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GATE 2017 ME (ME2) – Question 36

Engineering Mathematics · Calculus: Gradient, divergence, curl, vector identities, line, surface and volume integrals · 2 marks · Numerical answer

Evaluate $\iint_S\mathbf F\cdot\mathbf n\,dS$ on $x^2+y^2+z^2=9$, where $\mathbf F=(x+y)\hat i+(x+z)\hat j+(y+z)\hat k$ and n is the unit outward normal.

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Correct answer: 225 to 227

Explanation

Apply the divergence theorem to the closed sphere. Divergence is ∂(x+y)/∂x+∂(x+z)/∂y+∂(y+z)/∂z=1+0+1=2. Sphere radius is3, so volume is4π·27/3=36π. Outward flux is2·36π=72π=**226.195**, which lies in the official225–227 range.