Project ForgeEngineering Mathematics
Calculus
GATE 2013
Hard
The following surface integral is to be evaluated over a sphere for the given steady velocity vector field F = xi + yj + zk defined with respect to a Cartesian coordinate system having i, j and k as unit base vectors: ∬ₛ (1/4)(F·n) dA, where S is the sphere x²+y²+z²=1 and n is the outward unit normal vector to the sphere. The value of the surface integral is
- π
- B2π
- C3π/4
- D4π
Explanation
By the divergence theorem, ∬ₛ F·n dA = ∭ᵥ (∇·F) dV. Here ∇·F = ∂x/∂x+∂y/∂y+∂z/∂z = 3, so ∬ₛF·n dA = 3·(volume of unit sphere) = 3·(4π/3) = 4π. The requested integral is (1/4) of this: (1/4)(4π) = π.
Source: GATE ME 2013 (Booklet Code A) (2013)
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