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GATE 2022 ME (ME2) – Question 12

Engineering Mathematics · Calculus: Gradient, divergence, curl, vector identities, line, surface and volume integrals · 1 mark · Multiple choice

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 $\int_A\mathbf F\cdot d\mathbf A$ of a vector field $\mathbf F=3x\hat i+5y\hat j+6z\hat k$ over the entire surface $A$ of the cube is ______.

  1. 14
  2. 27
  3. 28
  4. 31

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Correct answer: (A) 14

Explanation

**Step 1: use the divergence theorem.** The flux of $\mathbf F$ out of the closed surface equals the integral of its divergence over the enclosed volume:
$$\oint_A\mathbf F\cdot d\mathbf A=\int_V\nabla\cdot\mathbf F\,dV.$$

**Step 2: compute the divergence.** For $\mathbf F=3x\,\hat i+5y\,\hat j+6z\,\hat k$:
$$\nabla\cdot\mathbf F=\frac{\partial(3x)}{\partial x}+\frac{\partial(5y)}{\partial y}+\frac{\partial(6z)}{\partial z}=3+5+6=14.$$

**Step 3: integrate.** The divergence is a constant, so the integral is $14\times\text{volume}$. A cube of unit edge has volume 1, wherever its centre is, so the flux is **14**.