Allied & ICAR Exam Bank • ALLIED_1046 NAT • 2 Marks

Section 1: Engineering Mathematics

Vector Calculus

Subtopic: Surface integrals

Question Statement:

If $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ and $r = |\vec{r}|$, then $\nabla(r^3)$ is equal to $k r \vec{r}$. The value of $k$ is:

Answer Key:

3

Step-by-Step Solution & Explanation

$\nabla(f(r)) = f'(r) \nabla r = f'(r) \frac{\vec{r}}{r}$. Here $f(r) = r^3 \implies f'(r) = 3r^2$. Thus $\nabla(r^3) = 3r^2 \frac{\vec{r}}{r} = 3r\vec{r}$. Hence $k = 3$.
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