GATE EE 2015 Set 1 — Question 24
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Engineering Mathematics → Vector Calculus → Gradient, Divergence & Curl
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Question
Consider a function , where is the distance from the origin and is the unit vector in the radial direction. The divergence of this function over a sphere of radius , which includes the origin, is
Correct answer
(C) 4π
Solution
The given function is . According to Gauss's Divergence Theorem, the integral of the divergence over a volume is equal to the flux through the bounding surface : For a sphere of radius centered at the origin, the surface element is . The flux is: Thus, the divergence of this function over the sphere is . Note that in this case.
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