Question

Find the curl for \(\vec{r}=x^2 yz\vec{i}+(3x+2y)z\vec{j}+21z^2 x\vec{k}\).

(a) \(\vec{i}(3x+2y)-\vec{j}(11z^2-x^2 y)+\vec{k}(3z-x^2 z)\)

(b) \(\vec{i}(x+2y)-\vec{j}(21z^2-x^2 y)+\vec{k}(3z-x^2 z)\)

(c) \(-\vec{i}(3x+2y)-\vec{j}(21z^2-x^2 y)+\vec{k}(3z-x^2 z)\)

(d) \(\vec{i}(3x+2y)-\vec{j}(21z^2-x^2 y)+\vec{k}(3z-x^2 z)\)

This question was addressed to me in an interview for internship.

The doubt is from Using Properties of Divergence and Curl in section Vector Differential Calculus of Engineering Mathematics

Answer 1

Right option is (c) \(-\vec{i}(3x+2y)-\vec{j}(21z^2-x^2 y)+\vec{k}(3z-x^2 z)\)

To explain I would say: We know that the curl for any vector quantity is given by

\(\bigtriangledown.\vec{r}=\begin{bmatrix}\vec{i}&\vec{j}&\vec{k}\\\frac{\partial}{\partial x}&\frac{\partial}{\partial y}&\frac{\partial}{\partial z}\\x^2 yz&(3x+2y)z&21z^2 x\end{bmatrix}\)

\(\bigtriangledown.\vec{r}=\vec{i}\left (\frac{\partial(21z^2 x)}{\partial y}-\frac{\partial((3x+2y)z)}{\partial z}\right)-\vec{j}\left (\frac{\partial(21z^2 x)}{\partial x}-\frac{\partial(x^2 yz)}{\partial z}\right)\)

\(+\vec{k}\left (\frac{\partial((3x+2y)z)}{\partial x}-\frac{\partial(x^2 yz)}{\partial y}\right)\)

\(\bigtriangledown.\vec{r}=-\vec{i}(3x+2y)-\vec{j}(21z^2-x^2 y)+\vec{k}(3z-x^2 z)\).