Project ForgeEngineering Mathematics
Calculus
GATE 2022
Easy
Consider a cube of unit edge length and sides parallel to co-ordinate axes, with its centroid at the point (1, 2, 3). The surface integral ∫∫_A F⃗·dA⃗ of a vector field F⃗ = 3xî + 5yĵ + 6zk̂ over the entire surface A of the cube is ______.
- 14
- B27
- C28
- D31
Explanation
By the divergence theorem, the surface integral equals ∫∫∫_V (∇·F⃗) dV. ∇·F⃗ = ∂(3x)/∂x + ∂(5y)/∂y + ∂(6z)/∂z = 3+5+6 = 14, a constant. The cube has unit volume, so the integral = 14×1 = 14.
Source: GATE ME 2022 Session 2 (ME-2) (2022)
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