91影视

Imagine the vector field below:

The goal is to demonstrate that this vector field is irrotational at the site in question.

Irrotational: If an electric field vector is irrotational at a fixed location $P$, the field is said to be irrotational at that point.

A scalar field's curl is described this way:

$=\left[\frac{\mathrm{鈭�}{F}_3}{\mathrm{鈭�}y}鈭�\frac{\mathrm{鈭�}{F}_2}{\mathrm{鈭�}z}\right]\mathbf{i}鈭�\left[\frac{\mathrm{鈭�}{F}_3}{\mathrm{鈭�}x}鈭�\frac{\mathrm{鈭�}{F}_1}{\mathrm{鈭�}z}\right]\mathbf{j}+\left[\frac{\mathrm{鈭�}{F}_2}{\mathrm{鈭�}x}鈭�\frac{\mathrm{鈭�}{F}_1}{\mathrm{鈭�}y}\right]\mathbf{k}$

Get the curl of the vector fieldusing the following formula:

$\begin{array}{r}=\left[\frac{\mathrm{鈭�}}{\mathrm{鈭�}y}(4z+12)鈭�\frac{\mathrm{鈭�}}{\mathrm{鈭�}z}\left(3y^3\right)\right]\mathbf{i}鈭�\left[\frac{\mathrm{鈭�}}{\mathrm{鈭�}x}(4z+12)\right.\left.鈭�\frac{\mathrm{鈭�}}{\mathrm{鈭�}z}(鈭�\sin 鈦�x)\right]\mathbf{j}\\ +\left[\frac{\mathrm{鈭�}}{\mathrm{鈭�}x}\left(3y^3\right)鈭�\frac{\mathrm{鈭�}}{\mathrm{鈭�}y}(鈭�\sin 鈦�x)\right]\mathbf{k}\end{array}$

$=[0鈭�0]\mathbf{i}鈭�[0鈭�0]\mathbf{j}+[0鈭�0]\mathbf{k}$

$=0i+0j+0k$

$=0$

Take note of the following for the vector field: ;

$curlF=0$

The answer at the position $P=(2,3,4)$, the vector field $F(x,y,z)$ is irrotational.