Homework Answers
(20) Let S be the sphere x2 + y2 + x2 = 4 with outward normal...
(20) Let S be the sphere x2 + y2 + x2 = 4 with outward normal vector. Let F(x, y, z) = (2-3 + tan-'(yz), ex2+2 + y3, cosh(xy) + 23). Use the Divergence Theorem to find the flux of Ě out of S.
1. Suppose F = (-y,x,z) and S is the part of the sphere x2 + y2...
1. Suppose F = (-y,x,z) and S is the part of the sphere x2 + y2 + z = 25 below the plane z = 4, oriented with the outward-pointing normal (so that the normal at (5,0,0) is 1). Compute the flux integral curl F.ds using Stoke's theorem.
Let V be the solid sphere of radius a centred at the origin. Let S be...
Let V be the solid sphere of radius a centred at the origin. Let S be the surface of V oriented with outward unit normal. Consider the vector field F(x, y, z) (xi + yj + zk) (x2 + y2 + z2)3/2 (a) Evaluate the flux integral Sle F:ñ ds by direct calculation. (b) Evaluate SIL, VF DV by direct calculation. (c) Compare your answers to parts (a) and (b) and explain why Gauss' theorem does not apply.
9. The upper half of the ellipsoid tr + ty? + Z2-1 intersects the cylinder x2 + y2-y 0 in a curves C. Calculate tfe circulation of v y'i+y+3i k around C by using Stokes Theorem. x2 + y2 inter...
9. The upper half of the ellipsoid tr + ty? + Z2-1 intersects the cylinder x2 + y2-y 0 in a curves C. Calculate tfe circulation of v y'i+y+3i k around C by using Stokes Theorem. x2 + y2 intersec ts the plane z y in a curve C. Calculate the circulation 10. The paraboloid z of v 2zi+ x j + y k around C by using Stokes Theorem.
9. The upper half of the ellipsoid tr + ty?...
Let P = (0,0, 2)and let S be the unit sphere with equation x2 + y2...
Let P = (0,0, 2)and let S be the unit sphere with equation x2 + y2 + z2 = 1.The collection of points on the sphere where the tangent plane of the sphere contains the point Pforms a curve. Parametrize this curve.
x where F = (x, xy, zy), and S is the upper half of the sphere...
x where F = (x, xy, zy), and S is the upper half of the sphere x2 + y2 + x2 = 1 oriented so that ñ points in the positive z direction.
Consider the vector field F(x, y, z) -(z,2x, 3y) and the surface z- /9 - x2 -y2 (an upper hemisph...
Consider the vector field F(x, y, z) -(z,2x, 3y) and the surface z- /9 - x2 -y2 (an upper hemisphere of radius 3). (a) Compute the flux of the curl of F across the surface (with upward pointing unit normal vector N). That is, compute actually do the surface integral here. V x F dS. Note: I want you to b) Use Stokes' theorem to compute the integral from part (a) as a circulation integral (c) Use Green's theorem (ie...
Q4. 8pnts]If you haven't explored it yet, here is a magical property of the Stoke's theorem Suppose we have a v...
Q4. 8pnts]If you haven't explored it yet, here is a magical property of the Stoke's theorem Suppose we have a vector field F(x,y, z) = -yi+ xj+ zk. Also, let C: x2y2 R2 for some R 0 be the curve in the xy-plane. Now, verify the Stoke's theorem when: (a) The surface S is given by the upper hemisphere 2y z2= R2,z0. R2 - y2, z 2 0. (b) The surface S is given by the paraboloid (c) The surface...
Let F(r, y, z)(z4+ 5y3)i + (y2 surface of the solid octant of the sphere x2+yj2...
Let F(r, y, z)(z4+ 5y3)i + (y2 surface of the solid octant of the sphere x2+yj2 + 22 = 9 for x> 0, y> 0 and z> 0 )j+ (3z + 7)k be the velocity field of a fluid. Let B be the Determine the flux of F through B in the direction of the outward unit normal
Let F(r, y, z)(z4+ 5y3)i + (y2 surface of the solid octant of the sphere x2+yj2 + 22 = 9 for x>...
Let S be the part of the sphere x^2 + y^2 + z^2 = 4 that lies between the cones z = √x^2 + y^2 an...
Let S be the part of the sphere x^2 + y^2 + z^2 = 4 that lies
between the cones z = √x^2 + y^2 and z = √3x^2 + 3y^2.
(1) Let S be the part of the sphere x2 + y2 + Z2-4 that lies between the cones X +y and z a) Find a differentiable parametrization of S b) Find the area of S c) Find 22 dS.
(1) Let S be the part of the sphere...
(20) Let S be the sphere x2 + y2 + x2 = 4 with outward normal vector. Let F(x, y, z) = (2-3 + tan-'(yz), ex2+2 + y3, cosh(xy) + 23). Use the Divergence Theorem to find the flux of Ě out of S.
1. Suppose F = (-y,x,z) and S is the part of the sphere x2 + y2 + z = 25 below the plane z = 4, oriented with the outward-pointing normal (so that the normal at (5,0,0) is 1). Compute the flux integral curl F.ds using Stoke's theorem.
Let V be the solid sphere of radius a centred at the origin. Let S be the surface of V oriented with outward unit normal. Consider the vector field F(x, y, z) (xi + yj + zk) (x2 + y2 + z2)3/2 (a) Evaluate the flux integral Sle F:ñ ds by direct calculation. (b) Evaluate SIL, VF DV by direct calculation. (c) Compare your answers to parts (a) and (b) and explain why Gauss' theorem does not apply.
9. The upper half of the ellipsoid tr + ty? + Z2-1 intersects the cylinder x2 + y2-y 0 in a curves C. Calculate tfe circulation of v y'i+y+3i k around C by using Stokes Theorem. x2 + y2 intersec ts the plane z y in a curve C. Calculate the circulation 10. The paraboloid z of v 2zi+ x j + y k around C by using Stokes Theorem.
9. The upper half of the ellipsoid tr + ty?...
Let P = (0,0, 2)and let S be the unit sphere with equation x2 + y2 + z2 = 1.The collection of points on the sphere where the tangent plane of the sphere contains the point Pforms a curve. Parametrize this curve.
x where F = (x, xy, zy), and S is the upper half of the sphere x2 + y2 + x2 = 1 oriented so that ñ points in the positive z direction.
Consider the vector field F(x, y, z) -(z,2x, 3y) and the surface z- /9 - x2 -y2 (an upper hemisphere of radius 3). (a) Compute the flux of the curl of F across the surface (with upward pointing unit normal vector N). That is, compute actually do the surface integral here. V x F dS. Note: I want you to b) Use Stokes' theorem to compute the integral from part (a) as a circulation integral (c) Use Green's theorem (ie...
Q4. 8pnts]If you haven't explored it yet, here is a magical property of the Stoke's theorem Suppose we have a vector field F(x,y, z) = -yi+ xj+ zk. Also, let C: x2y2 R2 for some R 0 be the curve in the xy-plane. Now, verify the Stoke's theorem when: (a) The surface S is given by the upper hemisphere 2y z2= R2,z0. R2 - y2, z 2 0. (b) The surface S is given by the paraboloid (c) The surface...
Let F(r, y, z)(z4+ 5y3)i + (y2 surface of the solid octant of the sphere x2+yj2 + 22 = 9 for x> 0, y> 0 and z> 0 )j+ (3z + 7)k be the velocity field of a fluid. Let B be the Determine the flux of F through B in the direction of the outward unit normal
Let F(r, y, z)(z4+ 5y3)i + (y2 surface of the solid octant of the sphere x2+yj2 + 22 = 9 for x>...
Let S be the part of the sphere x^2 + y^2 + z^2 = 4 that lies
between the cones z = √x^2 + y^2 and z = √3x^2 + 3y^2.
(1) Let S be the part of the sphere x2 + y2 + Z2-4 that lies between the cones X +y and z a) Find a differentiable parametrization of S b) Find the area of S c) Find 22 dS.
(1) Let S be the part of the sphere...
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