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The value of the integral ∯_S r · n dS over the closed surface S bounding a volume V, where r = x î + y ĵ + z k̂ is the position vector and n is the normal to the surface S, is

1. V
2. 2V
3. 3V
4. 4V

Correct answer: 3V

Solution

The integral of the position vector dotted with the outward normal over a closed surface can be evaluated using the divergence theorem, which relates the surface integral to a volume integral. In this case, the divergence of the position vector r is constant and equal to 3, leading to the result being 3 times the volume V.

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