Project ForgeEngineering Mathematics
Linear Algebra
GATE 2009
Medium
Consider the integral ∬(2xi−2yj+5zk)·n̂dS over the surface of a sphere of radius=3 with center at the origin, and surface unit normal n̂ pointing away from the origin. Using the Gauss divergence theorem, the value of this integral is
- A−180π
- B0
- C90π
- 180π
Explanation
div F=∂(2x)/∂x+∂(−2y)/∂y+∂(5z)/∂z=2−2+5=5 (constant). Volume of sphere r=3 is (4/3)π(27)=36π. Integral=5×36π=180π.
Source: GATE CH 2009 (2009)
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