# Consider the vector field F = <<5z, x, 5y>> and the surface which is the part of the elliptic paraboloid z = x^2+5y^2 that lies below the plane z = 5.

Consider the vector field $F=⟨5z,x,5y⟩$ and the surface which is the part of the elliptic paraboloid $z={x}^{2}+5{y}^{2}$ that lies below the plane z = 5. Calculate curl(F) and then apply Stokes' Theorem to compute the exact magnitude of the flux of curl(F) through the surface using line integral. You do not need to cinfirm your answer by evaluating the double integral of curl(F) over the surface(the right-hand side of Stokes' Theorem).
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Step 1
Curl:
$CurlF=<\left(\frac{\partial }{\partial y}\left(5y\right)-\frac{\partial }{\partial z}x\right),\left(\frac{\partial }{\partial z}\left(5z\right)-\frac{\partial }{\partial x}\left(5y\right)\right),\left(\frac{\partial }{\partial x}\left(x\right)-\frac{\partial }{\partial y}\left(5z\right)\right)>$
$=<5,5,1>$
Step 2
So using stokes theorem: $\oint {\oint }_{S}curlF.ds={\oint }_{C}F.dr$ where C is the curve and S is the surface enclosed by the curve.
At S , z=5 so the parabloid $z={x}^{2}+5{y}^{2}$ will become an ellipse, $5={x}^{2}+5{y}^{2}$.
Step 3
To find the flux of the curl across S:
$\oint {\oint }_{S}curlF.ds={\oint }_{C}F.dr$
$={\oint }_{C}<5z,x,5y>.$
$={\oint }_{C}<25,x,5y>.$
$={\oint }_{C}25dx+xdy$
Step 4
Now using Gauss Theorem for 2D:
$\oint {\oint }_{S}curlF.ds={\oint }_{C}25dx+xdy$
$=\oint {\oint }_{S}\left(\frac{\partial }{\partial x}x-\frac{\partial }{\partial y}25\right)dxdy$
$=\oint {\oint }_{S}dxdy$
=Area of the ellipse
$=\pi ×\sqrt{5}×1$
$=\sqrt{5\pi }$
Step 5
Thus, the flux of the curl across the surface is $\sqrt{5\pi }$.