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All Textbook Solutions for Calculus Volume 3

For the following exercises, use Green’s theorem to find the area. 166. Use Green’s theorem to prove the area of a disk with radius a is A=a2 For the following exercises, use Green’s theorem to find the area. 167. Use Green’s theorem to find the area of one loop of a four-leaf rose r = 3sin2 . (Hint: xdyydx=r2d).For the following exercises, use Green’s theorem to find the area. 168. Use Green’s theorem to find the area under one arch of the cycloid given by parametric plane x=tsint,y=1cost,t0.For the following exercises, use Green’s theorem to find the area. 169. Use Green’s theorem to find the area of the region enclosed by curve r(t)=t2i+(t33t)j,3t3..[T] Evaluate Green’s theorem using a computer algebra system to evaluate the integral cxeydx+exdy,where C is the circle given by x + y = 4 and is oriented in the counterclockwise direction.Evaluate c(x2y2xy+y2)ds,where C is the boundary of the unit square 0x1,0y1, traversed counterclockwise.Evaluate ( y+2)dx+( x1)dyc ( x1 ) 2+ ( y+2 ) 2,where C is any simple closed curve with an interior that does not contain point (1, -2) traversed counterclockwise.173. Evaluate xdx+ydy c x 2 + y 2 , . where C is any piecewise, smooth simple closed cuive enclosing the oligin, traversed counterclockwise.For the following exercises, use Green’s theorem to calculate the work done by force F on a particle that is moving counterclockwise around closed path C. 174. F(x,y)=xyi+(x+y)j, C : x2+y2= 4 For the following exercises, use Green’s theorem to calculate the work done by force F on a particle that is moving counterclockwise around closed path C. 175. F(x,y)=(x3/23y)i+(6x+5y)j,C: boundary of a triangle with vertices (0, 0), (5, 0), and (0, 5)For the following exercises, use Green’s theorem to calculate the work done by force F on a particle that is moving counterclockwise around closed path C. 176. Evaluate c( 2 x 3 y 3 )dx+( x 3 + y 3 )dy,where C is a unit circle oriented in the counterclockwise direction.A particle starts at point (-2, 0), moves along the x-axis to (2, 0), and then travels along semicircle y=4x2 to the stalling point. Use Green’s theorem to find the work done on this particle by force field F(x,y)=xi+(x3+3xy2)j.David and Sandra are skating on a frictionless pond in the wind. David skates on the inside, going along a circle of radius 2 in a counterclockwise direction. Sandia skates once around a circle of radius 3, also in the counterclockwise direction. Suppose the force of the wind at point (x, y) (x, y) (x, y) is F(x,y)=(x2y+10y)i+(x3+2xy2)j. Use Green’s theorem to determine who does more work.Use Green’s theorem to find the work done by force field F(x, y) = (3y-4x)i + (4x — y)j when an object moves once counterclockwise around ellipse 4x2+y2= 4.Use Green’s theorem to evaluate line integral ce2xsin2ydx+e2xcos2ydy,where C is ellipse 9(x- 1)2 + 4(y-3)2 = 36 oriented counterclockwise.Evaluate line integral c y 2dx+x2dy,where C is the boundary of a triangle with vertices (0, 0), (1, 1), and (1, 0), with the counterclockwise orientation.Use Green’s theorem to evaluate line integral ch.dr if h(x,y)=eyi-sinxj, where C is a triangle with vertices (1, 0), (0, 1), and (-1. 0) (-1. 0) traversed counterclockwise.Use Green’s theorem to evaluate line integral c1+ x 3dx+2xydy where C is a triangle with vertices (0, 0), (1, 0), and (1, 3) oriented clockwise.Use Green’s theorem to evaluate line integral cx2ydxxy2dywhere C is a circle x2+y2= 4 C oriented counterclockwise.Use Green’s theorem to evaluate line integral c(3y e sinx)dx+(7x+y2+1)dy where C is circle x2+ y2= 9 oriented in the counterclockwise direction.Use Green’s theorem to evaluate line integral c(3x5y)dx+(x6y)dy, where C is ellipse x24+y2=1 and is oriented in the counterclockwise direction.Let C be a tiiangulai closed curve from (0, 0) to (1, 0) to (1, 1) and finally back to (0, 0). Let.F (x, y) = 4yi + 6x2j. Use Green s theorem to evaluate cF.ds Use Green’s theoiem to evaluate line integral cydxxdy, where C is circle x2+y2= a2 oriented in the clockwise direction.Use Green’s theorem to evaluate line integral c(y+x)dx+(xsiny)dy,where C is any smooth simple closed curve joining the origin to itself oriented in the counterclockwise direction.Use Green’s theorem to evaluate line integral c(yIn( x 2 + y 2 ))dx+(2arctanyx)dy,where C is the positively oriented circle (x — 2)2 + (y- 3)2 = 1.Use Green’s theorem to evaluate cxydx+ x 3 y 3dy, where C is a triangle with vertices (0, 0), (1, 0), and (1, 2) with positive orientation.Use Green’s theorem to evaluate line integral csinydx+xcosydy. where C is ellipse x2+ xy + y2= 1 oriented in the counter clockwise direction.Let F(x,y)=(cos(x5))13y3i+13x3j.Find the counterclockwise circulation cF.dr, where C is a curve consisting of the line segment joining (-2, 0) and (-1,0), half circle y = 1x2 , the line segment joining (1, 0) and (2, 0), and half circle y = 4x2 .Use Green’s theorem to evaluate line integral csin(x3)dx+2ye x 2dy, where C is a triangular closed curve that connects the points (0, 0), (2, 2), and (0, 2) counterclockwise.Let C be the boundary of square 0x,0y, traversed counterclockwise. Use Green’s theorem to find csin(x+y)dx+cos(x+y)dy.Use Green’s theorem to evaluate line integral, ch.dr, where F(x, y) = (y2- x2)i + (x2+y2)j, and C is a triangle bounded by y = 0, x = 3, and = x, oriented counter clockwise.Use Green’s Theorem to evaluate integial cF.dr,where F(x, y) = (xy2)i + xj. and C is a unit circle oriented in the counterclockwise direction.Use Green’s theorem in a plane to evaluate line integral c(xy+ y 2)dx+x2dy,where C is a closed curve of a region bounded by y = x and y = x2oriented in the counterclockwise direction.Calculate the outward flux of F = -xi + 2yj over a square with cotiiers (± 1, ± 1). where the unit normal is outward pointing and oriented in the counterclockwise direction.200. [T] Let C be circle x2+ y2= 4 oriented in the counterclockwise direction. Evaluate c[( 3y e tan 1 x )dx+( 7x+ y 2 +1 )dy]using a computer algebra system.Find the flux of field F = -xi + yj across x2+ y2 = 16 oriented in the counterclockwise direction.Let F = (y2— x2)i + (x2+y2)j, and let C be a triangle bounded by y = 0, x = 3, and y = x oriented in the counterclockwise direction. Find the outward flux of F through C.[T] Let C be unit circle x2+ y2 = 1 traversed once counter clockwise. Evaluate c[y3+sin(xy)+xycos(xy)]dx+[x3+x2cos(xy)]dy by using a computer algebra system.[T] Find the outward flux of vector field F = xy2i + x2yj across the boundaiy of annulus R=(x,y):1x2+y24=(r,):1r2,02 using a computer algebra system.Consider region R bounded by parabolas y= x2and x =y2. Let C be the boundary of R oriented counterclockwise. Use Green’s theorem to evaluate c(x+ e x )dx+(2xcos( y 2))dy.For the following exercises, find the curl of F at the given point. 252.F(x,y,z)=exyi+exzj+eyzkat(3,2,0)For the following exercises, determine whether the statement is true or false. 206. If the coordinate functions of F: 33 have continuous second partial derivatives, then curl (div(F)) equals zero.For the following exercises, determine whether the statement is true or false. 207. .(xi+yj+zk)=1.For the following exercises, determine whether the statement is true or false. 209. If curl F = 0, then F is conservative.For the following exercises, determine whether the statement is true or false. 208. All vector fields of the form F(x,y,z)=f(x)i+g(y)j+h(z)kare conservative.For the following exercises, determine whether the statement is true or false. 210. If F is a constant vector field then div F = 0.For the following exercises, determine whether the statement is true or false. 211. If F is a constant vector field then curl F = 0.For the following exercises, find the curl of F. 212. F(x,y,z)=xy2z4i+(2x2y+z)j+y3z2k For the following exercises, find the curl of F. 213.F(x,y,z)=x2zi+y2xj+(y+2z)k For the following exercises, find the curl of F. 214.F(x,y,z)=3xyz2i+y2sinzj+xe2zk For the following exercises, find the curl of F. 215.F(x,y,z)=x2yzi+xy2zj+xyz2k For the following exercises, find the curl of F. 216.F(x,y,z)=(xcosy)i+xy2j For the following exercises, find the curl of F. 217.F(x,y,z)=(xy)i+(yz)j+(zx)k For the following exercises, find the curl of F. 218.F(x,y,z)=xyzi+x2y2z2j+y2z3k For the following exercises, find the curl of F. 219.F(x,y,z)=xyi+yzj+xzk For the following exercises, find the curl of F. 220.F(x,y,z)=x2i+y2j+z2k For the following exercises, find the divergence of F. 223.F(x,y,z)=3xyz2i+y2sinzj+xe2zk For the following exercises, find the divergence of F. 222.F(x,y,z)=x2zi+y2xj+(y+2z)k For the following exercises, find the divergence of F. 224.F(x,y)=(sinx)i+(cosy)j For the following exercises, find the divergence of F. 225.F(x,y,z)=x2i+y2j+z2k For the following exercises, find the divergence of F. 226.F(x,y,z)=(xy)i+(yz)j+(zx)k For the following exercises, find the divergence of F. 227.F(x,y)=xx2+y2i+yx2y2j For the following exercises, find the divergence of F. 228.F(x,y)=xiyj For the following exercises, find the divergence of F. 229.F(x,y,z)=axi+byj+ck For the following exercises, find the divergence of F. 230.F(x,y,z)=xyzi+x2y2z2j+y2z2k For the following exercises, find the divergence of F. 231.F(x,y,z)=xyi+yzj+xzk 233.w(x,y,z)=(x2+y2+z2)1/2 232.u(x,y,z)=ex(cosysiny) G(x,y,z)=xiyj+zk,findcurl(FG).234.IfF(x,y,z)=2i+2xj+3yk 235.IfF(x,y,z)=2i+2xj+3ykG(x,y,z)=xiyj+zk,finddiv(FG).Find div F, given that F = f, where f(x,y,z)=xy3z2 .237. Find the divergence of F for vector field F(x,y,z)=(y2+z2)(x+y)i+(z2+x2)(y+z)j+(x2+y2)(z+x)k.F(x, y. z) = (y2 + + y1 + (z2 + + z)j + (x2 + + x)k.Find the divergence of F for vector field F(x,y,z)=f1(y,z)i+f2(x,z)j+f3(x,y)k.For the following exercises, use r = |r|and r = (x, y,z). 239. Find the curl r.For the following exercises, use r = |r|and r = (x, y,z). 240. Find the curl rr .For the following exercises, use r = |r|and r = (x, y,z). 241. Find the curl rr3.For the following exercises, use r = |r| and r = (x, y,z). 242. Let F(x,y) = yi+xjx2+y2, where F is defined on (x,y)|(x,y)(0,0)F. Find curl F.For the following exercises, use a computer algebra system to find the curl of the given vector fields. 243.[T]F(x,y,z)=arctan(xy)i+Inx2+y2j+k For the following exercises, use a computer algebra system to find the curl of the given vector fields. 244.[T]F(x,y,z)=sin(xy)i+sin(yz)j+sin(zx)k For the following exercises, find the divergence of F at the given point. 246. F(x,y,z)=xyzi+yj+zkat(1,2,3)For the following exercises, find the divergence of F at the given point. 247. F(x,y,z)=exyi+exzj+eyzkat(3,2,0)For the following exercises, find the divergence of F at the given point. 248. F(x,y,z)=xyzi+yz+zkat(1,2,3)For the following exercises, find the divergence of F at the given point. 249. F(x,y,z)=exsinyi-excosyjat(0,0,3)For the following exercises, find the divergence of F at the given point. 245. F(x,y,z)=i+j+kat(2,1,3)For the following exercises, find the curl of F at the given point. 250.F(x,y,z)=i+j+kat(2,1,3)For the following exercises, find the curl of F at the given point. 251.F(x,y,z)=xyzi+yj+xkat(1,2,3)For the following exercises, find the curl of F at the given point. 252. F(x,y,z)=exyi+eyzj+eyzk at (3,2,0)For the following exercises, find the curl of F at the given point. 253. F(x,y,z)=xyzi+yj+zk at (1,2,1)For the following exercises, find the curl of F at the given point. 254. F(x,y,z)=exsinyiexcosyj at (0,0,3)For the following exercises, find the curl of F at the given point. 255. Let F(x,y,z)=(3x2y+az)i+x3j+(3x+3z2)k . For what value of a is F conservative?For the following exercises, find the curl of F at the given point. 256. Given vector field F(x,y)=1x2+y2(y,x) on domain D=2(0,0)=(x,y)2(x,y)(0,0) , is F conservative?For the following exercises, find the curl of F at the given point. 257. Given vector field F(x,y)=1x2+y2(x,y) on domain D=2(0,0) , is F conservative?For the following exercises, find the curl of F at the given point. 258. Find the work done by force field F(x,y)=eyixeyj in moving an object from P(0,1) to Q(2,0) . Is the force field conservative?For the following exercises, find the curl of F at the given point. 259. Compute divergence F=(sinx)+(cosy)jxyzk .For the following exercises, find the curl of F at the given point. 260. Compute curl F=(sinhx)i+(coshy)jxyzk .For the following exercises, consider a rigid body that is rotating about the x-axis counterclockwise with constant angular velocity =a,b,c . If P is a point in the body located at r=xi+yj+zk , the velocity at P is given by vector field F=r . 261. Express F in terms of i , j , and k vectors.For the following exercises, consider a rigid body that is rotating about the x-axis counterclockwise with constant angular velocity =a,b,c . If P is a point in the body located at r=xi+yj+zk , the velocity at P is given by vector field F=r . 262. Find div F .For the following exercises, consider a rigid body that is rotating about the x-axis counterclockwise with constant angular velocity =a,b,c . If P is a point in the body located at r=xi+yj+zk , the velocity at P is given by vector field F=r . 263. Find curl F .In the following exercises, suppose that F=0 and G=0 . 264. Does F+G necessarily have zero divergence?In the following exercises, suppose that F=0 and G=0 . 265. Does FG necessarily have zero divergence?In the following exercises, suppose a solid object in 3 has a temperature distribution given by T(x,y,z) . The heat flow vector field in the object is F=kT , where k0 is a property of the material. The heat flow vector points in the direction opposite to that of the gradient, which is the direction of greatest temperature decrease. The divergence of the heat flow vector is F=kT=k2T . 266. Compute the heat flow vector field.In the following exercises, suppose a solid object in 3 has a temperature distribution given by T(x,y,z) . The heat flow vector field in the object is F=kT , where k0 is a property of the material. The heat flow vector points in the direction opposite to that of the gradient, which is the direction of greatest temperature decrease. The divergence of the heat flow vector is F=kT=k2T . 267. Compute the divergence.Consider rotational velocity field v=0,10z,-10y. If a paddlewheel is placed in plane x+y+z=1 with its axis normal to this plane, using a computer algebra system, calculate how fast the paddlewheel spins in revolutions per unit time.For the following exercises, determine whether the statements are true or false. 269. If surface S is given by (x,y,z):0x1,0y1,z=10 , then Sf(x,y,z)dS=01 0 1 f(x,y,10)dxdy .wFor the following exercises, determine whether the statements are true or false. 270. If surface S is given by (x,y,z):0x1,0y1,zx , then Sf(x,y,z)dS=01 0 1 f(x,y,x)dxdy .For the following exercises, determine whether the statements are true or false. 271. Surface r=vcosu,vsinu,v2 , for 0u,0v2 , is the same as surface r=vcos2u,vsin2u,v , for 0u2,ov4 .For the following exercises, determine whether the statements are true or false. 272. Given the standard parameterization of a sphere, normal vectors tutv are outward normal vectors.For the following exercises, find parametric descriptions for the following surfaces. 273. Plane 3x-2y+z=2 For the following exercises, find parametric descriptions for the following surfaces. 274. Paraboloid z=x2+y2 , for 0z9 .For the following exercises, find parametric descriptions for the following surfaces. 275. Plane 2x-4y+3z=16 For the following exercises, find parametric descriptions for the following surfaces. 276. The frustum of cone z2=x2+y2 , for 2z8 For the following exercises, find parametric descriptions for the following surfaces. 277. The portion of cylinder x2+y2=9 in the first octant, for 0z3 For the following exercises, find parametric descriptions for the following surfaces. 278. A cane with base radius r and height h , where r and h are positive constants For the following exercises, use a computer algebra system to approximate the area of the following surfaces using a parametric description of the surface. 279. [T] Half cylinder (r,,z):r=4,0r,0z7 For the following exercises, use a computer algebra system to approximate the area of the following surfaces using a parametric description of the surface. 280. [T] Plane z=10-x-y above square |x|2,|y|2 For the following exercises, let S be the hemisphere x2+y2+z2=4 , with z0 , and evaluate each surface integral, in the counterclockwise direction. 281. szds For the following exercises, let S be the hemisphere x2+y2+z2=4 , with z0 , and evaluate each surface integral, in the counterclockwise direction. 282. s(x-2y)dS For the following exercises, let S be the hemisphere x2+y2+z2=4 , with z0 , and evaluate each surface integral, in the counterclockwise direction. 283. s(x2+y2)zdS wFor the following exercises, evaluate sFNds for vector field F, where N is an outward normal vector to surface S. 284. F(x,y,z)=xi+2yj3zk , and S is that part of plane 15x12y+3z=6 that lies above unit square 0x1 , 0y1 .For the following exercises, evaluate sFNds for vector field F, where N is an outward normal vector to surface S. 285. F(x,y,z)=xi+yj , and S is hemisphere z=1x2y2 .For the following exercises, evaluate sFNds for vector field F, where N is an outward normal vector to surface S. 286. F(x,y,z)=x2i+y2j+z2k , and S is the portion of plane z=y+1 that lies inside cylinder x2+y2=1 .For the following exercises, approximate the mass of the homogeneous lamina that has the shape of given surface S. Round to four decimal places. 287. [T] S is surface z=4x2y , with z0 , x0 , y0 , =x .For the following exercises, approximate the mass of the homogeneous lamina that has the shape of given surface S. Round to four decimal places. 288. [T] S is surface z=x2+y2 , with z1 ; =z .For the following exercises, approximate the mass of the homogeneous lamina that has the shape of given surface S. Round to four decimal places. 289. [T] S is surface x2+y2+x2=5 , with z1 ; =2 .For the following exercises, approximate the mass of the homogeneous lamina that has the shape of given surface S. Round to four decimal places. 290. Evaluate s ( y 2 zi+ y 3 j+xzk )dS , where S is the surface of cube 1x1 , 1y1 , and 0z2 , in a counterclockwise direction.For the following exercises, approximate the mass of the homogeneous lamina that has the shape of given surface S. Round to four decimal places. 291. Evaluate surface integral sgdS , where g(x,y,z)=xz+2x23xy and S is the portion of plane 2x3y+z=6 that lies over unit square R: 0x1 , 0y1 .wFor the following exercises, approximate the mass of the homogeneous lamina that has the shape of given surface S. Round to four decimal places. 292. Evaluate s(x+y+z)dS , where S is the surface defined parametrically by R(u,v)=(2u+v)i+(u2v)j+(u+3v)k for 0u1 , and 0v2 .For the following exercises, approximate the mass of the homogeneous lamina that has the shape of given surface S. Round to four decimal places. 293. [T] Evaluate s(x y 2+z)dS , where S is the surface defined by R(u,v)=u2i+vj+uk , 0u1 , 0v1 .For the following exercises, approximate the mass of the homogeneous lamina that has the shape of given surface S. Round to four decimal places. 295. Evaluate s( x 2+ y 2)dS , where S is the surface bounded above hemisphere z=1x2y2 , and below by plane z=0 .For the following exercises, approximate the mass of the homogeneous lamina that has the shape of given surface S. Round to four decimal places. 296. Evaluate s( x 2+ y 2+ z 2)dS , where S is the portion of plane z=x+1 that lies inside cylinder x2+y2=1 .For the following exercises, approximate the mass of the homogeneous lamina that has the shape of given surface S. Round to four decimal places. 297. [T] Evaluate sx2zdS , where S is the portion of cone z2=x2+y2 that lies between planes z=1 and z=4 .For the following exercises, approximate the mass of the homogeneous lamina that has the shape of given surface S. Round to four decimal places. 298. [T] Evaluate s( xz/y)dS , where S is the portion of cylinder x=y2 that lies in the first octant between planes z=0 , z=5 , y=1 , and y=4 .For the following exercises, approximate the mass of the homogeneous lamina that has the shape of given surface S. Round to four decimal places. 299. [T] Evaluate s(z+y)dS , where S is the part of the graph of z=1x2 in the first octant between the xz-plane and plane y=3 .For the following exercises, approximate the mass of the homogeneous lamina that has the shape of given surface S. Round to four decimal places. 300. Evaluate sxyzdS if S is the part of plane z=x+y that lies over the triangular region in the xy-plane with vertices (0,0,0) , (1,0,0) , and (0,2,0) .For the following exercises, approximate the mass of the homogeneous lamina that has the shape of given surface S. Round to four decimal places. 301. Find the mass of a lamina of density (x,y,z)=z in the shape of hemisphere z=(a2x2y2)1/2 .For the following exercises, approximate the mass of the homogeneous lamina that has the shape of given surface S. Round to four decimal places. 302. Compute sFNdS , where F(x,y,z)=xi5yj+4zk : and N is an outward normal vector S, where S is the union of two squares S1:x=0,0y1,0z1 and S2:z=1,0x1,0y1 .For the following exercises, approximate the mass of the homogeneous lamina that has the shape of given surface S. Round to four decimal places. 303. Compute sFNdS , where F(x,y,z)=xyi+zj+(x+y)k and N is an outward normal vector S, where S is the triangular region cut off from plane x+y+z=1 by the positive coordinate axes.For the following exercises, approximate the mass of the homogeneous lamina that has the shape of given surface S. Round to four decimal places. 304. Compute sFNdS , where F(x,y,z)=2yzi+(tan1xz)j+exyk : and N is an outward normal vector S, where S is the surface of sphere x2+y2+z2=1 .For the following exercises, approximate the mass of the homogeneous lamina that has the shape of given surface S. Round to four decimal places. 305. Compute sFNdS , where F(x,y,z)=xyzi+xyzj+xyzk and N is an outward normal vector S, where S is the surface of the five faces of the unit cube 0x1 , 0y1 , 0z1 missing z=0 .For the following exercises, express the surface integral as an iterated double integral by using a projection on S on the yz -plane. 306. sxy2z3dS ; S is the first-octant portion of plane 2x+3y+4z=12 .For the following exercises, express the surface integral as an iterated double integral by using a projection on S on the yz -plane. 307. s(x2-2y+z)dS ; S is the portion of the graph of 4x+y=8 bounded by the coordinate planes and plane z=6 .For the following exercises, express the surface integral as an iterated double integral by using a projection on S on the xz -plane 308. sxy2z3dS ; S is the first-octant portion of plan 2x+3y+4z=12 .For the following exercises, express the surface integral as an iterated double integral by using a projection on S on the xz -plane 309. s(x22y+z)dS ; S is the portion of the graph of 4x+y=8 bounded by the coordinate planes and plane z=6 .For the following exercises, express the surface integral as an iterated double integral by using a projection on S on the xz -plane 310. Evaluate surface integral syzdS , where S is the first-octant part of plane x+y+z= , where is a positive constant.For the following exercises, express the surface integral as an iterated double integral by using a projection on S on the xz -plane 311. Evaluate surface integral S(x2z+y2)dS , where S is hemisphere x2+y2+z2=a2 , z0 .For the following exercises, express the surface integral as an iterated double integral by using a projection on S on the xz -plane 312. Evaluate surface integral szdA , where S is surface zx2+y2 , 0z2 .For the following exercises, express the surface integral as an iterated double integral by using a projection on S on the xz -plane 313. Evaluate surface integral sx2yzdS where S is the part of plane z=1+2x+3y that lies above rectangle 0x3 and 0y2 .For the following exercises, express the surface integral as an iterated double integral by using a projection on S on the xz -plane 314. Evaluate surface integral syzdS , where S is plane x+y+z=1 that lies in the first octant.For the following exercises, express the surface integral as an iterated double integral by using a projection on S on the xz -plane 315. Evaluate surface integral syzdS , where S is the part of plane z=y+3 that lies inside cylinder x2+y2=1 .For the following exercises, use geometric reasoning to evaluate the given surface integrals. 316. s x 2+ y 2+ z 2dS , where S is surface x2+y2+z2=4 , z0 For the following exercises, use geometric reasoning to evaluate the given surface integrals. 317. s(xi+yj)dS , where S is surface x2+y2=4 , 1z3 , oriented with unit normal vectors pointing outward For the following exercises, use geometric reasoning to evaluate the given surface integrals. 318. s(zk)dS , Where S is disc x2+y29 on plane z=4 , oriented with unit normal vectors pointing upward A lamina has the shape of a portion of sphere x2+y2+z2=a2 that lies within cone z=x2+y2 . Let S be the spherical shell centered at the origin with radius a , and let C be the right circular cone with a vertex at the origin and an axis of symmetry that coincides with the z-axis. Determine the mass of the lamina if (x,y,z)=x2y2z .A lamina has the shape of a portion of sphere x2+y2+z2=a2 that lies within cone z=x2+y2 . Let S be the spherical shell centered at the origin with radius a, and let C be the right circular cone with a vertex at the origin and an axis of symmetry that coincides with the z-axis. Suppose the vertex angle of the cone is 0 , with 002 . Determine the mass of that portion of the shape enclosed in the intersection of S and C . Assume (x,y,z)=x2y2z .A paper cup has the shape of an inverted right circular cone of height 6in. and radius of tap 3in. If the cup is full of water weighing 6.25lb/ft3 , find the total force exerted by the water on the inside surface of the cup.For the following exercises, the heat flow vector field for conducting objects F=kT , where T(x,y,z) is the temperature in the object and k0 is a constant that depends on the material. Find the outward flux of F across the following surfaces S for the given temperature distributions and assume k=1 . 322. T(x,y,z)=100exy consists of the faces of cube x1 , y1 , z1 .For the following exercises, the heat flow vector field for conducting objects F=kT , where T(x,y,z) is the temperature in the object and k0 is a constant that depends on the material. Find the outward flux of F across the following surfaces S for the given temperature distributions and assume k=1 . 323. T(x,y,z)=In(x2+y2+z2) ; S is sphere x2+y2+z2=a2 .For the following exercises, consider the radial fields F=x,y,z(x2+y2+z2)p2=rrP , where P is a real number. Let S consist of spheres A and B centered at the origin with radii 0ab . The total outward flux across S consists of the outward flux across the outer sphere B less the flux into S across inner sphere A. 324. Find the total flux across S with P=0 .For the following exercises, consider the radial fields F=x,y,z(x2+y2+z2)p2=rrP , where P is a real number. Let S consist of spheres A and B centered at the origin with radii 0ab . The total outward flux across S consists of the outward flux across the outer sphere B less the flux into S across inner sphere A. 325. Show that for P=3 the flux across S is independent of a and b .For the following exercises, without using Stokes’ theorem, calculate directly both the flux of curl FN over the given surface and the circulation integral around its boundary, assuming all boundaries are oriented clockwise as viewed from above. 326. F(x,y,z)=y2i+z2j+x2k ; S is the first-octant portion of plane x+y+z=1 .For the following exercises, without using Stokes’ theorem, calculate directly both the flux of curl FN over the given surface and the circulation integral around its boundary, assuming all boundaries are oriented clockwise as viewed from above. 327. F(x,y,z)=zi+xj+yk ; S is hemisphere z=(a2x2y2)12 .For the following exercises, without using Stokes’ theorem, calculate directly both the flux of curl FN over the given surface and the circulation integral around its boundary, assuming all boundaries are oriented clockwise as viewed from above. 328. F(x,y,z)=y2i+2xj+5k ; S is hemisphere z=(4x2y2)12 .For the following exercises, without using Stokes’ theorem, calculate directly both the flux of curl FN over the given surface and the circulation integral around its boundary, assuming all boundaries are oriented clockwise as viewed from above. 329. F(x,y,z)=zi+2xj+3yk ; S is upper hemisphere z=9x2y2 .For the following exercises, without using Stokes’ theorem, calculate directly both the flux of curl FN over the given surface and the circulation integral around its boundary, assuming all boundaries are oriented clockwise as viewed from above. 330. F(x,y,z)=(x+2z)i+(yx)j+(zy)k ; S is a triangular region with vertices (3,0,0) , (0,3/2,0) , and (0,0,3) .For the following exercises, without using Stokes’ theorem, calculate directly both the flux of curl FN over the given surface and the circulation integral around its boundary, assuming all boundaries are oriented clockwise as viewed from above. 331. F(x,y,z)=2yi6zj+3xk ; S is a portion of paraboloid z=4x2y2 and is above the xy -plane.For the following exercises, use Stokes’ theorem to evaluate s(curlFN)dS for the vector fields and surface. 332. F(x,y,z)=xyizj and S is the surface of the cube 0x1 , 0y1 , 0z1 , except for the face where z=0 , and using the outward unit normal vector.For the following exercises, use Stokes’ theorem to evaluate s(curlFN)dS for the vector fields and surface. 333. F(x,y,z)=xyix2j+z2k ; and C is the intersection of paraboloid z=x2+y2 and plane z=y , and using the outward normal vector.For the following exercises, use Stokes’ theorem to evaluate s(curlFN)dS for the vector fields and surface. 334. F(x,y,z)=4yi+zj+2yk and C is the intersection of sphere x2+y2+z2=4 with plane z=0 , and using the outward normal vector For the following exercises, use Stokes’ theorem to evaluate s(curlFN)dS for the vector fields and surface. 335. Use Stokes’ theorem to evaluate c[2xy2zdx+2x2yzdy+( x 2 y 2 2z)dz] , where C is the curve given by x=cost , y=sint , z=sint , 0t2 , traversed in the direction of increasing t.For the following exercises, use Stokes’ theorem to evaluate s(curlFN)dS for the vector fields and surface. 336. [T] Use a computer algebraic system (CAS) and Stokes’ theorem to approximate line integral c(ydx+zdy+xdz) , when C is the intersection of plane x+y=2 and surface x2+y2+z2=2(x+y) , traversed counterclockwise viewed from the origin.For the following exercises, use Stokes’ theorem to evaluate s(curlFN)dS for the vector fields and surface. 337. [T] Use a CAS and Stokes’ theorem to approximate line integral c(3ydx+2zdy5xdz) , where C is the intersection of the xy-plane and hemisphere z=1x2y2 , traversed counterclockwise viewed from the top—that is, from the positive z-axis toward the xy-plane.For the following exercises, use Stokes’ theorem to evaluate s(curlFN)dS for the vector fields and surface. 338. [T] Use a CAS and Stokes’ theorem to approximate line integral c[( 1+y)zdx+( 1+z)xdy+( 1+x)ydz] , where C is a triangle with vertices (1,0,0) , (0,1,0) , and (0,0,1) oriented counterclockwise.For the following exercises, use Stokes’ theorem to evaluate s(curlFN)dS for the vector fields and surface. 339. Use Stokes’ theorem to evaluate scurlFdS , where F(x,y,z)=exycoszi+x2zj+xyk , and S is half of sphere x=1y2z2 , oriented out toward the positive x-axis.For the following exercises, use Stokes’ theorem to evaluate s(curlFN)dS for the vector fields and surface. 340. [T] Use a CAS and Stokes’ theorem to evaluate s(curlFN)dS , where F(x,y,z)=x2yi+xy2j+z3k and C is the curve of the intersection of plane 3x+2y+z=6 and cylinder x2+y2=4 , oriented clockwise when viewed from above.For the following exercises, use Stokes’ theorem to evaluate s(curlFN)dS for the vector fields and surface. 341. [T] Use a CAS and Stokes’ theorem to evaluate scurlFdS , where F(x,y,z)=(sin(y+z)yx2y33)i+xcos(y+z)j+cos(2y)k and S consists of the top and the four sides but not the bottom of the cube with vertices (1,1,1) , oriented outward.For the following exercises, use Stokes’ theorem to evaluate s(curlFN)dS for the vector fields and surface. 342. [T] Use a CAS and Stokes’ theorem to evaluate scurlFdS , where F(x,y,z)=z2i3xyj+x3y3k and S is the top part of z=5x2y2 above plane z=1 , and S is oriented upward.For the following exercises, use Stokes’ theorem to evaluate s(curlFN)dS for the vector fields and surface. 343. Use Stokes’ theorem to evaluate s(curlFN)dS , where F(x,y,z)=z2i+y2j+xk and S is a triangle with vertices (1,0,0) , (0,1,0) and (0,0,1) with counterclockwise orientation.For the following exercises, use Stokes’ theorem to evaluate s(curlFN)dS for the vector fields and surface. 344. Use Stokes’ theorem to evaluate line integral c(zdx+xdy+ydz) , where C is a triangle with vertices (3,0,0) , (0,0,2) , and (0,6,0) traversed in the given order.For the following exercises, use Stokes’ theorem to evaluate s(curlFN)dS for the vector fields and surface. 345. Use Stokes’ theorem to evaluate c( 1 2 y 2dx+zdy+xdz) , where C is the curve of intersection of plane x+z=1 and ellipsoid x2+2y2+z2=1 , oriented clockwise from the origin.For the following exercises, use Stokes’ theorem to evaluate s(curlFN)dS for the vector fields and surface. 346. Use Stokes’ theorem to evaluate s(curlFN)dS , where F(x,y,z)=xi+y2j+zexyk and S is the part of surface z=1x22y2 with z0 , oriented counterclockwise.For the following exercises, use Stokes’ theorem to evaluate s(curlFN)dS for the vector fields and surface. 347. Use Stokes’ theorem for vector field F(x,y,z)=zi+3xj+2zk where S is surface z=1x22y2 , z0 , C is boundary circle x2+y2=1 , and S is oriented in the positive z-direction.For the following exercises, use Stokes’ theorem to evaluate s(curlFN)dS for the vector fields and surface. 348. Use Stokes’ theorem for vector field F(x,y,z)=32y2i2xyj+yzk , where S is that part of the surface of plane x+y+z=1 contained within triangle C with vertices (1,0,0) , (0,1,0) , and (0,0,1) , traversed counterclockwise as viewed from above.For the following exercises, use Stokes’ theorem to evaluate s(curlFN)dS for the vector fields and surface. 349. A certain closed path C in plane 2x+2y+z=1 is known to project unto unit circle x2+y2=1 in the xy-plane. Let c be a constant and let R(x,y,z)=xi+yj+zk . Use Stokes’ theorem to evaluate c(ckR)dS .For the following exercises, use Stokes’ theorem to evaluate s(curlFN)dS for the vector fields and surface. 350. Use Stokes’ theorem and let C be the boundary of surface z=x2+y2 with 0x2 and 0y1 , oriented with upward facing normal. Define F(x,y,z)=[sin(x3)+xz]i+(xyz)j+cos(z4)k and evaluate cFdS .For the following exercises, use Stokes’ theorem to evaluate s(curlFN)dS for the vector fields and surface. 351. Let S be hemisphere x2+y2+z2=4 with z0 , oriented upward. Let F(x,y,z)=x2eyzi+y2exzj+z2exyk be a vector field. Use Stokes’ theorem to evaluate scurlFdS .For the following exercises, use Stokes’ theorem to evaluate s(curlFN)dS for the vector fields and surface. 352. Let F(x,y,z)=xyi+(ez2+y)j+(x+y)k and let S be the graph of function y=x29+z291 with z0 oriented so that the normal vector S has a positive y component. Use Stokes’ theorem to compute integral scurlFdS .For the following exercises, use Stokes’ theorem to evaluate s(curlFN)dS for the vector fields and surface. 353. Use Stokes’ theorem to evaluate FdS , where F(x,y,z)=yi+zj+xk and C is a triangle with vertices (0,0,0) , (2,0,0) and (0,2,0) oriented counterclockwise when viewed from above.For the following exercises, use Stokes’ theorem to evaluate s(curlFN)dS for the vector fields and surface. 354. Use the surface integral in Stokes’ theorem to calculate the circulation of field F, F(x,y,z)=x2y3i+j+zk around C, which is the intersection of cylinder x2+y2=4 and hemisphere x2+y2+z2=16 , z0 , oriented counterclockwise when viewed from above.For the following exercises, use Stokes’ theorem to evaluate s(curlFN)dS for the vector fields and surface. 355. Use Stokes’ theorem to compute scurlFdS , where F(x,y,z)=i+xy2j+xy2k and S is a part of plane y+z=2 inside cylinder x2+y2=1 and oriented counterclockwise.For the following exercises, use Stokes’ theorem to evaluate s(curlFN)dS for the vector fields and surface. 356. Use Stokes’ theorem to evaluate scurlFdS where F(x,y,z)=y2i+xj+z2k and S is the part of plane x+y+z=1 in the positive octant and oriented counterclockwise x0 , y0 , z0 .For the following exercises, use Stokes’ theorem to evaluate s(curlFN)dS for the vector fields and surface. 357. Let F(x,y,z)=xyi+2zj2yk and let C be the intersection of plane x+z=5 and cylinder x2+y2=9 , which is oriented counterclockwise when viewed from the top. Compute the line integral of F over C using Stokes’ theorem.For the following exercises, use Stokes’ theorem to evaluate s(curlFN)dS for the vector fields and surface. 358. [T] Use a CAS and let F(x,y,z)=xy2i+(yzx)j+eyxzk . Use Stokes’ theorem to compute the surface integral of curl F over surface S with inward orientation consisting of cube [0,1][0,1][0,1] with the right side missing.For the following exercises, use Stokes’ theorem to evaluate s(curlFN)dS for the vector fields and surface. 359. Let S be ellipsoid x24+y29+z2=1 oriented counterclockwise and let F be a vector field with component functions that have continuous partial derivatives.For the following exercises, use Stokes’ theorem to evaluate s(curlFN)dS for the vector fields and surface. 360. Let S be the part of paraboloid z=9x2y2 with z0 . Verify Stokes’ theorem for vector field F(x,y,z)=3zi+4xj+2yk .For the following exercises, use Stokes’ theorem to evaluate s(curlFN)dS for the vector fields and surface. 361. [T] Use a CAS and Stokes’ theorem to evaluate cFdS , if F(x,y,z)=(3zsinx)i+(x2+ey)j+(y3cosz)k , where C is the curve given by x=cost , y=sint , z=1 ; 0t2 .For the following exercises, use Stokes’ theorem to evaluate s(curlFN)dS for the vector fields and surface. 362. [T] Use a CAS and Stokes’ theorem to evaluate F(x,y,z)=2yi+ezjarctanxk with S as a portion of paraboloid z=4x2y2 cut off by the xy-plane oriented counterclockwise.For the following exercises, use Stokes’ theorem to evaluate s(curlFN)dS for the vector fields and surface. 363. [T] Use a CAS to evaluate scurl(F)dS , where F(x,y,z)=2zi+3xj+5yk and S is the surface parametrically by r(r,)=rcosi+rsinj+(4r2)k (02,0r3) .For the following exercises, use Stokes’ theorem to evaluate s(curlFN)dS for the vector fields and surface. 364. Let S be paraboloid z=a(1x2y2) , for z0 , where a0 is a real number. Let F=xy,y+z,zx . For what value(s) of a (if any) does s(F)ndS have its maximum value?For the following application exercises, the goal is to evaluate A=s(F)ndS , where F=xz,xz,xy and S is the upper half of ellipsoid x2+y2+8z2=1 , where z0 . 365. Evaluate a surface integral over a more convenient surface to find the value of A .For the following application exercises, the goal is to evaluate A=s(F)ndS , where F=xz,xz,xy and S is the upper half of ellipsoid x2+y2+8z2=1 , where z0 . 366. Evaluate A using a line integral.For the following application exercises, the goal is to evaluate A=s(F)ndS , where F=xz,xz,xy and S is the upper half of ellipsoid x2+y2+8z2=1 , where z0 . 367. Take paraboloid z=x2+y2 , and slice it with plane y=0 . Let S he the surface that remains for y0 , including the planar surface in the xz -plane. Let C be the semicircle and line segment that bounded the cap of S in plane z=4 with counterclockwise orientation. Let F=2z+y,2x+z,2y+x . Evaluate s(F)ndS .For the following exercises, let S he the disk enclosed by curve C:r(t)=coscost,sintsincost , for 0t2 , where 02 is a fixed angle. 368. What is the length of C in terms of ?For the following exercises, let S he the disk enclosed by curve C:r(t)=coscost,sintsincost , for 0t2 , where 02 is a fixed angle. 369. What is the circulation of C of vector field as F=yz,x a function of ?For the following exercises, let S he the disk enclosed by curve C:r(t)=coscost,sintsincost , for 0t2 , where 02 is a fixed angle. 370. For what value of is the Circulation 3 maximum?For the following exercises, let S he the disk enclosed by curve C:r(t)=coscost,sintsincost , for 0t2 , where 02 is a fixed angle. 371. Circle C in plane x+y+z=8 has radius 4 and center (2,3,3) . Evaluate cFdr for F=0,z,2y , where C has a counterclockwise orientation when viewed from above.For the following exercises, let S he the disk enclosed by curve C:r(t)=coscost,sintsincost , for 0t2 , where 02 is a fixed angle. 372. Velocity field v=0,1x2,0 , for x1 and z1 , represents a horizontal flow in the y -direction. Compute the curl of v in a Clockwise rotation.For the following exercises, let S he the disk enclosed by curve C:r(t)=coscost,sintsincost , for 0t2 , where 02 is a fixed angle. 373. Evaluate integral s(F)ndS , where F=xzi+yzj+xyezk and S is the cap of paraboloid z=5=x2y2 above plane z=3 , and n points in the positive z -direction on S .For the following exercises, use Stokes’ theorem to find the circulation of the following vector fields around any smooth, simple closed curve C . 374. F=(xsinyez)For the following exercises, use Stokes’ theorem to find the circulation of the following vector fields around any smooth, simple closed curve C . 375. F=y2z3,z2xyz3,3xy2z2 For the following exercises, use a computer algebraic system (CAS) and the divergence theorem to evaluate surface integral sFndS for the given choice of F and the boundary surface S. For each closed surface, assume N is the outward unit normal vector. 376. [T] F(x,y,z)=xi+yj+zk ; S is the surface of cube 0x1 , 0y1 , 0z1 .For the following exercises, use a computer algebraic system (CAS) and the divergence theorem to evaluate surface integral sFndS for the given choice of F and the boundary surface S. For each closed surface, assume N is the outward unit normal vector. 377. [T] F(x,y,z)=(cosyz)i+exzj+3z2k ; S is the surface of hemisphere z=4x2y2 together with disk x2+y24 in the xy-plane.For the following exercises, use a computer algebraic system (CAS) and the divergence theorem to evaluate surface integral sFndS for the given choice of F and the boundary surface S. For each closed surface, assume N is the outward unit normal vector. 378. [T] F(x,y,z)=(x2+y2x2)i+x2yj+3zk ; S is the surface of the five faces of unit cube 0x1 , 0y1 , 0z1 .For the following exercises, use a computer algebraic system (CAS) and the divergence theorem to evaluate surface integral sFndS for the given choice of F and the boundary surface S. For each closed surface, assume N is the outward unit normal vector. 379. [T] F(x,y,z)=xi+yj+zk ; S is the surface of paraboloid z=x2+y2 for 0z9 .For the following exercises, use a computer algebraic system (CAS) and the divergence theorem to evaluate surface integral sFndS for the given choice of F and the boundary surface S. For each closed surface, assume N is the outward unit normal vector. 380. [T] F(x,y,z)=x2i+y2j+z2k ; S is the surface of sphere x2+y2+z2=4 .For the following exercises, use a computer algebraic system (CAS) and the divergence theorem to evaluate surface integral sFndS for the given choice of F and the boundary surface S. For each closed surface, assume N is the outward unit normal vector. 381. [T] F(x,y,z)=xi+yj+(z21)k ; S is the surface of the solid bounded by cylinder x2+y2=4 and planes z=0 and z=1 .For the following exercises, use a computer algebraicsystem (CAS) and the divergence theorem to evaluate surface integral sFndS for the given choice of F andthe boundary surface S. For each closed surface, assume Nis the outward unit normal vector. 382. [T] F(x,y,z)=xy2i+yz2j+x2zk ; S is the surface bounded above by sphere =2 and below by cone =4 in spherical coordinates. (Think of S as thesurface of an “ice cream cone”)For the following exercises, use a computer algebraic system (CAS) and the divergence theorem to evaluate surface integral sFndS for the given choice of F and the boundary surface S. For each closed surface, assume N is the outward unit normal vector. 383. [T] F(x,y,z)=x3i+y3j+3a2zk(constanta0) ; S is the surface bounded by cylinder x2+y2=a2 and planes z=0 and z=1 .For the following exercises, use a computer algebraic system (CAS) and the divergence theorem to evaluate surface integral sFndS for the given choice of F and the boundary surface S. For each closed surface, assume N is the outward unit normal vector. 384. [T] Surface integral sFdS , where S is the solid bounded by paraboloid z=x2+y2 and plane z=4 , and F(x,y,z)=(x+y2z2)i+(y+z2x2)j+(z+x2y2)k .For the following exercises, use a computer algebraic system (CAS) and the divergence theorem to evaluate surface integral sFndS for the given choice of F and the boundary surface S. For each closed surface, assume N is the outward unit normal vector. 385. Use the divergence theorem to calculate surface integral sFdS , where F(x,y,z)=(ey2)i+(y+sin(z2))j+(z1)k and S is upper hemisphere x2+y2+z2=1 , z0 , oriented upward.For the following exercises, use a computer algebraic system (CAS) and the divergence theorem to evaluate surface integral sFndS for the given choice of F and the boundary surface S. For each closed surface, assume N is the outward unit normal vector. 386. Use the divergence theorem to calculate surface integral sFdS , where F(x,y,z)=x4ix3z2j+4xy2zk and S is the surface bounded by cylinder x2+y2=1 and planes z=x+2 and z=0 .For the following exercises, use a computer algebraic system (CAS) and the divergence theorem to evaluate surface integral sFndS for the given choice of F and the boundary surface S. For each closed surface, assume N is the outward unit normal vector. 387. Use the divergence theorem to calculate surface integral sFdS when F(x,y,z)=x2z3i+2xyz3j+xz4k and S is the surface of the box with vertices (1,2,3) .For the following exercises, use a computer algebraic system (CAS) and the divergence theorem to evaluate surface integral sFndS for the given choice of F and the boundary surface S. For each closed surface, assume N is the outward unit normal vector. 388. Use the divergence theorem to calculate surface integral sFdS when F(x,y,z)=ztan1(y2)i+z3In(x2+1)j+zk and S is a part of paraboloid x2+y2+z=2 that lies above plane z=1 and is oriented upward.For the following exercises, use a computer algebraic system (CAS) and the divergence theorem to evaluate surface integral sFndS for the given choice of F and the boundary surface S. For each closed surface, assume N is the outward unit normal vector. 389. [T] Use a CAS and the divergence theorem to calculate flux sFdS , where F(x,y,z)=(x3+y3)i+(y3+z3)j+(z3+x3)k and S is a sphere with (0,0) and radius 2.For the following exercises, use a computer algebraic system (CAS) and the divergence theorem to evaluate surface integral sFndS for the given choice of F and the boundary surface S. For each closed surface, assume N is the outward unit normal vector. 390. Use the divergence theorem to compute the value sFdS , where F(x,y,z)=(y3+3x)i+(xz+y)j+[z+x4cos(x2y)]k and S is the area of the region bounded by x2+y2=1 , x0 , y0 , and 0z1 .For the following exercises, use a computer algebraic system (CAS) and the divergence theorem to evaluate surface integral sFndS for the given choice of F and the boundary surface S. For each closed surface, assume N is the outward unit normal vector. 391. Use the divergence theorem to compute flux integral sFdS , where F(x,y,z)=yjzk and S consists of the union of paraboloid y=x2+z2 , 0y1 , and disk x2+z21 , y=1 , oriented outward. What is the flux through just the paraboloid?For the following exercises, use a computer algebraic system (CAS) and the divergence theorem to evaluate surface integral sFndS for the given choice of F and the boundary surface S. For each closed surface, assume N is the outward unit normal vector. 392. Use the divergence theorem to compute flux integral sFdS , where F(x,y,z)=x+yj+z4k and S is a part of cone z=x2+y2 beneath top plane z=1 , oriented downward.For the following exercises, use a computer algebraic system (CAS) and the divergence theorem to evaluate surface integral sFndS for the given choice of F and the boundary surface S. For each closed surface, assume N is the outward unit normal vector. 393. Use the divergence theorem to calculate surface sFdS for F(x,y,z)=x4ix3z2j+4xy2zk , where S is the surface bounded by cylinder x2+y2=1 and planes z=x+2 and z=0 .For the following exercises, use a computer algebraic system (CAS) and the divergence theorem to evaluate surface integral sFndS for the given choice of F and the boundary surface S. For each closed surface, assume N is the outward unit normal vector. 394. Consider F(x,y,z)=x2i+xyj+(z+1)k . Let E be the solid enclosed by paraboloid z=4x2y2 and plane z=0 with normal vectors painting outside E. Compute flux F across the boundary of E using the divergence theorem.For the following exercises, use a CAS along with the divergence theorem to compute the net onward flux for the fields across the given surfaces S . 395. [T]F=x,2y,3z ; S is sphere (x,y,z):x2+y2+z2=6 .For the following exercises, use a CAS along with the divergence theorem to compute the net onward flux for the fields across the given surfaces S . 396. [T]F=x,2y,z ; S is the boundary of the tetrahedron in the first octant formed by plane x+y+z=1 .For the following exercises, use a CAS along with the divergence theorem to compute the net onward flux for the fields across the given surfaces S . 397. [T]F=y2x,x3y,y2z ; S is sphere (x,y,z):x2+y2+z2=4 .`For the following exercises, use a CAS along with the divergence theorem to compute the net onward flux for the fields across the given surfaces S . 398. [T]F=x,y,z ; S is the surface of paraboloid z=4x2y2 , for z0 , plus its base in the xy -plane.For the following exercises, use a CAS and the divergence theorem to compute the net outward flux for the vector fields across the boundary of the given regions D. 399. [T] F=zx,xy,2yz ; D is the region between spheres of radius 2 and 4 centered at the origin.For the following exercises, use a CAS and the divergence theorem to compute the net outward flux for the vector fields across the boundary of the given regions D. 400. [T] F=rr=x,y,zx2+y2+z2 ; D is the region between spheres of radius 1 and 2 centered at the origin.For the following exercises, use a CAS and the divergence theorem to compute the net outward flux for the vector fields across the boundary of the given regions D. 401. [T] F=x2,y2,z2 , D is the region in the first octant between planes z=4xy and z=2xy .For the following exercises, use a CAS and the divergence theorem to compute the net outward flux for the vector fields across the boundary of the given regions D. 402. Let F(x,y,z)=2xi3xyj+xz2k . Use the divergence theorem to calculate sFdS , where S is the surface of the cube with corners at (0,0,0) , (1,0,0) , (0,1,0) , (1,1,0) , (0,0,1) , (1,0,1) , (0,1,1) , and (1,1,1) , oriented outward.For the following exercises, use a CAS and the divergence theorem to compute the net outward flux for the vector fields across the boundary of the given regions D. 403. Use the divergence theorem to find the outward flux of field F(x,y,z)=(x33y)i+(2yz+1)j+xyzk through the cube bounded by planes x=1 , y=1 , and z=1 .For the following exercises, use a CAS and the divergence theorem to compute the net outward flux for the vector fields across the boundary of the given regions D. 404. Let F(x,y,z)=2xi3yj+5zk and let S be hemisphere z=9x2y2 together with disk x2+y29 in the xy-plane. Use the divergence theorem.For the following exercises, use a CAS and the divergence theorem to compute the net outward flux for the vector fields across the boundary of the given regions D. 405. Evaluate sFNds , where F(x,y,z)=x2i+xyj+x3y3k and S is the surface consisting of all faces except the tetrahedron bounded by plane x+y+z=1 and the coordinate planes, with outward unit normal vector N.For the following exercises, use a CAS and the divergence theorem to compute the net outward flux for the vector fields across the boundary of the given regions D. 406. Find the net outward flux of field F=bzcy,cxaz,aybx across any smooth closed surface in R3 , where a, b, and c are constants.For the following exercises, use a CAS and the divergence theorem to compute the net outward flux for the vector fields across the boundary of the given regions D. 407. Use the divergence theorem to evaluate sRRnds , where R(x,y,z)=xi+yj+zk and S is sphere x2+y2+z2=a2 , with constant a0 .For the following exercises, use a CAS and the divergence theorem to compute the net outward flux for the vector fields across the boundary of the given regions D. 408. Use the divergence theorem to evaluate sFdS , where F(x,y,z)=y2zi+y3j+xzk and S is the boundary of the cube defined by 1x1 , 1y1 , and 0z2 .For the following exercises, use a CAS and the divergence theorem to compute the net outward flux for the vector fields across the boundary of the given regions D. 409. Let R be the region defined by x2+y2+z21 . Use the divergence theorem to find Rz2dV .For the following exercises, use a CAS and the divergence theorem to compute the net outward flux for the vector fields across the boundary of the given regions D. 410. Let E be the solid bounded by the xy-plane and paraboloid z=4x2y2 so that S is the surface of the paraboloid piece together with the disk in the xy-plane that farms its bottom. If F(x,y,z)=(xzsin(yz)+x3)i+cos(yz)j+(3zy2ex2+y2)k , find sFdS using the divergence theorem.For the following exercises, use a CAS and the divergence theorem to compute the net outward flux for the vector fields across the boundary of the given regions D. 411. Let E be the solid unit cube with diagonally opposite corners at the origin and (1,1,1) , and faces parallel to the coordinate planes. Let S be the surface of E, oriented with the outward-pointing normal. Use a CAS to find sFdS using the divergence theorem if F(x,y,z)=2xyi+3yezj+xsinzk .For the following exercises, use a CAS and the divergence theorem to compute the net outward flux for the vector fields across the boundary of the given regions D. 412. Use the divergence theorem to calculate the flux of F(x,y,z)=x3i+y3j+z3k through sphere x2+y2+z2=1 .For the following exercises, use a CAS and the divergence theorem to compute the net outward flux for the vector fields across the boundary of the given regions D. 413. Find sFdS , where F(x,y,z)=xi+yj+zk and S is the outwardly oriented surface obtained by removing cube [1,2][1,2][1,2] from cube [0,2][0,2][0,2] .For the following exercises, use a CAS and the divergence theorem to compute the net outward flux for the vector fields across the boundary of the given regions D. 414. Consider radial vector field F=rr=x,y,z( x 2 + y 2 + z 2 )1/2 . Compute the surface integral, where S is the surface of a sphere of radius a centered at the origin.For the following exercises, use a CAS and the divergence theorem to compute the net outward flux for the vector fields across the boundary of the given regions D. 415. Compute the flux of water through parabolic cylinder S:y=x2 , from 0x2 , 0z3 , if the velocity vector is F(x,y,z)=3z2i+6j+6xzk .For the following exercises, use a CAS and the divergence theorem to compute the net outward flux for the vector fields across the boundary of the given regions D. 416. [T] Use a CAS to find the flux of vector field F(x,y,z)=zi+zj+x2+y2k across the portion of hyperboloid x2+y2=z2+1 between planes z=0 and z=33 , oriented so the unit normal vector points away from the z-axis.For the following exercises, use a CAS and the divergence theorem to compute the net outward flux for the vector fields across the boundary of the given regions D. 417. [T] Use a CAS to find the flux of vector field F(x,y,z)=(ey+x)i+(3cos(xz)y)j+zk through surface S, where S is given by z2=4x2+4y2 from 0z4 , oriented so the unit normal vector points downward.For the following exercises, use a CAS and the divergence theorem to compute the net outward flux for the vector fields across the boundary of the given regions D. 418. [T] Use a CAS to compute sFdS , where F(x,y,z)=xi+yj+2zk and S is a part of sphere x2+y2+z2=2 with 0z1 .For the following exercises, use a CAS and the divergence theorem to compute the net outward flux for the vector fields across the boundary of the given regions D. 419. Evaluate sFdS , where F(x,y,z)=bxy2i+bx2yj+(x2+y2)z2k and S is a closed surface bounding the region and consisting of solid cylinder x2+y2a2 and 0zb .For the following exercises, use a CAS and the divergence theorem to compute the net outward flux for the vector fields across the boundary of the given regions D. 420. [T] Use a CAS to calculate the flux of F(x,y,z)=(x3+ysinz)i+(y3+zsinx)j+3zk across surface S, where S is the boundary of the solid bounded by hemispheres z=4x2y2 and z=1x2y2 , and plane z=0 .For the following exercises, use a CAS and the divergence theorem to compute the net outward flux for the vector fields across the boundary of the given regions D. 421. Use the divergence theorem to evaluate sFdS , where F(x,y,z)=xyi12y2j+zk and S is the surface consisting of three pieces: z=43x23y2 , 1z4 on the top; x2+y2=1 , 0z1 on the sides; and z=0 on the bottom.For the following exercises, use a CAS and the divergence theorem to compute the net outward flux for the vector fields across the boundary of the given regions D. 422. [T] Use a CAS and the divergence theorem to evaluate sFdS , Where F(x,y,z)=(2x+ycosz)i+(x2y)j+y2zk and S is sphere x2+y2+z2=4 orientated outward.For the following exercises, use a CAS and the divergence theorem to compute the net outward flux for the vector fields across the boundary of the given regions D. 423. Use the divergence theorem to evaluate sFdS , where F(x,y,z)=xi+yj+zk and S is the boundary of the solid enclosed by paraboloid y=x2+z22 , cylinder x2+z2=1 , and plane x+y=2 , and S is oriented outward.For the following exercises, Fourier’s law of heat transfer states that the heat flow vector F at a point is proportional to the negative gradient of the temperature; that is, F=kT , which means that heat energy flows hot regions to cold regions. The constant k0 is called the conductivity, which has metric units of joules per meter per second-kelvin or watts per meter-kelvin. A temperature function for region D is given. Use the divergence theorem to find net outward heat flux sFNdS=ksTNdS across the boundary S of D where k=1 . 424. T(x,y,z)=100+x+2y+z ; D=(x,y,z):0x1,0y1,0z1 For the following exercises, Fourier’s law of heat transfer states that the heat flow vector F at a point is proportional to the negative gradient of the temperature; that is, F=kT , which means that heat energy flows hot regions to cold regions. The constant k0 is called the conductivity, which has metric units of joules per meter per second-kelvin or watts per meter-kelvin. A temperature function for region D is given. Use the divergence theorem to find net outward heat flux sFNdS=ksTNdS across the boundary S of D where k=1 . 425. ; T(x,y,z)=100+ez ; D=(x,y,z):0x1,0y1,0z1 For the following exercises, Fourier’s law of heat transfer states that the heat flow vector F at a point is proportional to the negative gradient of the temperature; that is, F=kT , which means that heat energy flows hot regions to cold regions. The constant k0 is called the conductivity, which has metric units of joules per meter per second-kelvin or watts per meter-kelvin. A temperature function for region D is given. Use the divergence theorem to find net outward heat flux sFNdS=ksTNdS across the boundary S of D where k=1 . 426. T(x,y,z)=100+ex2y2z2 ; D is the sphere of radius a centered at the origin.True or False? Justify your answer with a proof or a counterexample. 427. Vector field F(x,y)=x2yi+y2xj is conservative.True or False? Justify your answer with a proof or a counterexample. 428. For vector field F(x,y)=P(x,y)i+Q(x,y)j , if Py(x,y)=Qx(x,y) in open region D , then DPdx+Qdy=0 .True or False? Justify your answer with a proof or a counterexample. 429. The divergence of a vector field is a vector field.True or False? Justify your answer with a proof or a counterexample. 430. If curl F=0 , then F is a conservative vector field.Draw the following vector fields. 431. F(x,y)=12i+2xj Draw the following vector fields. 432. F(x,y)=yi+3xjx2+y2 Are the following the vector fields conservative? If so, find the potential function f such that F=f . 433. F(x,y)yi+(x2ey)j Are the following the vector fields conservative? If so, find the potential function f such that F=f . 434. F(x,y)(6xy)i+(3x2yey)j Are the following the vector fields conservative? If so, find the potential function f such that F=f . 435. F(x,y,z)(2xy+z2)i+(x2+2yz)j+(2xz+y2)k Are the following the vector fields conservative? If so, find the potential function f such that F=f . 436. F(x,y,z)(exy)i+(ex+z)j+(ex+y2)k Evaluate the following integrals. 437. Cx2dy+(2x3xy)dx , along C:y=12x from (0,0) to (4,2)Evaluate the following integrals. 438. Cydx+xy2dy , where C:x=t,y=t1,0t1 Evaluate the following integrals. 439. Sxy2dS , where S is surface z=x2y,0x1,0y4 Find the divergence and curl for the following vector fields. 440. F(x,y,z)=3xyzi+xyezj3xyk Find the divergence and curl for the following vector fields. 441. F(x,y,z)=exi+exyj+exyzk Use Green’s theorem to evaluate the following integrals. 442. C3xydx+2xy2dy , Where C is a square with vertices (0,0) , (0,2) , (2,2) and (2,0)Use Green’s theorem to evaluate the following integrals. 443. C3ydx+(x+ey)dy , where C is a circle centered at the origin with radius 3 Use Stokes’ theorem to evaluate ScurlFdS . 444. F(x,y,z)=yixj+zk , where S is the upper half of the unit sphere Use Stokes’ theorem to evaluate ScurlFdS . 445. F(x,y,z)=yixyzj+2zxk , where S is the upward-facing paraboloid z=x2+y2 lying in cylinder x2+y2=1 Use the divergence theorem to evaluate SFdS . 446. F(x,y,z)=(x3y)i+(3yex)j+(z+x)k , over cube S defined by 1x1 , 0y2 , 0z2 Use the divergence theorem to evaluate SFdS . 447. F(x,y,z)=(2xy)i+(y2)j+(2z3)k , where S is bounded by paraboloid z=x2+y2 and plane z=2 Find the amount of work perfumed by a 50 -kg woman ascending a helical staircase with radius 2m and height 100m . The woman completes five revolutions during the climb.Find the total mass of a thin wire in the shape of a semicircle with radius 2 , and a density function of P(z,y)=y+x2 .Find the total mass of a thin sheet in the shape of a hemisphere with radius 2 for z0 with a density function P(x,y,z)=x+y+z .Use the divergence theorem to compute the value of the flux integral over the unit sphere with F(x,y,z)=3zi+2yj+2xk .Classify each of the following equations as linear or nonlinear. If the equation is linear, determine whether it is homogeneous or nonhomogeneous. 1. x3y+(x1)y8y=0 Classify each of the following equations as linear or nonlinear. If the equation is linear, determine whether it is homogeneous or nonhomogeneous. 2. (1+y2)y+xy3y=cosx Classify each of the following equations as linear or nonlinear. If the equation is linear, determine whether it is homogeneous or nonhomogeneous. 3. xy+eyy=x Classify each of the following equations as linear or nonlinear. If the equation is linear, determine whether it is homogeneous or nonhomogeneous. 4. y+4xy8xy=5x2+1 Classify each of the following equations as linear or nonlinear. If the equation is linear, determine whether it is homogeneous or nonhomogeneous. 5. y+(sinx)yxy=4y Classify each of the following equations as linear or nonlinear. If the equation is linear, determine whether it is homogeneous or nonhomogeneous. 6. y+(x+3y)y=0 For each of the following problems, verify that the given function is a solution to the differential equation. Use a graphing utility to graph the particular solutions for several values of c1and c2. What do the solutions have in common? 7. [T] y+2y3y=0 ; y(x)=c1ex+c2e3x For each of the following problems, verify that the given function is a solution to the differential equation. Use a graphing utility to graph the particular solutions for several values of c1and c2. What do the solutions have in common? 8. [T] x2y2y3x2+1=0 ; y(x)=c1x2+c2x1+x2In(x)+12 For each of the following problems, verify that the given function is a solution to the differential equation. Use a graphing utility to graph the particular solutions for several values of c1and c2. What do the solutions have in common? 9. [T] y+14y+49y=0 ; y(x)=c1e7x+c2xe7x For each of the following problems, verify that the given function is a solution to the differential equation. Use a graphing utility to graph the particular solutions for several values of c1and c2. What do the solutions have in common? 10. [T] 6y49y+8y=0 ; y(x)=c1ex/6+c2e8x Find the general solution to the linear differential equation. 11. y3y10y=0 Find the general solution to the linear differential equation. 12. y7y+12y=0 Find the general solution to the linear differential equation. 13. y+4y+4y=0 Find the general solution to the linear differential equation. 14. 4y12y+9y=0 Find the general solution to the linear differential equation. 15. 2y3y5y=0

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