The GATE Grind

GATE 2023 CE (CE2) – Question 11

Engineering Mathematics · Calculus: Integrals, partial and total derivatives, gradient, divergence, curl, line, surface and volume integrals · 1 mark · Multiple choice

Let $\phi$ be a scalar field, and $\mathbf u$ be a vector field. Which of the following identities is true for $\operatorname{div}(\phi\mathbf u)$?

  1. $\operatorname{div}(\phi\mathbf u)=\phi\operatorname{div}\mathbf u+\mathbf u\cdot\operatorname{grad}\phi$
  2. $\operatorname{div}(\phi\mathbf u)=\phi\operatorname{div}\mathbf u+\mathbf u\times\operatorname{grad}\phi$
  3. $\operatorname{div}(\phi\mathbf u)=\phi\operatorname{grad}\mathbf u+\mathbf u\cdot\operatorname{grad}\phi$
  4. $\operatorname{div}(\phi\mathbf u)=\phi\operatorname{grad}\mathbf u+\mathbf u\times\operatorname{grad}\phi$

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Correct answer: (A) $\operatorname{div}(\phi\mathbf u)=\phi\operatorname{div}\mathbf u+\mathbf u\cdot\operatorname{grad}\phi$

Explanation

**Write $\operatorname{div}(\phi\mathbf u)$ in components** with $\mathbf u=(u_1,u_2,u_3)$:
$$\operatorname{div}(\phi\mathbf u)=\sum_{i}\frac{\partial(\phi u_i)}{\partial x_i}.$$

**Apply the product rule** to each term:
$$\frac{\partial(\phi u_i)}{\partial x_i}=\phi\frac{\partial u_i}{\partial x_i}+u_i\frac{\partial\phi}{\partial x_i}.$$

**Sum over $i$:**
$$\operatorname{div}(\phi\mathbf u)=\phi\sum_i\frac{\partial u_i}{\partial x_i}+\sum_iu_i\frac{\partial\phi}{\partial x_i}=\phi\operatorname{div}\mathbf u+\mathbf u\cdot\operatorname{grad}\phi .$$

Options B and D use a cross product (which would give a vector, but the divergence is a scalar), and C uses $\operatorname{grad}\mathbf u$ (a tensor). Answer **A**.