Project ForgeEngineering Mathematics
Calculus
GATE 2017
Hard
The surface integral ∬ₛ F·n dS over the surface S of the sphere x²+y²+z²=9, where F = (x+y)î + (x+z)ĵ + (y+z)k̂ and n is the unit outward surface normal, yields __________.
226.19(± 1 accepted)
Explanation
By the divergence theorem, the surface integral equals ∭ div F dV. div F = ∂(x+y)/∂x + ∂(x+z)/∂y + ∂(y+z)/∂z = 1+0+1 = 2 (constant). So the integral = 2 × Volume of sphere (radius 3) = 2 × (4/3)π(3)³ = 2 × 36π = 72π ≈ 226.19.
Source: GATE ME 2017 Session 2 (ME-2) (2017)
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