Question

The expression curl $\left(gradf\right)$, where $f$ is a scalar function, is

• A. euqal to ${▽}^{2}f$
• B. equal to div $\left(gradf\right)$
• C. a scalar of zero magnitude
• D. a vector of zero magnitude

Answer 1

The correct option is D a vector of zero magnitude

$gradf=\frac{\partial f}{\partial x}i+\frac{\partial f}{\partial y}j+\frac{\partial f}{\partial z}k$
$curl\left(gradf\right)=\left|\begin{array}{ccc}i& j& k\\ \frac{\partial }{\partial x}& \frac{\partial }{\partial y}& \frac{\partial }{\partial z}\\ \frac{\partial f}{\partial x}& \frac{\partial f}{\partial y}& \frac{\partial f}{\partial z}\end{array}\right|$
$=(\frac{{\partial }^{2}f}{\partial y\partial z}-\frac{{\partial }^{2}f}{\partial z\partial y})i+(\frac{{\partial }^{2}f}{\partial z\partial x}-\frac{{\partial }^{2}f}{\partial x\partial z})j+(\frac{{\partial }^{2}f}{\partial x\partial y}-\frac{{\partial }^{2}f}{\partial y\partial x})k$
$curl\left(gradf\right)=o\hat{i}+o\hat{j}+o\hat{k}$
Note: Important Repeated Operations by Operator $(▽)$
$\left(1\right)Curl\left(gradf\right)=▽\times ▽f=0$
$\left(2\right)div.\left(gradf\right)=▽.▽f={▽}^{2}f$
$\left(3\right)div.\left(curlf\right)=▽.(▽\times f)=0$