The surface integral \( \int\!\!\!\int \limits_s \frac{1}{\pi }\left( {9xi - 3yj} \right)\).nds over the sphere given by x2 + y2 + z2 = 9 is
By Gauss divergence theorem
\( \smallint \limits_s \bar F.\bar nds\; = \smallint \limits_v div\bar Fdv\)
Here
div F̅ = 9 – 3 =6
\(\therefore \int\!\!\!\int \limits_s \frac{1}{\pi }\left( {9x\bar i - 3y\bar j} \right).\bar nds = \frac{1}{\pi } \smallint \limits_v 6dv = \frac{1}{\pi }6v\)
\(= \frac{1}{\pi }6\left( {\frac{4}{3}\pi {r^3}} \right)\)
= 8(3)3
= 216
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