Let C be the intersection curve of the cylinder x ^ 2 + y ^ 2 = 4 and the plane z = x + 5. Calculate the circulation of the vector field F (x, y, z) = yi-xj + sin (z ^ 2) k around C
Added by Stacy W.
Step 1
To parameterize this curve, we can use the following parameterization: x = 2cos(t) y = 2sin(t) z = x + 5 = 2cos(t) + 5 where t is the parameter that varies from 0 to 2π. Show more…
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Compute the flux of the vector field F(x, y, z) =( x, y, z ) across the vertical the part of the cylinder x^2 +y^2 = 4 between the planes z = 1 and z = 4 whose orientation is pointing outside the cylinder.
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Use the surface integral in Stokes' Theorem to calculate the circulation of the field $\mathbf{F}$ around the curve $C$ in the indicated direction. $$\mathbf{F}=x^{2} y^{3} \mathbf{i}+\mathbf{j}+z \mathbf{k}$$ $C:$ The intersection of the cylinder $x^{2}+y^{2}=4$ and the hemisphere $x^{2}+y^{2}+z^{2}=16, z \geq 0,$ counterclockwise when viewed from above
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Stokes’ Theorem
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