Question

prove that the curl of the gradient of
(scalar function) is zero​

Answer 1

SOLUTION :

TO PROVE

The curl of the gradient of (scalar function) is zero

In vector form for function f

$\sf{ \nabla \times ( \nabla f \: ) = \vec{ 0} }$

PROOF

$\sf{ \nabla \times ( \nabla f \: ) }$

$\displaystyle = \sf{ \nabla \times \bigg( \: \hat{i} \: \frac{ \partial f}{ \partial x} + \hat{j} \: \frac{ \partial f}{ \partial y} + \hat{k} \: \frac{ \partial f}{ \partial z} \: \bigg) }$

$= \sf{ \displaystyle\begin{vmatrix} \hat{i} & \hat{j} & \hat{k}\\ \frac{ \partial}{ \partial x} & \frac{ \partial}{ \partial y} & \frac{ \partial}{ \partial z} \\ \frac{ \partial f}{ \partial x} & \frac{ \partial f}{ \partial y} & \frac{ \partial f}{ \partial z} \end{vmatrix} }$

$= \displaystyle \sf\sum \: \hat{i} \bigg( \: \frac{ { \partial}^{2} f}{ \partial y \partial z} - \frac{ { \partial}^{2} f}{ \partial y \partial z} \: \bigg)$

$= \vec{0}$

Hence proved

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LEARN MORE FROM BRAINLY

Divergence of r / r^3 is

(a) zero at the origin

(b) zero everywhere

(c) zero everywhere except the origin

(d) nonzero

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