Question

The surface integral \(\mathop \int\!\!\!\int \limits_s \frac{1}{\pi }\left( {9xi - 3yj} \right)\).nds over the sphere given by x² + y²+ z² = 9 is

Answer 1

By Gauss divergence theorem

\(\mathop \smallint \limits_s \bar F.\bar nds\; = \mathop \smallint \limits_v div\bar Fdv\)

Here

div F̅ = 9 – 3 =6

\(\therefore \mathop \int\!\!\!\int \limits_s \frac{1}{\pi }\left( {9x\bar i - 3y\bar j} \right).\bar nds = \frac{1}{\pi }\mathop \smallint \limits_v 6dv = \frac{1}{\pi }6v\)

\(= \frac{1}{\pi }6\left( {\frac{4}{3}\pi {r^3}} \right)\)

= 8(3)³

= 216