 1), Q(0, 4, 6)Determine whether the points P and Q lie on the given surface. 2. r(u, v) = 1 + u v, u + v2, u2 v2 P(1, 2, 1), Q(2, 3, 3)Identify the surface with the given vector equation. 3. r(u, v) = (u + v) i + (3 v) j + (1 + 4u + 5v) k 4E 5E 6E Match the equations with the graphs labeled IVI and give reasons for your answers. Determine which families of grid curves have u constant and which have v constant. 13. r(u, v) = u cos v i + u sin v j + v k Match the equations with the graphs labeled IVI and give reasons for your answers. Determine which families of grid curves have u constant and which have v constant. 14. r(u, v) = uv2 i + u2v j + (u2 v2) k Match the equations with the graphs labeled IVI and give reasons for your answers. Determine which families of grid curves have u constant and which have v constant. 15. r(u, v) = (u3 u) i + v2 j + u2 k Match the equations with the graphs labeled IVI and give reasons for your answers. Determine which families of grid curves have u constant and which have v constant. 16. x = (1 u)(3 + cos v) cos 4u, y = (1 u)(3 + cos v) sin 4u, z = 3u + (1 u) sin v 17E Match the equations with the graphs labeled IVI and give reasons for your answers. Determine which families of grid curves have u constant and which have v constant. 18. x = sin u, y = cos u sin v, z = sin v Find a parametric representation for the surface. 19. The plane through the origin that contains the vectors i j and j k 20E Find a parametric representation for the surface. 21. The part of the hyperboloid 4x2 4y2 z2 = 4 that lies in front of the yz-plane Find a parametric representation for the surface. 22. The part of the ellipsoid x2 + 2y2 + 3z2 = 1 that lies to the left of the xz-plane Find a parametric representation for the surface. 23. The part of the sphere x2 + y2 + z2 = 4 that lies above the cone z=x2+y2 Find a parametric representation for the surface. 24. The part of the cylinder x2 + z2 = 9 that lies above the xy-plane and between the planes y = 4 and y = 4 25E Find a parametric representation for the surface. 26. The part of the plane z = x + 3 that lies inside the cylinder x2 + y2 = 1 29E Find parametric equations for the surface obtained by rotating the curve x = 1/y, y 1, about the y-axis and use them to graph the surface.33E 34E Find an equation of the tangent plane to the given parametric surface at the specified point. 35.r(u, v) = ucosv i+usinvj +vk;u= 1,v=/3 36E Find an equation of the tangent plane to the given parametric surface at the specified point. Graph the surface andthe tangent plane. 37. r(u, v)=u2i+2usinv j + ucos v k;u =1,v= 0 38E 39E 40E 41E Find the area of the surface. 42. The part of the conez = that lies between the planey=xand the cylindery=x2 43E 44E Find the area of the surface. 45. The part of the surfacez = xythat lies within thecylinderx2+y2= 1 Find the area of the surface. 46. The part of the surfacex= z2+ythat lies between theplanesy = 0, y = 2, z = 0,andz = 2 Find the area of the surface. 47. The part of the paraboloidy=x2+ z2that lies within thecylinderx2+ z2= 16 Find the area of the surface. 48.The helicoid (or spiral ramp) with vector equation r(u,v)=ucos v i + u sin v j + v k, 0 u 1, 0 v Find the area of the surface. 49. The surface with parametric equationsx =u2, y=uv, z =v2,0 u 1, 0 v 2 50E 51E Find the area of the surface correct to four decimal places by expressing the area in terms of a single integral andusing your calculator to estimate the integral. 52.The part of the surface z = cos(x2+y2)that lies inside thecylinderx2+y2= 1 Find the area of the surface correct to four decimal places by expressing the area in terms of a single integral andusing your calculator to estimate the integral. 53.The part of the surface z = ln(x2+y2+ 2) that lies abovethe diskx2+y2 1 54E 56E 59E 60E 61E The figure shows the surface created when the cylindery2+z2= 1 intersects the cylinderx2+ z2 = 1. Find thearea of this surface.63E LetSbe the surface of the box enclosed by the planesx= 1,y= 1, z = 1. Approximate Scos(x + 2y+ 3z) dSbyusing a Riemann sum as in Definition 1, taking the patchesSij to be the squares that are the faces of the boxSand the pointsto be the centers of the squares.2E 3E 4E Evaluate the surface integral. 5. s (x + y + z) dS, S is the parallelogram with parametric equationsx=u+v, y = u v, z =1 +2u+ v, 0 u 2, 0 v 1 Evaluate the surface integral. 6. s xyz dS, Sis the cone with parametric equationsx = ucosv, y = u sin v, z = u, 0 u 1, 0 v /2 Evaluate the surface integral. 7. s y dS,Sis the helicoid with vector equation r(u,v) = ucosv, usinv, v,0 u 1, 0 v Evaluate the surface integral. 8.s (x2+ y2)dS, Sis the surface with vector equation r(u,v)=2uv, u2 v2, u2+v2,u2+ v2 1 Evaluate the surface integral. 9. s x2yz dS, Sis the part of the planez = 1 + 2x + 3ythat lies above the rectangle [0, 3][0, 2]Evaluate the surface integral. 10. s xz dS, S is the part of the plane 2x + 2y + z = 4 that lies in the first octant Evaluate the surface integral. 11. s x dS, S is the triangular region with vertices (1, 0, 0), (0, -2, 0), and (0, 0, 4)Evaluate the surface integral. 12. s y dS, S is the surface z=23(x3/2+y3/2),0x1,0y1 Evaluate the surface integral. 13. s z2dS, S is the part of the paraboloid x = y2 + z2 given by 0 x 1 Evaluate the surface integral. 14. s y2z2 dS, S is the part of the cone y=x2+z2 given by 0 y 5 Evaluate the surface integral. 15. s x dS, S is the surface y = x2 + 4z, 0 x 1, 0 z 1 Evaluate the surface integral. 16 s y2 dS, S is the part of the sphere x2 + y2 + z2 = 1 that lies above the cone z=x2+y2 Evaluate the surface integral. 17. s (x2z + y2z)dS, S is the hemisphere x2 + y2 + z2 = 4, z 0 Evaluate the surface integral. 18. s (x + y + z) dS, S is the part of the half-cylinder x2 + z2 = 1. z 0, that lies between the planes y = 0 and y = 2 Evaluate the surface integral. 19. s xz dS, S is the boundary of the region enclosed by the cylinder y2 + z2 = 9 and the planes x = 0 and x + y = 5 Evaluate the surface integral. 20. s (x2 + y2 + z2) dS, S is the part of the cylinder x2 + y2 = 9 between the planes z = 0 and z = 2, together with its top and bottom disks Evaluate the surface integral s F dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. 21. F(x, y, z) = zexy i - 3zexy j + xy k, S is the parallelogram of Exercise 5 with upward orientation Evaluate the surface integral s F dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. 22. F(x, y, z) = z i + y j + x k, S is the helicoid of Exercise 7 with upward orientation Evaluate the surface integral s F dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. 23. F(x, y, z) = xy i + yz j + zx k, S is the part of the paraboloid z = 4 - x2 - y2 that lies above the square 0 x 1, 0 y 1, and has upward orientation Evaluate the surface integral s F dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. 24. F(x, y, z) = -x i - y j + z3 k, S is the part of the cone z=x2+y2 between the planes z = 1 and z = 3 with downward orientation Evaluate the surface integral s F dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. 25. F(x, y, z) = x i + y j + z2 k, S is the sphere with radius 1 and center the origin Evaluate the surface integral s F dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. 26. F(x, y, z) = y i - x j + 2z k, S is the hemisphere x2 + y2 + z2 = 4, z 0, oriented downward Evaluate the surface integral s F dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. 28. F(x, y, z) = yz i + zx j + xy k, S is the surface z = x sin y, 0 x 2, 0 y , with upward orientation Find a formula for s F dS similar to Formula 10 for the case where S is given by y = h(x, z) and n is the unit normal that points toward the left.Find a formula for s F dS similar to Formula 10 for the case where S is given by x = k(yy z) and n is the unit normal that points forward (that is, toward the viewer when the axes arc drawn in the usual way).Find the center of mass of the hemisphere x2 + y2 + z2 = a2, z 0, if it has constant density.Find the mass of a thin funnel in the shape of a cone z=x2+y2, 1 z 4, if its density function is (x, y, z) = 10 - z.(a) Give an integral expression for the moment of inertia I about the z-axis of a thin sheet in the shape of a surface S if the density function is . (b) Find the moment of inertia about the z-axis of the funnel in Exercise 40.Let S be the part of the sphere x2 + y2 + z2 = 25 that lies above the plane z = 4. If S has constant density k, find (a) the center of mass and (b) the moment of inertia about the z-axis.43E 44E 45E 46E The temperature at the point (x, y, z) in a substance with conductivity K = 6.5 is u(x, y, z) = 2y2 + 2z2. Find the rate of heat flow inward across the cylindrical surface y2 + z2 - 6, 0 x 4.48E 49E 1. A hemisphere H and a portion P of a paraboloid are shown. Suppose F is a vector field on 3 whose components have continuous partial derivatives. Explain why HcurlFdS=PcurlFdS Use Stokes Theorem to evaluate s curl F dS. 2. F(x, y, z) = x2sin z i + y2 j + xy k. S is the part of the paraboloid z = 1 - x2 - y2 that lies above the xy-plane, oriented upward Use Stokes Theorem to evaluate s curl F dS. 3. F(x, y, z) = zey i + x cos y j + xz sin y k, S is the hemisphere x2 + y2 + z2 = 16, y 0, oriented in the direction of the positive y-axis 4E (x, y, z) = xyz i + xy j + x2yz k. S consists of the top and the four sides (but not the bottom) of the cube with vertices (1, 1, 1), oriented outward Use Stokes Theorem to evaluate s curl F dS. 6. F(x, y, z) = exy i + exz j + x2z k, S is the half of the ellipsoid 4x2 + y2 + 4z2 = 4 that lies to the right of the xz-plane, oriented in the direction of the positive y-axis Use Stokes Theorem to evaluate c F dr. In each case C is oriented counterclockwise as viewed from above. 7. F(x, y, z) = (x + y2) i + (y + z2) j + (z + x2) k, C is the triangle with vertices (1, 0, 0), (0, 1, 0), and (0, 0, 1)Use Stokes Theorem to evaluate c F dr. In each case C is oriented counterclockwise as viewed from above. 8. F(x, y, z) = i + (x + yz) j + (xy - z) k, C is the boundary of the part of the plane 3x + 2y + z = 1 in the first octant Use Stokes Theorem to evaluate c F dr. In each Case C is oriented counterclockwise as viewed from above. 9. F(x, y, z) = xy i + yz j + zx k, C is the boundary of the part of the paraboloid z = 1 - x2 - y2 in the first octant 10E (a) Use Stokes Theorem to evaluate c F dr, where F(x, y, z) = x2z i + xy2 j + z2 k and C is the curve of intersection of the plane x + y + z = 1 and the cylinder x2 + y2 = 9, oriented counterclockwise as viewed from above. (b) Graph both the plane and the cylinder with domains chosen so that you can see the curve C and the surface that you used in part (a). (c) Find parametric equations for C and use them to graph C.12E Verify that Stokes Theorem is true for the given vector field F and surface S. 13. F(x, y, z) = -y i + x j -2 k, S is the cone z2 = x2 + y2, 0 z 4, oriented downward Verify that Stokes Theorem is true for the given vector field F and surface S. 14. F(x, y, z) = -2yz i + y j + 3x k, S is the part of the paraboloid z = 5 - x2 - y2 that lies above the plane z = 1, oriented upward Verify that Stokes Theorem is true for the given vector field F and surface S. 15. F(x, y, z) = y i + z j + x k, S is the hemisphere x2 + y2 + z2 = 1, y 0, oriented in the direction of the positive y-axis A particle moves along line segments from the origin to the points (1, 0, 0), (1, 2, 1), (0, 2, 1), and back to the origin under the influence of the force field F(x, y, z) = z2 i + 2xy j + 4y2 k Find the work done.Evaluate c (y + sin x) dx + (z2 + cos y) dy + x3 dz where C is the curve r(t) = (sin t, cos t, sin 2t, 0 t 2. [Hint: Observe that C lies on the surface z = 2xy.]If S is a sphere and F satisfies the hypotheses of Stokes Theorem, show that s curl F dS = 0.20E Verify that the Divergence Theorem is true for the vector field F on the region E. 1. F(x, y, z) = 3x i + xy j + 2xz k, E is the cube bounded by the planes x =0, x = 1, y = 0, y = 1, z = 0, and z = 1 Verify that the Divergence Theorem is true for the vector field F on the region E. 2. F(x, y, z) = y2z3 i + 2yz j + 4z2 k, E is the solid enclosed by the paraboloid z = x2 + y2 and the plane z = 9 Verify that the Divergence Theorem is true for the vector field F on the region E. 3. F(x, y, z) = z, y, x, E is the solid ball x2 + y2 + z2 16 Verify that the Divergence Theorem is true for the vector field F on the region E. 4. F(x, y, z) = x2, -y, z, E is the solid cylinder y2 + z2 9, 0 x 2 Use the Divergence Theorem to calculate the surface integral s F dS; that is, calculate the flux of F across S. 5. F(x, y, z) = xye2 i + xy2z3 j - yez k, S is the surface of the box bounded by the coordinate planes and the planes x = 3, y = 2, and z = 1 Use the Divergence Theorem to calculate the surface integral s F dS; that is, calculate the flux of F across S. 6. F(x, y, z) = x2yz i + xy2z j + xyz2 k, S is the surface of the box enclosed by the planes x = 0, x = a, y = 0, y = b, z = 0, and z = c, where a, b, and c are positive numbers Use the Divergence Theorem to calculate the surface integral s F dS; that is, calculate the flux of F across S. 7. F(x, y, z) = 3xy2 i + xez j+ z3 k, S is the surface of the solid bounded by the cylinder y2 + z2 = 1 and the planes x = -1 and x = 2 Use the Divergence Theorem to calculate the surface integral s F dS; that is, calculate the flux of F across S. 8. F(x, y, z) = (x3 + y3) i + (y3 + z3) j + (z3 + x3) k, S is the sphere with center the origin and radius 2 Use the Divergence Theorem to calculate the surface integral s F dS; that is, calculate the flux of F across S. 9. F(x, y, z) = xey i + (z - ey) j - xy k, S is the ellipsoid x2 + 2y2 + 3z2 = 4 10E 11E 12E 13E Use the Divergence Theorem to calculate the surface integral s F dS; that is, calculate the flux of F across S. 14. F = |r|2r, where r = x i + y j+ z k, S is the sphere with radius R and center the origin 17E Let F(x, y, z) = z tan-1(y2) i + z3 ln(x2 + 1) j + z k. Find the flux of F across the part of the paraboloid x2 + y2 + z = 2 that lies above the plane z = 1 and is oriented upward.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 P1 and at P2.(a) Are the points P1 and P2 sources or sinks for the vector field F shown in the figure? Give an explanation based solely on the picture. (b) Given that F(x, y) = x, y2, use the definition of divergence to verify your answer to part (a).23E Use the Divergence Theorem to evaluate S(2x+2y+z2)dS where S is the sphere x2 + y2 + z2 = 1.25E 26E 27E 28E 29E Prove each identity, assuming that S and E satisfy the conditions of the Divergence Theorem and the scalar functions and components of the vector fields have continuous second-order partial derivatives. 30. S(fggf)ndS=E(f2gg2f)dV Suppose S and E satisfy the conditions of the Divergence Theorem and f is a scalar function with continuous partial derivatives. Prove that SfndS=EfdV These surface and triple integrals of vector functions are vectors defined by integrating each component function. [Hint: Start by applying the Divergence Theorem to F = fc, where c is an arbitrary constant vector.]32E What is a vector field? Give three examples that have physical meaning.2RCC 3RCC (a) Define the line integral of a vector field F along a smooth curve C given by a vector function r(t). (b) If F is a force field, what does this line integral represent? (c) If F = P,Q, R,what is the connection between the lineintegral of F and the line integrals of the component functionsP, Q,andR?5RCC (a) What does it mean to say that C F dris independent of path? (b) If you know thatC F dr is independent of path, what you say about F?7RCC 8RCC 9RCC 10RCC 11RCC 12RCC 13RCC 14RCC 15RCC In what ways are the Fundamental Theorem for Line Integrals, Greens Theorem, Stokes Theorem, and the Divergence Theorem similar?1RQ 2RQ 3RQ 4RQ 5RQ 6RQ 7RQ Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 8. The work done by a conservative force field in moving a particle around a closed path is zero.9RQ 10RQ 11RQ Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 12. There is a vector field F such that curl F = xi + yj + zk 13RQ 1RE Evaluate the line integral. 2. C x ds, C is the arc of the parabola y = x2 from (0, 0) to (1, 1)3RE Evaluate the line integral. 4. C y dx + (x + y2) dy, C is the ellipse 4x2 + 9y2 = 36 with counterclockwise orientation 5RE Evaluate the line integral. 6. C xy dx + ey dy + xz dz, C is given by r(t) = t4 i + t2 j + t3 k, 0 t 1 Evaluate the line integral. 7. C xy dx + y2 dy + yz dz, C is line segment from (1,0, 1), to (3,4, 2)8RE 9RE 10RE 11RE 12RE 13RE Show that F is a conservative and use this fact to evaluate C F dr along the given curve. 14. F(x, y, z) = ey i + (xey + ez) j + yez k, C is the line segment from (0, 2, 0) to (4, 0, 3)Verify that Greens Theorem is true for the line integral C xy2 dx x2y dy, where C consists of the parabola y = x2 from (-1, 1) to (1, 1) and the line segment from (1, 1) to (-1, 1).16RE 17RE 18RE 19RE If F and G are vector fields whose component functions have continuous first partial derivatives, show that curl(F G) = F div G G div F + (G )F (F )G 21RE If f and g are twice differentiable functions, show that 2(fg) = f 2 g + g2 f + 2f g If f is a harmonic function, that is, 2f = 0, show that the line integral fy dx fx dy is independent of path in any simple region D.24RE 25RE 27RE 28RE 29RE 30RE 31RE 32RE Use Stokes Theorem to evaluate C F dr, where F(x, y, z) = xy i + yz j + zx k, and C is the triangle with vertices (1, 0. 0), (0, 1, 0), and (0, 0, 1), oriented counterclockwise as viewed from above.34RE 35RE Compute the outward flux of F(x, y, z) = xi+yj+zk(x2+y2+z2)32 through the ellipsoid 4x2 + 9y2 + 6z2 = 36.Let F(x, y, z) = (3x2 yz 3y) i + (x3z 3x) j + (x3y + 2z) k Evaluate C F dr, where C is the curve with initial point (0, 0, 2) and terminal point (0, 3, 0) shown in the figure.38RE Find S F n dS, where F(x, y, z) = x i + y j + z k and S is the outwardly oriented surface shown in the figure (the boundary surface of a cube with a unit corner cube removed).40RE

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