Using the Divergence Theorem, evaluate int int_S F.NdS, where F(x,y,z)=(z^3i-x^3j+y^3k) and S is the sphere x^2+y^2+z^2=a^2, with outward unit normal vector N.

usagirl007A 2021-01-19 Answered
Using the Divergence Theorem, evaluate SF.NdS, where F(x,y,z)=(z3ix3j+y3k) and S is the sphere x2+y2+z2=a2, with outward unit normal vector N.
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Expert Answer

casincal
Answered 2021-01-20 Author has 82 answers
Step 1
Divergence theorem relates surface integrals and volume integrals. By using the Gauss divergence theorem we can evaluate this surface integral.
The Gauss divergence theorem formula can be stated as follows:
SF.NdS=V÷F.dV
÷F=(z3ix)+(x3jy)+(y3kz)
Step 2
div F = 0, since the partial derivative of z3 with respect to x is zero and partial derivative of x3 with respect to y is zero and the partial derivative of y3 with respect to z is zero.
Then V0.dV=0
Since SF.NdS=V÷F.dV
Then by Gauss Divergence theorem, we can say SF.NdS=0.
Step 3
So the final answer is zero by the Gauss Divergence theorem.
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