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All Textbook Solutions for Calculus: Early Transcendentals (3rd Edition)
78EGeneral volume formulas Use integration to find the volume of the following solids. In each case, choose a convenient coordinate system, find equations for the bounding surfaces, set up a triple integral, and evaluate the integral. Assume that a, b, c, r, R, and h are positive constants. 79.Frustum of a cone Find the volume of a truncated solid cone of height h whose ends have radii r and R.80EIntersecting spheres One sphere is centered at the origin and has a radius of R. Another sphere is centered at (0, 0, r) and has a radius of r, where r R/2. What is the volume of the region common to the two spheres?A 90-kg person sits 2 m from the balance point of a seesaw. How far from that point must a 60-kg person sit to balance the seesaw? Assume the seesaw has no mass.
Solve the equation m1(x1x)+m2(x2x)=0 for x to verify the preceding expression for the center of mass.3QCExplain why the integral for My has x in the integrand. Explain why the density drops out of the center-of-mass calculation if it is constant.
5QCExplain how to find the balance point for two people on opposite ends of a (massless) plank that rests on a pivot.If a thin 1-m cylindrical rod has a density of = 1 g/cm for its left half and a density of = 2 g/cm for its right half, what is its mass and where is its center of mass?Explain how to find the center of mass of a thin plate with a variable densityIn the integral for the moment Mx of a thin plate, why does y appear in the integrand?Explain how to find the center of mass of a three-dimensional object with a variable density.In the integral for the moment Mxz with respect to the xz-plane of a solid, why does y appear in the integrand?Individual masses on a line Sketch the following systems on a number line and find the location of the center of mass. 7.m1 = 10 kg located at x = 3 m; m2 = 3 kg located at x = 1 mIndividual masses on a line Sketch the following systems on a number line and find the location of the center of mass. 8.m1 = 8 kg located at x = 2 m; m2 = 4 kg located at x = 4 m; m3 = 1 kg located at x = 0 mOne-dimensional objects Find the mass and center of mass of the thin rods with the following density functions. 9.(x) = 1 + sin x, for 0 xOne-dimensional objects Find the mass and center of mass of the thin rods with the following density functions. 10.(x) = 1 + x3, for 0 x 1One-dimensional objects Find the mass and center of mass of the thin rods with the following density functions. 11.(x)=2x2/16,for0x4One-dimensional objects Find the mass and center of mass of the thin rods with the following density functions. 12.(x)=2+cosx,for0xOne-dimensional objects Find the mass and center of mass of the thin rods with the following density functions. 13.(x)={1if0x21+xif2x414ECentroid calculations Find the mass and centroid (center of mass) of the following thin plates, assuming constant density. Sketch the region corresponding to the plate and indicate the location of the center of mass. Use symmetry when possible to simplify your work. 15.The region bounded by y = sin x and y = 1 sin x between x = /4 and x = 3/4Centroid calculations Find the mass and centroid (center of mass) of the following thin plates, assuming constant density. Sketch the region corresponding to the plate and indicate the location of the center of mass. Use symmetry when possible to simplify your work. 16.The region in the first quadrant bounded by x2 + y2 = 16Centroid calculations Find the mass and centroid (center of mass) of the following thin plates, assuming constant density. Sketch the region corresponding to the plate and indicate the location of the center of mass. Use symmetry when possible to simplify your work. 17.The region bounded by y = 1 |x| and the x-axisCentroid calculations Find the mass and centroid (center of mass) of the following thin plates, assuming constant density. Sketch the region corresponding to the plate and indicate the location of the center of mass. Use symmetry when possible to simplify your work. 18.The region bounded by y = ex, y = ex, x = 0, and x = ln 2Centroid calculations Find the mass and centroid (center of mass) of the following thin plates, assuming constant density. Sketch the region corresponding to the plate and indicate the location of the center of mass. Use symmetry when possible to simplify your work. 19.The region bounded by y = ln x, the x-axis, and x = eCentroid calculations Find the mass and centroid (center of mass) of the following thin plates, assuming constant density. Sketch the region corresponding to the plate and indicate the location of the center of mass. Use symmetry when possible to simplify your work. 20.The region bounded by x2 + y2 = 1 and x2 + y2 = 9, for y 0Variable-density plates Find the center of mass of the following plane regions with variable density. Describe the distribution of mass in the region. 21.R = {(x, y): 0 x 4, 0 y 2}; (x, y) = 1 + x/2Variable-density plates Find the center of mass of the following plane regions with variable density. Describe the distribution of mass in the region. 22.R = {(x, y): 0 x 1, 0 y 5}; (x, y) = 2ey/2Variable-density plates Find the center of mass of the following plane regions with variable density. Describe the distribution of mass in the region. 23.The triangular plate in the first quadrant bounded by x + y = 4 with (x, y) = 1 + x + yVariable-density plates Find the center of mass of the following plane regions with variable density. Describe the distribution of mass in the region. 24.The upper half (y 0) of the disk bounded by the circle x2 + y2 = 4 with (x, y) = 1 + y/2Variable-density plates Find the center of mass of the following plane regions with variable density. Describe the distribution of mass in the region. 25.The upper half (y 0) of the plate bounded by the ellipse x2 + 9y2 = 9 with (x, y) = 1 + yVariable-density plates Find the center of mass of the following plane regions with variable density. Describe the distribution of mass in the region. 26.The quarter disk in the first quadrant bounded by x2 + y2 = 4 with (x, y) = 1 + x2 + y2Center of mass of constant-density solids Find the center of mass of the following solids, assuming a constant density of 1. Sketch the region and indicate the location of the centroid. Use symmetry when possible and choose a convenient coordinate system. 27.The upper half of the ball x2 + y2 + z2 16 (for z 0)Center of mass of constant-density solids Find the center of mass of the following solids, assuming a constant density of 1. Sketch the region and indicate the location of the centroid. Use symmetry when possible and choose a convenient coordinate system. 28.The solid bounded by the paraboloid z = x2 + y2 and the plane z = 25Center of mass of constant-density solids Find the center of mass of the following solids, assuming a constant density of 1. Sketch the region and indicate the location of the centroid. Use symmetry when possible and choose a convenient coordinate system. 29.The tetrahedron in the first octant bounded by z = 1 x y and the coordinate planes.Center of mass of constant-density solids Find the center of mass of the following solids, assuming a constant density of 1. Sketch the region and indicate the location of the centroid. Use symmetry when possible and choose a convenient coordinate system. 30.The solid bounded by the cone z = 16 r and the plane z = 0Center of mass of constant-density solids Find the center of mass of the following solids, assuming a constant density of 1. Sketch the region and indicate the location of the centroid. Use symmetry when possible and choose a convenient coordinate system. 31.The sliced solid cylinder bounded by x2 + y2 = 1, z = 0, and y + z = 1Center of mass of constant-density solids Find the center of mass of the following solids, assuming a constant density of 1. Sketch the region and indicate the location of the centroid. Use symmetry when possible and choose a convenient coordinate system. 32.The sliced solid cylinder bounded by x2 + y2 = 1, z = 0, and y + z = 1Variable-density solids Find the coordinates of the center of mass of the following solids with variable density. 33.R = {(x, y, z): 0 x 4, 0 y 1, 0 z 1}; (x, y, z) = 1 + x/2Variable-density solids Find the coordinates of the center of mass of the following solids with variable density. 34.The solid bounded by paraboloid z = 4 x2 y2 and z = 0 with (x, y, z) = 5 zVariable-density solids Find the coordinates of the center of mass of the following solids with variable density. 35.The solid bounded by the upper half of the sphere = 6 and z = 0 with density f(, , ) = 1 + /4Variable-density solids Find the coordinates of the center of mass of the following solids with variable density. 36.The interior of the cube in the first octant formed by the planes x = 1, y = l, and z = 1 with (x, y, z) = 2 + x + y + zVariable-density solids Find the coordinates of the center of mass of the following solids with variable density. 37.The interior of the prism formed by z = x, x = 1, y = 4, and the coordinate planes with (x, y, z) = 2 + yVariable-density solids Find the coordinates of the center of mass of the following solids with variable density. 38.The solid bounded by the cone by z = 9 r and z = 0 with (r, , z) = 1 + zExplain why or why not Determine whether the following statements are true and give an explanation or counterexample. a.A thin plate of constant density that is symmetric about the x-axis has a center of mass with an x-coordinate of zero. b.A thin plate of constant density that is symmetric about both the x-axis and the y-axis has its center of mass at the origin. c.The center of mass of a thin plate must lie on the plate. d.The center of mass of a connected solid region (all in one piece) must lie within the region.Limiting center of mass A thin rod of length L has a linear density given by (x) = 2ex/3 on the interval 0 x L. Find the mass and center of mass of the rod. How does the center of mass change as L ?Limiting center of mass A thin rod of length L has a linear density given by (x)=101+x2on the interval 0 x L. Find the mass and center of mass of the rod. How does the center of mass change as L ?42ETwo-dimensional plates Find the mass and center of mass of the thin constant-density plates shown in the figure. 43.Two-dimensional plates Find the mass and center of mass of the thin constant-density plates shown in the figure. 44.Centroids Use polar coordinates to find the centroid of the following constant-density plane regions. 45.The semicircular disk R = {(r, ): 0 r 2, 0 }46ECentroids Use polar coordinates to find the centroid of the following constant-density plane regions. 47.The region bounded by the cardioid r = 1 + cosCentroids Use polar coordinates to find the centroid of the following constant-density plane regions. 48.The region bounded by the cardioid r = 3 3 cos49E50E51E52E53E54ECenters of mass for general objects Consider the following two- and three-dimensional regions. Specify the surfaces and curves that bound the region, choose a convenient coordinate system, and compute the center of mass assuming constant density. All parameters are positive real numbers. 55.A solid cone has a base with a radius of a and a height of h. How far from the base is the center of mass?Centers of mass for general objects Consider the following two- and three-dimensional regions. Specify the surfaces and curves that bound the region, choose a convenient coordinate system, and compute the center of mass assuming constant density. All parameters are positive real numbers. 56.A solid is enclosed by a hemisphere of radius a. How far from the base is the center of mass?57ECenters of mass for general objects Consider the following two- and three-dimensional regions. Specify the surfaces and curves that bound the region, choose a convenient coordinate system, and compute the center of mass assuming constant density. All parameters are positive real numbers. 58.A tetrahedron is bounded by the coordinate planes and the plane x/a + y/a + z/a = 1. What are the coordinates of the center of mass?Centers of mass for general objects Consider the following two- and three-dimensional regions. Specify the surfaces and curves that bound the region, choose a convenient coordinate system, and compute the center of mass assuming constant density. All parameters are positive real numbers. 59.A solid is enclosed by the upper half of an ellipsoid with a circular base of radius r and a height of a. How far from the base is the center of mass?Geographic vs. population center Geographers measure the geographical center of a country (which is the centroid) and the population center of a country (which is the center of mass computed with the population density). A hypothetical country is shown in the figure with the location and population of five towns. Assuming no one lives outside the towns, find the geographical center of the country and the population center of the country.Center of mass on the edge Consider the thin constant-density plate {(r, ): 0 a r 1, 0 } bounded by two semicircles and the x-axis. a.Find and graph the y-coordinate of the center of mass of the plate as a function of a. b.For what value of a is the center of mass on the edge of the plate?Center of mass on the edge Consider the constant-density solid {(, , ): 0 a 1, 0 /2, 0 2 } bounded by two hemispheres and the xy-plane. a.Find and graph the z-coordinate of the center of mass of the plate as a function of a. b.For what value of a is the center of mass on the edge of the solid?Draining a soda can A cylindrical soda can has a radius of 4 cm and a height of 12 cm. When the can is full of soda, the center of mass of the contents of the can is 6 cm above the base on the axis of the can (halfway along the axis of the can). As the can is drained, the center of mass descends for a while. However, when the can is empty (filled only with air), the center of mass is once again 6 cm above the base on the axis of the can. Find the depth of soda in the can for which the center of mass is at its lowest point. Neglect the mass of the can and assume the density of the soda is 1 g/cm3 and the density of air is 0.001 g/cm3.Triangle medians A triangular region has a base that connects the vertices (0, 0) and (b, 0), and a third vertex at (a, h), where a 0, b 0, and h 0. a.Show that the centroid of the triangle is (a+b3,h3). b.Recall that the three medians of a triangle extend from each vertex to the midpoint of the opposite side. Knowing that the medians of a triangle intersect in a point M and that each median bisects the triangle, conclude that the centroid of the triangle is M.The golden earring A disk of radius r is removed from a larger disk of radius R to form an earring (see figure). Assume the earring is a thin plate of uniform density. a.Find the center of mass of the earring in terms of r and R. (Hint: Place the origin of a coordinate system either at the center of the large disk or at Q: either way, the earring is symmetric about the x-axis.) b.Show that the ratio R/r such that the center of mass lies at the point P (on the edge of the inner disk) is the golden mean (1+5)/21.618. (Source: P. Glaister, Golden Earrings, Mathematical Gazette, 80, 1996)1QC2QCSolve the equations u = x + y, v = –x + 2y for x and y.
4QCInterpret a Jacobian with a value of 1 (as in Example 6).
Example 6 A triple integral
Use a change of variables to evaluate , where D is a parallelepiped bounded by the planes
y = x, y = x + 2, z = x, z = x + 3, z = 0, and z = 4
(Figure 16.83)
Suppose S is the unit square in the first quadrant of the uv-plane. Describe the image of the transformation T: x = 2u, y = 2v.Explain how to compute the Jacobian of the transformation T: x = g(u, v), y = h(u, v).Using the transformation T: x = u + v, y = u v, the image of the unit square S = {(u, v): 0 u 1, 0 v 1} is a region R in the xy-plane. Explain how to change variables in the integral Rf(x, y) dA to find a new integral over S.Suppose S is the unit cube in the first octant of uvw-space with one vertex at the origin. What is the image of the transformation T:x = u/2, y = v/2, z = w/2?Transforming a square Let S = {(u, v): 0 u l, 0 v 1} be a unit square in the uv-plane. Find the image of S in the xy-plane under the following transformations. 5.T: x = 2u, y = v/2Transforming a square Let S = {(u, v): 0 u l, 0 v 1} be a unit square in the uv-plane. Find the image of S in the xy-plane under the following transformations. 6.T: x = u, y = vTransforming a square Let S = {(u, v): 0 u l, 0 v 1} be a unit square in the uv-plane. Find the image of S in the xy-plane under the following transformations. 7.T: x = (u + v)/2, y = (u v)/2Transforming a square Let S = {(u, v): 0 u l, 0 v 1} be a unit square in the uv-plane. Find the image of S in the xy-plane under the following transformations. 8.T: x = 2u + v, y = 2uTransforming a square Let S = {(u, v): 0 u l, 0 v 1} be a unit square in the uv-plane. Find the image of S in the xy-plane under the following transformations. 9.T: x = u2 v2, y = 2uvTransforming a square Let S = {(u, v): 0 u l, 0 v 1} be a unit square in the uv-plane. Find the image of S in the xy-plane under the following transformations. 10.T: x = 2uv, y = u2 v2Transforming a square Let S = {(u, v): 0 u l, 0 v 1} be a unit square in the uv-plane. Find the image of S in the xy-plane under the following transformations. 11.T: x = u cos v, y = u sin vTransforming a square Let S = {(u, v): 0 u l, 0 v 1} be a unit square in the uv-plane. Find the image of S in the xy-plane under the following transformations. 12.T: x = v sin u, y = v cos uImages of regions Find the image R in the xy-plane of the region S using the given transformation T. Sketch both R and S. 13.S = {(u, v): v 1 u, u 0, v 0}; T: x = u, y = v2Images of regions Find the image R in the xy-plane of the region S using the given transformation T Sketch both R and S. 14.S = {(u, v): u2 + v2 1}; T: x = 2u, y = 4vImages of regions Find the image R in the xy-plane of the region S using the given transformation T. Sketch both R and S. 15.S= {(u, v): 1 u 3, 2 v 4}; T: x = u/v, y = vImages of regions Find the image R in the xy-plane of the region S using the given transformation T. Sketch both R and S. 16.S = {(u, v): 2 u 3, 3 v 6}; T: x = u, y = v/uComputing Jacobians Compute the Jacobian J(u, v) for the following transformations. 17.T: x = 3u, y = 3vComputing Jacobians Compute the Jacobian J(u, v) for the following transformations. 18.T: x = 4v, y = 2uComputing Jacobians Compute the Jacobian J(u, v) for the following transformations. 19.T: x = 2uv, y = u2 v2Computing Jacobians Compute the Jacobian J(u, v) for the following transformations. 20.T: x = u cos v, y = u sin vComputing Jacobians Compute the Jacobian J(u, v) for the following transformations. 21.T:x=(u+v)/2,y=(uv)/2Computing Jacobians Compute the Jacobian J(u, v) for the following transformations. 22.T: x = u/v, y = vSolve and compute Jacobians Solve the following relations for x and y, and compute the Jacobian J(u, v). 23.u = x + y, v = 2x ySolve and compute Jacobians Solve the following relations for x and y, and compute the Jacobian J(u, v). 24.u = xy, v = xSolve and compute Jacobians Solve the following relations for x and y, and compute the Jacobian J(u, v). 25.u = 2x 3y, v = y xSolve and compute Jacobians Solve the following relations for x and y, and compute the Jacobian J(u, v). 26.u = x + 4y, v = 3x + 2yDouble integralstransformation given To evaluate the following integrals, carry out these steps. a.Sketch the original region of integration R in the xy-plane and the new region S in the uv-plane using the given change of variables. b.Find the limits of integration for the new integral with respect to u and v. c.Compute the Jacobian. d.Change variables and evaluate the new integral. 27.RxydA. where R is the square with vertices (0, 0), (1, 1), (2, 0), and (1, 1); use x = u + v, y = u v.Double integralstransformation given To evaluate the following integrals, carry out these steps. a.Sketch the original region of integration R in the xy-plane and the new region S in the uv-plane using the given change of variables. b.Find the limits of integration for the new integral with respect to u and v. c.Compute the Jacobian. d.Change variables and evaluate the new integral. 28.Rx2ydA, where R = {(x, y): 0 x 2, x y x + 4}; use x = 2u, y = 4v + 2u.Double integralstransformation given To evaluate the following integrals, carry out these steps. a.Sketch the original region of integration R in the xy-plane and the new region S in the uv-plane using the given change of variables. b.Find the limits of integration for the new integral with respect to u and v. c.Compute the Jacobian. d.Change variables and evaluate the new integral. 29.Rx2x+2ydA, where R = {(x, y): 0 x 2, x/2 y 1 x}; use x = 2u, y = v u.Double integralstransformation given To evaluate the following integrals, carry out these steps. a.Sketch the original region of integration R in the xy-plane and the new region S in the uv-plane using the given change of variables. b.Find the limits of integration for the new integral with respect to u and v. c.Compute the Jacobian. d.Change variables and evaluate the new integral. 30.RxydA, where R is bounded by the ellipse 9x2 + 4y2 = 36; use x = 2u, y = 3v.Double integralsyour choice of transformation Evaluate the following integrals using a change of variables. Sketch the original and new regions of integration, R and S. 31.01yy+2xydxdyDouble integralsyour choice of transformation Evaluate the following integrals using a change of variables. Sketch the original and new regions of integration, R and S. 32.Ry2x2dA, where R is the diamond bounded by y x = 0, y x = 2, y + x = 0, and y + x = 2Double integralsyour choice of transformation Evaluate the following integrals using a change of variables. Sketch the original and new regions of integration, R and S. 33.R(yxy+2x+1)4dA, where R is the parallelogram bounded by y x = 1, y x = 2, y + 2x = 0, and y + 2x = 4Double integralsyour choice of transformation Evaluate the following integrals using a change of variables. Sketch the original and new regions of integration, R and S. 34.RexydA, where R is the region bounded by the hyperbolas xy = 1 and xy = 4, and the lines y/x = 1 and y/x = 3Double integralsyour choice of transformation Evaluate the following integrals using a change of variables. Sketch the original and new regions of integration, R and S. 35.RxydA, where R is the region bounded by the hyperbolas xy = 1 and xy = 4, and the lines y = 1 and y = 3Double integralsyour choice of transformation Evaluate the following integrals using a change of variables. Sketch the original and new regions of integration, R and S. 36.R(xy)x2ydA, where R is the triangular region bounded by y = 0, x 2y = 0, and x y = 1Jacobians in three variables Evaluate the Jacobians J(u, v, w) for the following transformations. 37.x = v + w, y = u + w, z = u + v38EJacobians in three variables Evaluate the Jacobians J(u, v, w) for the following transformations. 39.x = vw, y = uw, z = u2 v2Jacobians in three variables Evaluate the Jacobians J(u, v, w) for the following transformations. 40.u = x y, v = x z, w = y + z (Solve for x, y, and z first.)Triple integrals Use a change of variables to evaluate the following integrals. 41.DxydV; D is bounded by the planes y x =0, y x =2, z y = 0, z y = 1, z = 0, and z = 3.Triple integrals Use a change of variables to evaluate the following integrals. 42.DdV; D is bounded by the planes y 2x = 0, y 2x = 1, z 3y = 0, z 3y = 1, z 4x = 0, and z 4x = 3.Triple integrals Use a change of variables to evaluate the following integrals. 43.DzdV; D is bounded by the paraboloid z = 16 x2 4y2 and the xy-plane. Use x = 4u cos v, y = 2u sin v, z = w.Triple integrals Use a change of variables to evaluate the following integrals. 44.DdV; D is bounded by the upper half of the ellipsoid x2/9 + y2/4 + z2 = 1 and the xy-plane. Use x = 3u, y = 2v, z = w.Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a.If the transformation T: x = g(u, v), y = h(u, v) is linear in u and v. then the Jacobian is a constant. b.The transformation x = au + bv, y = cu + dv generally maps triangular regions to triangular regions. c.The transformation x = 2v, y = 2u maps circles to circles.46E47EEllipse problems Let R be the region bounded by the ellipse x2/a 2 + y2/b2 = 1, where a 0 and b 0 are real numbers. Let T be the transformation x = au, y = bv. 48.Find the area of R.Ellipse problems Let R be the region bounded by the ellipse x2/a 2 + y2/b2 = 1, where a 0 and b 0 are real numbers. Let T be the transformation x = au, y = bv. 49. Evaluate R|xy|dA.Ellipse problems Let R be the region bounded by the ellipse x2/a 2 + y2/b2 = 1, where a 0 and b 0 are real numbers. Let T be the transformation x = au, y = bv. 50.Find the center of mass of the upper half of R (y 0) assuming it has a constant density.Ellipse problems Let R be the region bounded by the ellipse x2/a 2 + y2/b2 = 1, where a 0 and b 0 are real numbers. Let T be the transformation x = au, y = bv. 51.Find the average square of the distance between points of R and the origin.Ellipse problems Let R be the region bounded by the ellipse x2/a 2 + y2/b2 = 1, where a 0 and b 0 are real numbers. Let T be the transformation x = au, y = bv. 52.Find the average distance between points in the upper half of R and the x-axis.Ellipsoid problems Let D be the solid bounded by the ellipsoid x2/a2 + y2/b2 + z2/c2 = 1, where a 0, b 0, and c 0 are real numbers. Let T be the transformation x = au, y = bv, z = cw. 53.Find the volume of D.Ellipsoid problems Let D be the solid bounded by the ellipsoid x2/a2 + y2/b2 + z2/c2 = 1, where a 0, b 0, and c 0 are real numbers. Let T be the transformation x = au, y = bv, z = cw. 54.Evaluate D|xyz|dA.Ellipsoid problems Let D be the solid bounded by the ellipsoid x2/a2+y2/b2+z2/c2=1, where a 0, b 0, and c 0 are real numbers. Let T be the transformation x = au, y = bv, z =cw. 55.Find the center of mass of the upper half of D (z 0) assuming it has a constant density.Ellipsoid problems Let D be the solid bounded by the ellipsoid x2/a2+y2/b2+z2/c2=1, where a 0, b 0, and c 0 are real numbers. Let T be the transformation x = au, y = bv, z =cw. 56.Find the average square of the distance between points of D and the origin.Parabolic coordinates Let T be the transformation x = u2 v2, y = 2uv a.Show that the lines u = a in the uv-plane map to parabolas in the xy-plane that open in the negative x-direction with vertices on the positive x-axis. b.Show that the lines v = b in the uv-plane map to parabolas in the xy-plane that open in the positive x-direction with vertices on the negative x-axis. c.Evaluate J(u, v). d.Use a change of variables to find the area of the region bounded by x = 4 y2/16 and x = y2/4 1. e.Use a change of variables to find the area of the curved rectangle above the x-axis bounded by x = 4 y2/16, x = 9 y2/36, x = y2/4 1, and x = y2/64 16. f.Describe the effect of the transformation x = 2uv. y = u2 v2 on horizontal and vertical lines in the uv-plane.Shear transformations in 3 The transformation T in 3 given by x = au + bv + cw,y = dv + ew,z = w, where a, b, c, d, and e are positive real numbers, is one of many possible shear transformations in 3. Let S be the unit cube {(u, v, w): 0 u 1,0 v 1,0 w 1}. Let D = T(S) be the image of S. a. Explain with pictures and words the effect of T on S. b. Compute the Jacobian of T c. Find the volume of D and compare it to the volume of S (which is 1). d. Assuming a constant density, find the center of mass of D and compare it to the center of mass of S (which is 12,12,12).Linear transformations Consider the linear transformation T in 2: given by x = au + bv, y = cu + dv where a, b, c. and d are real numbers, with ad bc. a. Find the Jacobian of T b. Let S be the square in the uv-plane with vertices (0,0), (1,0), (0, 1), and (1,1), and let R = T(S). Show that area (R) = |J(u. v)|. c. Let l be the line segment joining the points P and Q in the uv-plane. Show that T(l) (the image of l under T) is the line segment joining T(P) and T(Q) in the xy-plane. (Hint: Use vectors.) d. Show that if S is a parallelogram in the uv-plane and R = T(S), then area(R) = |J(u, v)| area(S). (Hint: Without loss of generality, assume the vertices of S are (0,0), (A, 0), (B, C), and (A + B, C), where A, B, and C are positive, and use vectors.)Meaning of the Jacobian The Jacobian is a magnification (or reduction) factor that relates the area of a small region near the point (u, v) to the area of the image of that region near the point (x,y). a.Suppose S is a rectangle in the uv-plane with vertices O(0,0), P(u, 0), {u, v), and Q(0, v) (see figure). The image of S under the transformation x = g(u, v), y = h(u, v) is a region R in the xy-plane. Let O P and Q be the images of O, P, and Q, respectively, in the xy-plane, where O P and Q do not all lie on the same line. Explain why the coordinates of O, P, and Q are (g(0, 0), h(0, 0)), (g(u, 0), h(u, 0)), and (g(0, v), h(0, v)), respectively. b.Use a Taylor series in both variables to show that g(u,0)g(0,0)+gu(0,0)ug(0,v)g(0,0)+gv(0,0)vh(u,0)h(0,0)+hu(0,0)uh(0,v)h(0,0)+hv(0,0)v where gu (0,0) is xu evaluated at (0,0), with similar meanings for gv, hu, and hv. c.Consider the vectors OP and OQ and the parallelogram, two of whose sides are OP and OQ. Use the cross product to show that the area of the parallelogram is approximately |J(u,v)|uv. d.Explain why the ratio of the area of R to the area of S is approximately |J(u, v)|.Open and closed boxes Consider the region R bounded by three pairs of parallel planes: ax + by = 0, ax + by = 1, cx + dz = 0, cx + dz = 1, ey + fz = 0, and ey + fz = 1, where a, b, c, d, e, and f are real numbers. For the purposes of evaluating triple integrals, when do these six planes bound a finite region? Carry out the following steps. a.Find three vectors n1, n2. and n3, each of which is normal to one of the three pairs of planes. b.Show that the three normal vectors lie in a plane if their triple scalar product n1,(n2 n3) is zero. c.Show that the three normal vectors lie in a plane if ade + bcf = 0. d.Assuming n1, n2, and n3 lie in a plane P, find a vector N that is normal to P. Explain why a line in the direction of N does not intersect any of the six planes and therefore the six planes do not form a bounded region. e.Consider the change of variables u = ax + by, v = cx + dz, w = ey + ft Show that J(x,y,z)=(u,v,w)(x,y,z)=adebcf What is the value of the Jacobian if R is unbounded?Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. Assuming g is integrable and a, b, c, and d are constants, cdabg(x,y) dx dy = (abg(x,y)dx)(cdg(x,y)dy). b. The spherical equation = /2, the cylindrical equation z = 0, and the rectangular equation z = 0 all describe the same set of points. c. Changing the order of integration in Df(x,y,z) dx dy dz from dx dy dz to dy dz dx requires also changing the integrand from f(x, y, z) to f(y, z, x). d. The transformation T: x = v, y = u maps a square in the uv-plane to a triangle in the xy-plane.Evaluating integrals Evaluate the following integrals as they are written. 2.1214xy(x2+y2)2dxdyEvaluating integrals Evaluate the following integrals as they are written. 3.131e1xydydxEvaluating integrals Evaluate the following integrals as they are written. 4.120lnxx3eydydxChanging the order of integration Assuming f is integrable, change the order of integration in the following integrals. 5.11x21f(x,y)dydxChanging the order of integration Assuming f is integrable, change the order of integration in the following integrals. 6.02y11f(x,y)dxdyChanging the order of integration Assuming f is integrable, change the order of integration in the following integrals. 7.0101y2f(x,y)dxdyArea of plane regions Use double integrals to compute the area of the following regions. Make a sketch of the region. 8.The region bounded by the lines y = x 4, y = x, and y = 2x 4Area of plane regions Use double integrals to compute the area of the following regions. Make a sketch of the region. 9.The region bounded by y = |x| and y = 20 x2Area of plane regions Use double integrals to compute the area of the following regions. Make a sketch of the region. 10.The region between the curves y = x2 and y = 1 + x x2Miscellaneous double integrals Choose a convenient method for evaluating the following integrals. 11.R2x4+1dA; R is the region bounded by x = 1, x = 2, y=x3/2, and y = 0Miscellaneous double integrals Choose a convenient method for evaluating the following integrals. 12.Rx1/2eydA; R is the region bounded by x = 1, x = 4, y=x, and y = 0Miscellaneous double integrals Choose a convenient method for evaluating the following integrals. 13.R(x+y)dA; R is the disk bounded by the circle r = 4 sin .Miscellaneous double integrals Choose a convenient method for evaluating the following integrals. 14.R(x2+y2)dA; R is the region {(x, y): 0 x 2, 0 y x}.Miscellaneous double integrals Choose a convenient method for evaluating the following integrals. 15.01y1/31x10cos(x4y)dxdyMiscellaneous double integrals Choose a convenient method for evaluating the following integrals. 16.02y24x8y1+x4y4dxdyCartesian to polar coordinates Evaluate the following integrals over the specified region. 17.R3x2ydA;R={(r,):0r1,0/2}Cartesian to polar coordinates Evaluate the following integrals over the specified region. 18.RdA(1+x2+y2)2;R={(r,):1r4,0}Computing areas Sketch the following regions and use integration to find their areas. 19.The region bounded by all leaves of the rose r = 3 cos 2Computing areas Sketch the following regions and use integration to find their areas. 20.The region inside both the circles r = 2 and r = 4 cosComputing areas Sketch the following regions and use integration to find their areas. 21.The region that lies inside both the cardioids r = 2 2 cos and r = 2 + 2 cosAverage values 22.Find the average value of z=16x2y2 over the disk in the xy-plane centered at the origin with radius 4.Average values 23.Find the average distance from the points in the solid cone bounded by z=2x2+y2 to the z-axis, for 0 z 8.Changing order of integration Rewrite the following integrals using the indicated order of integration. 24. 0101z201x2f(x,y,z) dy dx dz in the order dz dy dxChanging order of integration Rewrite the following integrals using the indicated order of integration. 25. 0101x22x2+y22f(x,y,z) dz dy dx in the order dx dz dyChanging order of integration Rewrite the following integrals using the indicated order of integration. 26.0209x20xf(x,y,z)dydzdx in the order dz dx dyTriple integrals Evaluate the following integrals, changing the order of integration if needed. 27.01zz1x21x2dydxdzTriple integrals Evaluate the following integrals, changing the order of integration if needed. 28.00y0sinxdzdxdyTriple integrals Evaluate the following integrals, changing the order of integration if needed. 29.19012y24sinx2zdxdydzTriple integrals Evaluate the following integrals, changing the order of integration if needed. 30.022x2/22x2/2x2+3y28x2y2dzdydxTriple integrals Evaluate the following integrals, changing the order of integration if needed. 31.020y1/30y2yz5(1+x+y2+z6)2dxdzdyVolumes of solids Find the volume of the following solids. 32. The solid beneath the paraboloid f(x, y) = 12 x2 2y2 and above the region R = {(x, y) : l x 2, 0 : y l}Volumes of solids Find the volume of the following solids. 33. The solid bounded by the surfaces x = 0, z = 3 2y, and z = 2x2 + 1Volumes of solids Find the volume of the following solids. 32.The prism in the first octant bounded by the planes y = 3 3x and z = 2Volumes of solids Find the volume of the following solids. 33.One of the wedges formed when the cylinder x2 + y2 = 4 is cut by the planes z = 0 and y = zVolumes of solids Find the volume of the following solids. 34.The solid bounded by the parabolic cylinders z = y2 + 1 and z = 2 x2Volumes of solids Find the volume of the following solids. 35.The solid common to the two cylinders x2 + y2 = 4 and x2 + z2 4Volumes of solids Find the volume of the following solids. 36.The tetrahedron with vertices (0, 0, 0), (1, 0, 0), (1, 1, 0), and (1, 1, 1)Single to double integral Evaluate 01/2(sin12xsin1x)dx by converting it to a double integral.Tetrahedron limits Let D be the tetrahedron with vertices at (0,0,0), (1,0,0), (0, 2,0), and (0,0,3). Suppose the volume of D is to be found using a triple integral. Give the limits of integration for the six possible orderings of the variables.Polar to Cartesian Evaluate using rectangular coordinates, where (r, θ) are polar coordinates.
Average value 40.Find the average of the square of the distance between the origin and the points in the solid paraboloid D = {(x, y, z): 0 z 4 x2 y2}Average value 41.Find the average x-coordinate of the points in the prism D = {(x, y, z): 0 x 1, 0 y 3 3x, 0 z 2}.Integrals in cylindrical coordinates Evaluate the following integrals in cylindrical coordinates. 42.0309x203(x2+y2)3/2dzdydxIntegrals in cylindrical coordinates Evaluate the following integrals in cylindrical coordinates. 43.112201y21(1+x2+y2)dxdzdyVolumes in cylindrical coordinates Use integration in cylindrical coordinates to find the volume of the following solids. 46. The solid bounded by the hemisphere z=9x2y2 and the hyperdoloid z=1x2y2.Volumes in cylindrical coordinates Use integration in cylindrical coordinates to find the volume of the following solids. 45.The solid cylinder whose height is 4 and whose base is the disk {(r, ): 0 r 2 cos }Integrals in spherical coordinates Evaluate the following integrals in spherical coordinates. 46.020/202cos2sindddIntegrals in spherical coordinates Evaluate the following integrals in spherical coordinates. 47.00/42sec4sec2sindddVolumes in spherical coordinates Use integration in spherical coordinates to find the volume of the following solids. 48.The solid cardioid of revolution D={(,,):0(1cos)/2,0,02}Volumes in spherical coordinates Use integration in spherical coordinates to find the volume of the following solids. 49.The solid rose petal of revolution D={(,,):04sin2,0/2,02}Volumes in spherical coordinates Use integration in spherical coordinates to find the volume of the following solids. 50.The solid above the cone =/4 and inside the sphere = 4 cosCenter of mass of constant-density plates Find the center of mass (centroid) of the following thin constant-density plates. Sketch the region corresponding to the plate and indicate the location of the center of mass. Use symmetry whenever possible to simplify your work. 51.The region bounded by y = sin x and y = 0 between x = 0 and x =Center of mass of constant-density plates Find the center of mass (centroid) of the following thin constant-density plates. Sketch the region corresponding to the plate and indicate the location of the center of mass. Use symmetry whenever possible to simplify your work. 52.The region bounded by y = x3 and y = x2 between x = 0 and x = 1Center of mass of constant-density plates Find the center of mass (centroid) of the following thin constant-density plates. Sketch the region corresponding to the plate and indicate the location of the center of mass. Use symmetry whenever possible to simplify your work. 53.The half-annulus {(r, ): 2 r 4, 0 }Center of mass of constant-density plates Find the center of mass (centroid) of the following thin constant-density plates. Sketch the region corresponding to the plate and indicate the location of the center of mass. Use symmetry whenever possible to simplify your work. 54.The region bounded by y = x2 and y = a2 x2, where a 0Center of mass of constant-density solids Find the center of mass of the following solids, assuming a constant density. Use symmetry whenever possible and choose a convenient coordinate system. 55.The paraboloid bowl bounded by z = x2 + y2 and z = 36Center of mass of constant-density solids Find the center of mass of the following solids, assuming a constant density. Use symmetry whenever possible and choose a convenient coordinate system. 56.The tetrahedron bounded by z = 4 x 2y and the coordinate planes59REVariable-density solids Find the coordinates of the center of mass of the following solids with the given density. 58.The cube in the first octant bounded by the planes x = 2, y = 2, and z = 2, with (x, y, z) = 1 + x + y + zCenter of mass for general objects Consider the following two- and three-dimensional regions. Compute the center of mass assuming constant density. All parameters are positive real numbers. 59.A solid is bounded by a paraboloid with a circular base of radius R and height h. How far from the base is the center of mass?62REA sector of a circle in the first quadrant is bounded between the x-axis, the line y = x, and the circle x2 + y2 = a2. What are the coordinates of the center of mass?
An ice cream cone is bounded above by the sphere x2 + y2 + z2 = a2 and below by the upper half of the cone z2 = x2 + y2. What are the coordinates of the center of mass?
Volume and weight of a fish tank A spherical fish tank with a radius of 1 ft is filled with water to a level 6 in below the top of the tank. a.Determine the volume and weight of the water in the fish tank. (The weight density of water is about 62.5 lb/ft3.) b.How much additional water must be added to completely fill the tank?67RETransforming a square Let S = {(u, v): 0 u 1, 0 v 1} be a unit square in the uv-plane. Find the image of S in the xy-plane under the following transformations. 66.T: x = v, y = u67–70. Transforming a square Let S = {(u, v) : 0 ≤ u ≤ 1, 0 ≤ v ≤ 1} be a unit square in the uv-plane. Find the image of S in the xy-plane under the following transformations.
69. T : x = 3u + v, y = u + 3v
Transforming a square Let S = {(u, v): 0 u 1, 0 v 1} be a unit square in the uv-plane. Find the image of S in the xy-plane under the following transformations. 68.T: x = u, y = 2v + 271REComputing Jacobians Compute the Jacobian J(u, v) of the following transformations. 70.T: x = u + v, y = u v73RE74REDouble integralstransformation given To evaluate the following integrals, carry out the following steps. a.Sketch the original region of integration R and the new region S using the given change of variables. b.Find the limits of integration for the new integral with respect to u and v. c.Compute the Jacobian. d.Chance variables and evaluate the new integral. 73.xy2dA;R={(x,y):y/3x(y+6)/3,0y3};usex=u+v/3,y=v.Double integralstransformation given To evaluate the following integrals, carry out the following steps. a.Sketch the original region of integration R and the new region S using the given change of variables. b.Find the limits of integration for the new integral with respect to u and v. c.Compute the Jacobian. d.Chance variables and evaluate the new integral. 74.R3xy2dA;R={(x,y):0x2,xyx+4} use x = 2u, y = 4v + 2u75-78. Double integrals—transformation given To evaluate the following integrals, carry out these steps.
Sketch the original region of integration R and the new region S using the given change of variables.
Find the limits of integration for the new integral with respect to u and v.
Compute the Jacobian.
Change variables and evaluate the new integral.
77. , R = {(x, y) : 0 ≤ y ≤ x ≤ 1}; use x = u + v, y = v − u.
Double integralstransformation given To evaluate the following integrals, carry out the following steps. a.Sketch the original region of integration R and the new region S using the given change of variables. b.Find the limits of integration for the new integral with respect to u and v. c.Compute the Jacobian. d.Chance variables and evaluate the new integral. 76.Rxy2dA; R is the region between the hyperbolas xy = 1 and xy = 4 and the lines y = 1 and y = 4; use x = u/v, y = v.Double integrals Evaluate the following integrals using a change of variables. Sketch the original and new regions of integration. R and S. 77.Ry4dA; R is the region bounded by the hyperbolas xy = 1 and xy = 4 and the lines y/s = 1 y/x = 3.Double integrals Evaluate the following integrals using a change of variables. Sketch the original and new regions of integration. R and S. 78.R(y2+xy2x2)dA; R is the region bounded by the lines y = x, y = x 3, y = 2x + 3, and y = 2x 3.Triple integrals Use a change of variables to evaluate the following integrals. 79.DyzdV;D is bounded by the planes x + 2y = 1, x + 2y = 2, x z = 0, x z = 2, 2y z = 0, and 2y z = 3.Triple integrals Use a change of variables to evaluate the following integrals. 80.DxdV; D is bounded by the planes y 2x = 0, y 2x = 1, z 3y = 0, z 3y = 1, z 4x = 0, and z 4x = 3.If the vector field in Example 1c describes the velocity of a fluid and place a small cork in the plane at (2, 0), what path will it follow? Example 1 Vector fields Sketch representative vectors of the following vector fields. a. F(x, y) = 0,x = xj (a shear field) b. F(x, y) = 1y2,0 = (1 y2)i, for |y| 1(channel flow) c. F(x, y) = y,x = yi + xj (a rotation field)In Example 2, verify that g(x, y). G(x, y) = 0. In parts (a) and (b) of Example 2. Verify that |F| = 1 and |G| = 1 at all points excluding the origin.Find the gradient field associated with the function (x, y, z) = xyzHow is a vector field F = f, g, h used to describe the motion of air at one instant in time?Sketch the vector field F = x, y.3EGiven a differentiable, scalar-valued function , why is the gradient of a vector field?Interpret the gradient field of the temperature function T = f(x, y)Show that all the vectors in vector field F=2x,yx2+y2 have the same length, and state the length of the vectors.Sketch a few representative vectors of vector field F=0,1 along the line y = 2.Sketching vector fields Sketch the following vector fields. 8. F=1,0Sketching vector fields Sketch the following vector fields. 9. F=1,1Two-dimensional vector fields Sketch the following vector fields. 6.F = 1, yTwo-dimensional vector fields Sketch the following vector fields. 7.F = x, 0Two-dimensional vector fields Sketch the following vector fields. 8.F = x, yTwo-dimensional vector fields Sketch the following vector fields. 9.F = x, yTwo-dimensional vector fields Sketch the following vector fields. 10.F = 2x, 3yTwo-dimensional vector fields Sketch the following vector fields. 11.F = y, xTwo-dimensional vector fields Sketch the following vector fields. 12.F = x, y, yTwo-dimensional vector fields Sketch the following vector fields. 13.F = x, y, xTwo-dimensional vector fields Sketch the following vector fields. 14.F=xx2+y2,yx2+y2Two-dimensional vector fields Sketch the following vector fields. F=ex,020EThree-dimensional vector fields Sketch a few representative vectors of the following vector fields. 21.F = (1, 0, z)Three-dimensional vector fields Sketch a few representative vectors of the following vector fields. 24.F=(x,y,z)x2+y2+z2Three-dimensional vector fields Sketch a few representative vectors of the following vector fields. 23.F = (y, x, 0)Matching vector Fields with graphs Match vector Fields ad with graphs AD. a.F = 0, x2 b.F = x y, x c.F = 2x y d.F = y, xNormal and tangential components For the vector field F and curve C, complete the following:
Determine the points (if any) along the curve C at which the vector field F is tangent to C.
Determine the points (if any) along the curve C at which the vector field F is normal to C.
Sketch C and a few representative vectors of F on C.
F = ; C = {(x, y) : y – x2 = 1}
Normal and tangential components For the vector field F and curve C, complete the following:
Determine the points (if any) along the curve C at which the vector field F is tangent to C.
Determine the points (if any) along the curve C at which the vector field F is normal to C.
Sketch C and a few representative vectors of F on C.
F = ; C ={(x, y): y − x2 = 1}
Normal and tangential components For the vector field F and curve C, complete the following:
Determine the points (if any) along the curve C at which the vector field F is tangent to C.
Determine the points (if any) along the curve C at which the vector field F is normal to C.
Sketch C and a few representative vectors of F on C.
F = ; C = {(x, y) : x2 + y2 = 4}
Normal and tangential components For the vector field F and curve C, complete the following:
Determine the points (if any) along the curve C at which the vector field F is tangent to C.
Determine the points (if any) along the curve C at which the vector field F is normal to C.
Sketch C and a few representative vectors of F on C.
F = ; C = {(x, y): x2 + y2 = 1}
Normal and tangential components For the vector field F and curve C, complete the following:
Determine the points (if any) along the curve C at which the vector field F is tangent to C.
Determine the points (if any) along the curve C at which the vector field F is normal to C.
Sketch C and a few representative vectors of F on C.
F = ; C = {(x, y): x = 1}
Normal and tangential components For the vector field F and curve C, complete the following:
Determine the points (if any) along the curve C at which the vector field F is tangent to C.
Determine the points (if any) along the curve C at which the vector field F is normal to C.
Sketch C and a few representative vectors of F on C.
F = ; C = {(x, y) : x2 + y2 = 1}
Design your own vector field Specify the component functions of a vector field F in 2 with the following properties. Solutions are not unique. 45.F is everywhere normal to the line x = y.Design your own vector field Specify the component functions of a vector field F in 2 with the following properties. Solutions are not unique. 44.F is everywhere normal to the line x = 2.Design your own vector field Specify the component functions of a vector field F in 2 with the following properties. Solutions are not unique. 47.At all points except (0, 0), F has unit magnitude and points away from the origin along radial lines.Design your own vector field Specify the component functions of a vector field F in 2 with the following properties. Solutions are not unique. 46.The flow of F is counterclockwise around origin, increasing in magnitude with distance from the origin.Gradient fields Find the gradient field F = for the following potential functions . 29.(x,y)=x2yy2xGradient fields Find the gradient field F = for the following potential functions . 30.(x,y)=xyGradient fields Find the gradient field F = for the following potential functions . 31.(x,y)=x/yGradient fields Find the gradient field F = for the following potential functions . 32.(x,y)=tan1(y/x)Gradient fields Find the gradient field F = for the following potential functions . 33.(x,y,z)=(x2+y2+z2)/2Gradient fields Find the gradient field F = for the following potential functions . 34.(x,y)=ln(1+x2+y2+z2)Gradient fields Find the gradient field F = for the following potential functions . 35.(x,y)=(x2+y2+z2)1/2Gradient fields Find the gradient field F = for the following potential functions . 36.(x,y)=ezsin(x+y)Gradient fields on curves For the potential function and points A, B, C, and D on the level curve (x, y) = 0, complete the following steps. a. Find the gradient field F = . b. Evaluate F at the points A, B, C, and D. c. Plot the level curve (x, y) = 0 and the vectors F at the points A, B, C, and D. 43. (x, y) = y 2x; A(1, 2), B(0, 0), C(1,2), and D(2,4)Gradient fields on curves For the potential function and points A, B, C, and D on the level curve (x, y) = 0, complete the following steps. a. Find the gradient field F = b. Evaluate F at the points A, B, C, and D. c. Plot the level curve (x, y) = 0 and the vectors F at the points A, B, C, and D. 44. (x,y)=12x2y; A(2, 2), B(1, 1/2), C(1, 1/2), and D(2, 2)Gradient fields on curves For the potential function and points A, B, C, and D on the level curve (x, y) = 0, complete the following steps. a. Find the gradient field F =. b. Evaluate F at the points A, B, C, and D. c. Plot the level curve (x, y) = 0 and the vectors F at the points A, B, C, and D. 45. (x, y) = y + sin x; A(/2, 1), B(, 0), C(3/2, 1), and D(2, 0)Gradient fields on curves For the potential function and points A, B, C, and D on the level curve (x, y) = 0, complete the following steps. a. Find the gradient field F = . b. Evaluate F at the points A, B, C, and D. c. Plot the level curve (x, y) = 0 and the vectors F at the points A, B, C, and D. 46. (x,y)=32x4y432; A(2, 2), B(2, 2), C(2, 2), and D(2, 2)Gradient fields Find the gradient field F = for the potential function . Sketch a few level curves of and a few vectors of F. 25.(x,y)=x2+y2,forx2+y216Gradient fields Find the gradient field F = for the potential function . Sketch a few level curves of and a few vectors of F. 27.(x,y)=x+y,for|x|2,|y|2Equipotential curves Consider the following potential functions and graphs of their equipotential curves. a.Find the associated gradient field F = . b.Show that the vector field is orthogonal to the equipotential curve at the point (1, 1). Illustrate this result on the figure. c.Show that the vector field is orthogonal to the equipotential curve at all points (x, y). d.Sketch two flow curves representing F that are everywhere orthogonal to the equipotential curves. 37. (x, y) = 2x +3yEquipotential curves Consider the following potential functions and graphs of their equipotential curves. a.Find the associated gradient field F = . b.Show that the vector field is orthogonal to the equipotential curve at the point (1, 1). Illustrate this result on the figure. c.Show that the vector field is orthogonal to the equipotential curve at all points (x, y). d.Sketch two flow curves representing F that are everywhere orthogonal to the equipotential curves. 38. (x, y) = x + y2Equipotential curves Consider the following potential functions and graphs of their equipotential curves. a.Find the associated gradient field F = . b.Show that the vector field is orthogonal to the equipotential curve at the point (1, 1). Illustrate this result on the figure. c.Show that the vector field is orthogonal to the equipotential curve at all points (x, y). d.Sketch two flow curves representing F that are everywhere orthogonal to the equipotential curves. 39. (x, y) = ex yEquipotential curves Consider the following potential functions and graphs of their equipotential curves. a.Find the associated gradient field F = . b.Show that the vector field is orthogonal to the equipotential curve at the point (1, 1). Illustrate this result on the figure. c.Show that the vector field is orthogonal to the equipotential curve at all points (x, y). d.Sketch two flow curves representing F that are everywhere orthogonal to the equipotential curves. 40. (x, y) = x2+ 2y2Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a.The vector field F = 3x2, 1 is a gradient field for both 1(x,y)=x3+y and 2(x,y)=y+x3+100. b.The vector field F=y,xx2+y2 is constant in direction and magnitude on the unit circle. c.The vector field F=y,xx2+y2 is neither a radial field nor a rotation field.Electric field due to a point charge The electric field in the xy-plane due to a point charge at (0,0) is a gradient field with a potential function V(x,y)=kx2+y2 where k 0 is a physical constant. a.Find the components of the electric field in the x-and y-directions, where E(x,y)=(x,y) b.Show that the vectors of the electric field point in the radial direction (outward from the origin) and the radial component of E can be expressed as Er = k/r2, where x2+y2. c.Show that the vector field is orthogonal to the equipotential curves at all points in the domain of VElectric field due to a line of charge The electric field in the xy-plane due to an infinite line of charge along the z-axis is a gradient field with a potential function V(x,y)=cln(r0x2+y2). where c 0 is a constant and r0 is a reference distance at which the potential is assumed to be 0 (see figure). a.Find the components of the electric field in the x-and y directions, where E(x,y)=V(x,y). b.Show that the electric field at a point in the xy-plane is directed outward from the origin and has magnitude |E|=c/r,.where r=x2+y2. c.Show that the vector field is orthogonal to the equipotential curves at all points in the domain of V.Gravitational force due to a mass The gravitational force on a point mass m due to a point mass M at the origin is a gradient field with potential U(r)=GMmr, where G is the gravitational constant and r=x2+y2+z2 is the distance between the masses. a.Find the components of the gravitational force in the x-,y-, and z-directions, where F(x, y, z) = U(x, y z). b.Show that the gravitational force points in the radial direction (outward from point mass M) and the radial component is F(r)=GMmr2. c.Show that the vector field is orthogonal to the equipotential surfaces at all points in the domain of U.Flow curves in the plane Let F(x,y)=(f(x,y),g(x,y)) be defined on 2. 51.Explain why the flow curves or streamlines of F satisfy y=g(x,y)/f(x,y) and are everywhere tangent to the vector field.Flow curves in the plane Let F(x,y)=(f(x,y),g(x,y)) be defined on 2. 52.Find and graph the flow curves for the vector field F = 1, x.Flow curves in the plane Let F(x,y)=(f(x,y),g(x,y)) be defined on 2. 53.Find and graph the flow curves for the vector field F = x, x.Flow curves in the plane Let F(x,y)=(f(x,y),g(x,y)) be defined on 2. 54.Find and graph the flow curves for the vector field F = y, x. Note that d/dx(y2)=2yy(x)Flow curves in the plane Let F(x,y)=(f(x,y),g(x,y)) be defined on 2. 55.Find and graph the flow curves for the vector field F = y, x.62E63E64E65EVector fields in polar coordinates A vector field in polar coordinates has the form F(r, ) = F(r, ) ur + g(r, ) u, where the unit vectors are defined in Exercise 62. Sketch the following vector fields and express them in Cartesian coordinates. 62. Vectors in 2 may also be expressed in terms of polar coordinates. The standard coordinate unit vectors in polar coordinates are denoted ur and u (see figure). Unlike the coordinate unite vectors in Cartesian coordinates, ur and u change their direction depending on the point (r, ). Use the figure to show that for r 0, the following relationships among the unit vectors in Cartesian and polar coordinates hold: ur = cos i + sin j i = ur cos u sin u = sin i + cos j j = ur sin + u cos 66. F = r uCartesian-to-polar vector field Write the vector field F = y, x in polar coordinates, and sketch the field.Explain mathematically why differentiating the arc length integral leads to s'(t) = |r'(t)|.Suppose r(t) = t,0, for a t b, is a parametric description of C; note that C is the interval [a, b] on the xaxis. Show that C f(x, y) ds = abf(t,0)dt, which is an ordinary, single variable integral introduced in Chapter 5.3QCSuppose a two dimensional force field is everywhere directed outward from the origin, and C is a circle centered at the origin. What is the angle between the field and the unit vectors tangent to C?5QCHow does a line integral differ from the single-variable integral abf(x)dx?If a curve C is given by r(t) = t, t2, what is |r(t)|?3EFind a parametric description r(t) for the following curves. 4. The segment of the curve x = sin y from (0,0) to (0,3)