Use Stokes' Theorem to evaluate ?_C F · dr where C is oriented counterclockwise as viewed from above. F(x, y, z) = yzi + 5xzj + e^xyk, C is the circle x^2 + y^2 = 16, z = 4.
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First, we need to find the curl of the vector field F. Curl(F) = ∇ x F = (dFz/dy - dFy/dz)i + (dFx/dz - dFz/dx)j + (dFy/dx - dFx/dy)k F(x, y, z) = yzi + xzj + xyzk dFz/dy = x, dFy/dz = x, dFx/dz = 0, dFz/dx = y, dFy/dx = z, dFx/dy = z Curl(F) = (x - x)i + (0 Show more…
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Use Stokes' Theorem to evaluate $\oint_{C} \mathbf{F} \cdot d \mathbf{r}$ $$ \begin{array}{l}{\mathbf{F}(x, y, z)=(x-y) \mathbf{i}+(y-z) \mathbf{j}+(z-x) \mathbf{k} ; \quad C \text { is the }} \\ {\text { circle } x^{2}+y^{2}=a^{2} \text { in the } x y \text { -plane with counterclockwise }} \\ {\text { orientation looking down the positive } z \text { -axis. }}\end{array} $$
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Stokes’ Theorem
Use Stokes' Theorem to evaluate ∮C F · dr where C is oriented counterclockwise as viewed from above. F(x, y, z) = yzi + 5xzj + exyk, C is the circle x^2 + y^2 = 9, z = 2.
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