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All Textbook Solutions for Calculus: Early Transcendentals
Evaluate the line integral, where C is the given curve. 3. C xy4 ds, C is the right half of the circle x2 + y2 = 16Evaluate the line integral, where C is the given curve. 4. C xey ds, C is the line segment from (2, 0) to (5, 4)Evaluate the line integral, where C is the given curve. 5. C (x2y + sin x) dy, C is the arc of the parabola y = x2 from (0, 0) to (, 2)Evaluate the line integral, where C is the given curve. 6. C ex dx, C is the arc of the curve x = y3 from (1, 1) to (1, 1)Evaluate the line integral, where C is the given curve. 7. C (x + 2y) dx + x2dy, C consists of line segments from (0, 0) to (2, 1) and from (2, 1) to (3, 0)Evaluate the line integral, where C is the given curve. 8. C x2 dx + y2dy, C consists of the arc of the circle x2 + y2 = 4 from (2, 0) to (0, 2) followed by the line segment from (0, 2) to (4, 3)Evaluate the line integral, where C is the given curve. 9. C x2 y ds, C: x = cos t, y = sin t, z = t, 0 t /2Evaluate the line integral, where C is the given curve. 10. C y2z dS, C is the line segment from (3, 1, 2) to (1, 2, 5)Evaluate the line integral, where C is the given curve. 11. C xeyz ds, C is the line segment from (0, 0, 0) to (1, 2, 3)Evaluate the line integral, where C is the given curve. 12. C(x2 + y2 + z2) ds, C: x = t, y = cos 2t, z = sin 2t, 0 t 2Evaluate the line integral, where C is the given curve. 13. C xyeyz dy, C: x = t, y = t2, z = t3, 0 t 1Evaluate the line integral, where C is the given curve. 14. C y dx + z dy + x dz, C: x = t, y = t, z = t2, 1 t 4Evaluate the line integral, where C is the given curve. 15. C z2 dx + x2 dy + y2 dz, C is the line segment from (1, 0, 0) to (4, 1, 2)Evaluate the line integral, where C is the given curve. 16. C (y + z) dx + (x + z) dy + (x + y) dz, C consists of line segments from (0, 0, 0) to (1, 0, 1) and from (1, 0, 1) to (0, 1, 2)Let F be the vector field shown in the figure. (a) If C1 is the vertical line segment from (3, 3) to (3, 3), determine whether c1Fdr is positive, negative, or zero. (b) If C2 is the counterclockwise-oriented circle with radius 3 and center the origin, determine whether c2Fdr is positive, negative, or zero.The figure shows a vector field F and two curves C1 and C2. Are the line integrals of F over C1, and C2 positive, negative, or zero? Explain.19E20EEvaluate the line integral C F dr, where C is given by the vector function r(t). 21. F(x, y, z) = sin x i + cos y j + xz k, r(t) = t3i t2j + t k, 0 t 1Evaluate the line integral C F dr, where C is given by the vector function r(t). 22. F(x, y, z) = x i + y j + xy k, r(t) = cos t i + sin t j + t k, 0 tUse a calculator to evaluate the line integral correct to four decimal places. 23. C F dr, where F(x,y)=x+yi+(y/x)j and r(t) = sin2 t i + sin t cos t j, /6 t /3Use a calculator to evaluate the line integral correct to four decimal places. 24. C F dr, where F(x, y, z) = yzex i + zxey j + xyez k and r(t) = sin t i + cos t j + tan t k, 0 t /4Use a calculator to evaluate the line integral correct to four decimal places. 25. C xy arctan z ds, where C has parametric equations x = t2, y = t3, z=t, 1 t 2Use a calculator to evaluate the line integral correct to four decimal places. 26. C z ln(x + y) ds, where C has parametric equations x = 1 + 3t, y = 2 + t2, z = t4, 1 t 131E(a) Find the work done by the force field F(x, y) = x2 i + xy j on a particle that moves once around the circle x2 + y2 = 4 oriented in the counterclockwise direction. (b) Use a computer algebra system to graph the force field and circle on the same screen. Use the graph to explain your answer to part (a).A thin wire is bent into the shape of a semicircle x2 + y2 = 4, x 0. If the linear density is a constant k, find the mass and center of mass of the wire.A thin wire has the shape of the first-quadrant part of the circle with center the origin and radius a. If the density function is (x, y) = kxy, find the mass and center of mass of the wire.35E36EIf a wire with linear density (x, y) lies along a plane curve C, its moments of inertia about the x- and y-axes are defined as Ix = C y2 (x, y) ds Iy = C x2 (x, y) ds Find the moments of inertia for the wire in Example 3.If a wire with linear density (x, y, z) lies along a space curve C, its moments of inertia about the x-, y-, and z-axes are defined as Ix = C (y2 + z2) (x, y, z) ds Iy = C (x2 + z2) (x, y, z) ds Iz = C (x2 + y2) (x, y, z) ds Find the moments of inertia for the wire in Exercise 35. 35. (a) Write the formulas similar to Equations 4 for the center of mass (x,y,z) of a thin wire in the shape of a space curve C if the wire has density function (x, y, z). (b) Find the center of mass of a wire in the shape of the helix x = 2 sin t, y = 2 cos t, z = 3t, 0 t 2, if the density is a constant k.Find the work done by the force field F(x, y) = x i + (y + 2) j in moving an object along an arch of the cycloid r(t) = (t sin t) i + (1 cos t) j 0 t 2Find the work done by the force field F(x, y) = x2 i + yex j on a particle that moves along the parabola x = y2 + 1 from (1, 0) to (2, 1).Find the work done by the force field F(x, y, z) = x y2, y z2, z x2 on a particle that moves along the line segment from (0, 0, 1) to (2, 1, 0).The force exerted by an electric charge at the origin on a charged particle at a point (x, y, z) with position vector r = x, y, z is F(r) = Kr/| r |3 where K is a constant. (See Example 16.1.5.) Find the work done as the particle moves along a straight line from (2, 0, 0) to (2, 1, 5).The position of an object with mass m at time t is r(t) = at2 i + bt3j, 0 t 1. (a) What is the force acting on the object at time t? (b) What is the work done by the force during the time interval 0 t 1?44EA 160-lb man carries a 25-lb can of paint up a helical staircase that encircles a silo with a radius of 20 ft. If the silo is 90 ft high and the man makes exactly three complete revolutions climbing to the top, how much work is done by the man against gravity?46E(a) Show that a constant force field does zero work on a particle that moves once uniformly around the circle x2 + y2 = 1. (b) Is this also true for a force field F(x) = kx, where k is a constant and x = x, y?48E49E50EAn object moves along the curve C shown in the figure from (1, 2) to (9, 8). The lengths of the vectors in the force field F are measured in newtons by the scales on the axes. Estimate the work done by F on the object.Experiments show that a steady current I in a long wire produces a magnetic field B that is tangent to any circle that lies in the plane perpendicular to the wire and whose center is the axis of the wire (as in the figure). Ampres Law relates the electric current to its magnetic effects and states that C B dr = 0I where I is the net current that passes through any surface bounded by a closed curve C, and 0 is a constant called the permeability of free space. By taking C to be a circle with radius r, show that the magnitude B = |B| of the magnetic field at a distance r from the center of the wire is B=0I2rThe figure shows a curve C and a contour map of a function f whose gradient is continuous. Find C f dr.A table of values of a function f with continuous gradient is given. Find C f dr, where C has parametric equations x = t2 + 1 y = t3 + t 0 t 1Determine whether or not F is a conservative vector field. If it is, find a function f such that F = f. 3. F(x, y) = (xy + y2) i + (x2 + 2xy) jDetermine whether or not F is a conservative vector field. If it is, find a function f such that F = f. 4. F(x, y) = (y2 2x) i + 2xy jDetermine whether or not F is a conservative vector field. If it is, find a function f such that F = f. 5. F(x, y) = y2exy i + (1 + xy)exy jDetermine whether or not F is a conservative vector field. If it is, find a function f such that F = f. 6. F(x, y) = yex i + (ex + ey) jDetermine whether or not F is a conservative vector field. If it is, find a function f such that F = f. 7. F(x, y) = (yex + sin y) i + (ex + x cos y) jDetermine 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 0Determine 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) jDetermine whether or not F is a conservative vector field. If it is, find a function f such that F = f. 10. F(x, y) = (ln y + y/x) i + (ln x + x/y) jThe 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)16E17E(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 /2Show 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, )Suppose youre asked to determine the curve that requires the least work for a force field F to move a particle from one point to another point. You decide to check first whether F is conservative, and indeed it turns out that it is. How would you reply to the request?Suppose an experiment determines that the amount of work required for a force field F to move a particle from the point (1, 2) to the point (5, 3) along a curve C1 is 1.2 J and the work done by F in moving the particle along another curve C2 between the same two points is 1.4 J. What can you say about F? Why?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 = 1Show that if the vector field F = P i + Q j + R k is conservative and P, Q, R have continuous first-order partial derivatives, then Py=QxPz=RxQz=RyUse Exercise 29 to show that the line integral C y dx + x dy + xyz dz is not independent of path.Determine whether or not the given set is (a) open, (b) connected, and (c) simply-connected. 31. {(x, y) | 0 y 3}Determine whether or not the given set is (a) open, (b) connected, and (c) simply-connected. 32. {(x, y) | 1 |x| 2}33EDetermine whether or not the given set is (a) open, (b) connected, and (c) simply-connected. 34. {(x, y) | (x, y) (2, 3)}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 4Evaluate 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 = y2Use 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 = 2Use 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 = 4Use 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 = 9Use 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 clockwiseUse 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)17EA 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.20E(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=3Cx3dy26EUse 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 kFind (a) the curl and (b) the divergence of the vector field. 2. F(x, y, z) = x3yz2 j + y4z3 kFind (a) the curl and (b) the divergence of the vector field. 3. F(x, y, z) = xyez i + yzex kFind (a) the curl and (b) the divergence of the vector field. 4. F (x, y, z) = sin yz i + sin zx j + sin xy kFind (a) the curl and (b) the divergence of the vector field. 5. x, y, z) = i + j + k6EFind (a) the curl and (b) the divergence of the vector field. 7. F(x, y, z) = ex sin y, ey sin z, ez sin xFind (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))Determine whether or not the vector field is conservative. If it is conservative, find a function f such that F = f. 13. F(x, y, z) = y2z3 i + 2xyz3 j + 3xy2z2 kDetermine 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 kDetermine 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 kDetermine 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 kDetermine 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 kDetermine 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 kIs 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 G24EProve 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 fProve 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) FProve 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 GProve 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) = 0Prove 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) 2FLet r = x i + y j + z k and r = |r|. 30. Verify each identity. (a) r = 3 (b) (rr) = 4r (c) 2r3 = 12r31ELet 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.)34E35E36E37EMaxwells 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=1c22Ht239EDetermine 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) k4E5E6EMatch 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 kMatch 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) kMatch 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 kMatch 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 v17EMatch 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 vFind a parametric representation for the surface. 19. The plane through the origin that contains the vectors i j and j k20EFind a parametric representation for the surface. 21. The part of the hyperboloid 4x2 4y2 z2 = 4 that lies in front of the yz-planeFind a parametric representation for the surface. 22. The part of the ellipsoid x2 + 2y2 + 3z2 = 1 that lies to the left of the xz-planeFind a parametric representation for the surface. 23. The part of the sphere x2 + y2 + z2 = 4 that lies above the cone z=x2+y2Find 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 = 4Find a parametric representation for the surface. 25. The part of the cylinderx2+ y2+ z2 =36 that lies between the planes z = 0 and z = 3Find a parametric representation for the surface. 26. The part of the plane z = x + 3 that lies inside the cylinder x2 + y2 = 129EFind 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.Find an equation of the tangent plane to the given parametric surface at the specified point. 33. x = u + v, y = 3u2, z = u v; (2, 3, 0)Find an equation of the tangent plane to the given parametric surface at the specified point. 34.x=u2+ 1,y=v3+1,z = u + v;(5, 2, 3)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=/3Find an equation of the tangent plane to the given parametric surface at the specified point. 36. r(u, v)=sinui+ cosusinv j+ sinvk;u=/6,v =/637EFind an equation of the tangent plane to the given parametric surface at the specified point. Graph the surface andthe tangent plane. 38. r(u, v)=(1u2 v2) i vj uk; (1, 1, 1)Find the area of the surface. 39. The part of the plane 3x +2y+ z = 6 that lies in the first octant40EFind the area of the surface. 41. The part of the planex+2y + 3z= 1 that lies inside the cylinderx2+ y2=3Find the area of the surface. 42. The part of the conez = that lies between the planey=xand the cylindery=x2Find the area of the surface. 43. The surface z = (x3/2+ y3/2), 0 x 1, 0 y 1Find the area of the surface. 44. The part of the surface z =4 2x2+ ythat lies above thetriangle with vertices (0, 0), (1, 0), and (1, 1)Find the area of the surface. 45. The part of the surfacez = xythat lies within thecylinderx2+y2= 146EFind the area of the surface. 47. The part of the paraboloidy=x2+ z2that lies within thecylinderx2+ z2= 16Find 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 vFind the area of the surface. 49. The surface with parametric equationsx =u2, y=uv, z =v2,0 u 1, 0 v 2Find the area of the surface. 50. The part of the spherex2+ y2+z2=b2that lies inside thecylinderx2+ y2=a2,where 0 a b51EFind 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= 1Find 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 154EFind the area of the surface with vector equation r(u,v)= cos3u cos3v, sin3u cos3v, sin3v, 0 u , 0 v 2. State your answer correct to four decimalplaces.59E(a) Show that the parametric equationsx = acosh u cosv,y =bcoshusinv, z= c sinh u, represent a hyperboloid of one sheet. (b) Use the parametric equations in part (a) to graph thehyperboloid for the casea= 1,b= 2,c =3. (c) Set up, but do not evaluate, a double integral for the surface area of the part of the hyperboloid in part (b) thatlies between the planesz= 3 and z = 3.Find the area of the part of the spherex2+y2+ z2= 4zthat lies inside the paraboloidz=x2+y2.The figure shows the surface created when the cylindery2+z2= 1 intersects the cylinderx2+ z2 = 1. Find thearea of this surface.Find the area of the part of the spherex2+y2+ z2 = a2 that lies inside the cylinderx2+y2=ax.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.A surface S consists of the cylinderx2+ y2=1, 1 z 1, together with its top and bottom disks. Suppose you know thatf is a continuous function with f(1, 0, 0) =2 f(0, 1, 0) =3 f(0, 0, 1) =4 Estimate the value of S f(x, y, z) dSby using a Riemann sum, taking the patchesSijto be four quarter-cylinders and the topand bottom disks.3ESuppose thatf(x, y,z)=g(), where g is a function of one variable such thatg(2) =5. Evaluate S f(x, y,z) dS, whereSis the spherex2+y2+ z2= 4.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 1Evaluate 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 /2Evaluate the surface integral. 7. s y dS,Sis the helicoid with vector equation r(u,v) = ucosv, usinv, v,0 u 1, 0 vEvaluate the surface integral. 8.s (x2+ y2)dS, Sis the surface with vector equation r(u,v)=2uv, u2 v2, u2+v2,u2+ v2 1Evaluate 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 octantEvaluate 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),0x1,0y1Evaluate the surface integral. 13. s z2dS, S is the part of the paraboloid x = y2 + z2 given by 0 x 1Evaluate the surface integral. 14. s y2z2 dS, S is the part of the cone y=x2+z2 given by 0 y 515EEvaluate 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+y2Evaluate the surface integral. 17. s (x2z + y2z)dS, S is the hemisphere x2 + y2 + z2 = 4, z 0Evaluate 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 = 2Evaluate 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 = 5Evaluate 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 disksEvaluate 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 orientationEvaluate 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 orientationEvaluate 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 orientationEvaluate 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 orientationEvaluate 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 originEvaluate 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 downwardEvaluate 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 orientationFind 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.43E44EUse Gausss Law to find the charge contained in the solid hemisphere x2 + y2 + z2 a2, z 0, if the electric field is E(x, y, z) = x i + y j + 2z kUse Gausss Law to find the charge enclosed by the cube with vertices (1, 1, 1) if the electric field is E(x, y, z) = x i + y j + z kThe 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 4.48ELet F be an inverse square field, that is, |F(r) = cr/|r|3 for some constant c, where r = x i + y j + z k. Show that the flux of F across a sphere S with center the origin is independent of the radius of S.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=PcurlFdSUse 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 upwardUse 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-axisUse Stokes Theorem to evaluate s curl F dS. 4. F(x, y, z) = tan-1(x2yz2) i + x2y j + x2z2 k. S is the cone x=x2+z2,0x2, oriented in the direction of the positive x-axisF(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 outwardUse 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-axisUse 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 octantUse 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 octantUse Stokes Theorem to evaluate c F dr. In each Case C is oriented counterclockwise as viewed from above. 10. F(x, y, z) = 2y i + xz j + (x + y) k, C is the curve of intersection of the plane z = y + 2 and the cylinder x2 + y2 = l(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.(a) Use Stokes Theorem to evaluate c F dr, where F(x,y,z)=x2yi+13x3j+xykand C is the curve of intersection of the hyperbolic paraboloid z = y2 - x2 and the cylinder x2 + y2 = l, oriented counterclockwise as viewed from above. (b) Graph both the hyperbolic paraboloid 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.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 downwardVerify 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 upwardVerify 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-axisA 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.Suppose S and C satisfy the hypotheses of Stokes Theorem and f, g have continuous second-order partial derivatives. Use Exercises 24 and 26 in Section 16.5 to show the following. (a) c(f g) dr = s (f g) dS (b) c(ff) dr = 0 (c) c (fg + gf) dr = 0Verify 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 = 1Verify 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 = 9Verify 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 16Verify 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 2Use 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 = 1Use 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 numbersUse 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 = 2Use 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 2Use 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 = 4Use the Divergence Theorem to calculate the surface integral s F dS; that is, calculate the flux of F across S. 10. F(x, y, z) = z i + y j + zx k, S is the surface of the tetrahedron enclosed by the coordinate planes and the plane xa+yb+zc=1 where a, b, c and c are positive numbersUse the Divergence Theorem to calculate the surface integral s F dS; that is, calculate the flux of F across S. 11. F(x, y, z) = (2x3 + y3) i + (y3 + z3)j + 3y2z k, S is the surface of the solid bounded by the paraboloid z = 1 - x2 - y2 and the xy-planeUse the Divergence Theorem to calculate the surface integral s F dS; that is, calculate the flux of F across S. 12. F(x, y, z) = (xy + 2xz) i + (x2 + y2) j + (xy - z2) k, S is the surface of the solid bounded by the cylinder x2 + y2 = 4 and the planes z = y - 2 and z = 0Use the Divergence Theorem to calculate the surface integral s F dS; that is, calculate the flux of F across S. 13. F = |r| r, where r = x i + y j + z k, S consists of the hemisphere z=1x2+y2 and the disk x2 + y2 1 in the xy-plane14EUse the Divergence Theorem to evaluate s F dS, where F(x,y,z)=z2xi+(13y3+tanz)j+(x2z+y2)k and S is the top half of the sphere x2 + y2 + z2 = 1. [Hint: Note that S is not a closed surface. First compute integrals over S1 and S2, where S1 is the disk x2 + y2 1, oriented downward, and S2 = S S1.]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).Verify that div E = 0 for the electric field E(x)=Q|x|3x.24EProve 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. 25. SandS=0,where a is a constant vector26E27EProve 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. 28. SDnfdS=E2fdV29EProve 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)dV31E32EWhat is a vector field? Give three examples that have physical meaning.2RCC3RCC(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?5RCC6RCC7RCCWrite expressions for the area enclosed by a curve C in terms of line integrals around C.9RCC10RCC11RCC12RCC13RCC14RCC15RCC