(a) If $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$, then show that : $…

Question

(a) If $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$, then show that : $\nabla^2(r^m) = m(m - 1)r^{m-2}$.

(b) Find the divergence and curl of the vector function $\vec{F} = x^2 y\hat{i} + xy\hat{j} + 3yz\hat{k}$ at the point $(-1, 1, 1)$.

(c) Find the circulation of $\vec{F}$ around the curve $C$, where $\vec{F} = y\hat{i} + z\hat{j} + x\hat{k}$ and $C$ is the circle $x^2 + y^2 = 1, z = 0$.

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