Surface integral over a discontinuous flux

Calculus Level 5

Find the flux \(A\) over the surface region of the sphere of radius 5, centred in origin, limited by the plans \(z=3\) and \(z=-4\), where the field is: \(F(x,y,z)=\left(\frac{y}{\sqrt{I}},\dfrac{-x}{\sqrt{I}}+7y,-7z\right)\)

\( {I}=(4-\sqrt{x^{2}+y^{2}})^{2}+z^{2}\)

For that, do you will need to find the flux over the torus of radius \( R=4 \) and \(r=\epsilon\), centred in origin, and make the limit:

\(\displaystyle B= \lim_{\epsilon \to 0} \int \int_T F(x,y,z)\cdot \hat{n_{in}} \,dS. \)

Your answer for \(A\) should be something like:

\(\displaystyle A=a\pi+\lim_{\epsilon \to 0} (b\pi^{2} \epsilon^{2} )\)

Submit \(a+b\) as your answer.


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