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Gauss Divergence Theorem

The surface i...

Question

The surface integral $\int \int_{s} F . n d S$ over the surface S of the sphere $x^{2} + y^{2} + z^{2} = 9$ , where F =(x+y)i + (x + z)j + (y + z)k and n is the unit outward surface normal, yields_______
226.19

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Solution

The correct option is A 226.19
By Gauss-Divergence Theorem
$\int_{s} \int F . n d s = \int_{v} d i v . F d V$
$\int_{v} \nabla . [(x + y)^i + (x + z)^j + (y + z)^k] d V$
$= \int_{v} (1 + 0 + 1) d V$
$= 2 \int_{v} d V = 2 V$
$V = \frac{4}{3} π . {(3)}^{3} = 36 π$
$2 V = 2 \times 36 π = 226.19$

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