Let F(x, y, z) be the gradient vector field of f(x, y, z) = exyz ,...
1 Help Entering Answers 1 point) Use Stokes' Theorem to evaluateF.dr where F(x,y,z) 6yzi 3xzj +3e k and C is the circy4,z 5 oriented counterclockwise as viewed from above Since the circle is oriented counterclockwise as viewed from above the surface we attach to the circle is oriented upwards The easiest surface to attach to this curve is the disk x2 + y2 < 4, z-5. Using this surface in Stokes' Theorem evaluate the following. F-dr = where sqrt(4-xA2) sqrt(4-x^2)...
Help Entering Answers 1 point) Verify that Stokes' Theorem is true for the vector field F that lies above the plane z1, oriented upwards. 2yzi 3yj +xk and the surface S the part of the paraboloid z 5-x2-y To verify Stokes' Theorem we will compute the expression on each side. First computecurl F dS curl F0,3+2y,-2 Edy dx curl F dS- where x2 = curl F ds- Now compute F.dr The boundary curve C of the surface S can be...
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Use Stokes' Theorem to evaluate Sc F. dr where C is oriented counterclockwise as viewed from above. F(x, y, z) = xyl +22+ 4yk, C is the curve of intersection of the plane X + 2 = 10 and the cylinder x2 + y2 - 36.
Use Stoke's Theorem to evaluate ScF. dr, where F(x, y, z) = -xzzi + y2zj + zºk and C is the curve of intersection of the planez = 1 – X – Y and the cylinder x2 + y2 = 1, oriented counterclockwise as viewed from above.
F(x, y,z)=(y2 +e", 2xy + z sin y, cos y) is a gradient vector field. Compute Sc F. dr where C=GUC,, C işthe curve y = x^, z = 0 from (0,0,0) to (1,1,0) and C, is the straight line from (1,1,0) to (2,2,3)
F(x, y,z) = (y2 +e", 2xy + z sin y, cos y) is a gradient vector field. Compute Sc F. dr where C=GUC,, C işthe curve y = x^, z = 0 from (0,0,0) to (1,1,0) and C, is the straight line from (1,1,0) to (2,2,3).
the plane 7-1 with the cylinder Consider the vector field F(x, y, z) = (x²); + (x+y); + (4y2Z) K and the curve C defined by the intersection Counter clockwise as viewed from above. Evaluate the Work- SF. dr done by F along in the following ways (a) Directly, using parametrization of C (b) Using stakes theorem
Consider the vector field F(x, y, z) = (z arctan(y2), 22 In(22 +1), 32) Let the surface S be the part of the sphere x2 + y2 + x2 = 4 that lies above the plane 2=1 and be oriented downwards. (a) Find the divergence of F. (b) Compute the flux integral SS. F . ñ ds.
7. Find (a) the curl and (b) the divergence of the vector field F(x, y, z)= e' sin yi+e' cos yj+zk F.de where is the curve of intersection of the plane : = 5 - x and the cylinder rº + y2 = 9. 8. Use Stokes Theorem to evaluate F(x, y, - ) = xyi +2=j+3yk
F(x,y,z)= (y² +e",2xy +z sin y, -cos y) is a gradient vector field. Compute Sc F. dr where C=C UC2, C, is the curve y=x*, z =0 from (0,0,0) to (1,1,0) and C, is the straight line from (1,1,0) to (2,2,3).