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Question 3 (10 marks) Use Stokes' theorem to evaluate ff(VxG)•dS where G = 2x² yi +...
Let F = < x-eyz, xexx, z?exy >. Use Stokes' Theorem to evaluate slice curlĒ ds, where S is the hemisphere x2 + y2 + z2 = 1, 2 > 0, oriented upwards.
Use Stokes' Theorem to evaluate curl F. ds. F(x, y, z) = zeli + x cos(y)j + xz sin(y)k, S is the hemisphere x2 + y2 + z2 = 4, y 2 0, oriented in the direction of the positive y-axis.
6. Use the Divergence Theorem to evaluate SSF. ds, where Ể(x, y, z) = (x/x2 + y2 + z2 , yvx2 + y2 + z2 , z7 x2 + y2 + z2 ) and S consists of the hemisphere z V1 – x2 - y2 and the disk x2 + y2 = 1 in the xy-plane.
Please show work
Page 1 10. Use Stokes Theorem to evaluate S. curl F. ds F = (x, y, z) = z² i + 2xj + y2k, S:z = 1 - x2 - y2, z 20
Problem 4: Use the surface integral in Stokes' theorem to evaluate F.dr for the hemisphere S : x2 + y2 + z2 = 9; z > 0, its bounding circle C: 2+9 and the field F-yi- xj. You only have to compute the surface integral, not the line integral. (20 points)
Verify that Stokes' Theorem is true for the vector field
Help Entering Answers (1 point) Verify that Stokes' Theorem is true for the vector field F -yi+ zj + xkand the surface S the hemisphere x2 + y2 + z2-25, y > 0oriented in the direction of the positive y- axis To verify Stokes' Theorem we will compute the expression on each side. First compute curl F dS curl F The surface S can be parametrized by S(s, t) -...
Use Stokes' Theorem (in reverse) to evaluate Sf (curl F). n ds where F = 5yzi + 9x j +2yze+'k ,S is the portion of the paraboloid z = x2 + aby2 for 0 sz s 3, and the unit normal on S points away from the z-axis. 16 Enter your answer symbolically, as in these examples
3. Using Stokes theorem evaluate fa.dr. where A = (x² + y - 4)i + 3ryj + (2x2 + 2?)k and C is the curve bounding the surface S given by (a) the hemisphere IP + y2 + z2 = 16 above the ry plane (b) the paraboloid z = 4 - (z? + y²) above the ry plane.
Use Stokes' Theorem to evaluate.
8. Use Stokes, Theorem to evaluate J, ▽ x ที่ do, where F(x, y, z)-(z2yz,yz2,23ezy and s is part of the sphere x2 + y2 + z-5 that lies above the plane z-1. Also, s is oriented upward.
8. Use Stokes, Theorem to evaluate J, ▽ x ที่ do, where F(x, y, z)-(z2yz,yz2,23ezy and s is part of the sphere x2 + y2 + z-5 that lies above the plane z-1. Also, s is oriented...
Use the Divergence Theorem to evaluate ∬SF⋅dS∬SF⋅dS where
F=〈z2x,y33+3tan(z),x2z−1〉F=〈z2x,y33+3tan(z),x2z−1〉
and SS is the top half of the sphere x2+y2+z2=9x2+y2+z2=9.
(1 point) Use the Divergence Theorem to evaluate FdS where F2x +3 tan2).^z-1 and S is the top half of the sphere x2 +y2 + z2 -9 Hint: S is not a closed surface. First compute integrals overs, and S2 , where S, is the disk x2 + y2 < 9, z = 0 oriented downward and S2 = S U...