91Ó°ÊÓ

A vector field is an assigning of a vector to every point in a subset of space in vector calculus and physics. A vector field in the plane, for example, might be represented as a collection of arrows, including a certain magnitude and direction, each tied to a specific point in the plane.

Consider the vector field below:

The goal is to demonstrate that the vector field is rotating at$P=(1,1,1).$

Irrotational:

in the case of a vector field $F(x,y,z)$If curl $F=0$at point$P$, this field is referred to as irrotational at that location.

A vector field's curl is defined as follows:

$CurlF(x,y,z)=\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ \frac{∂}{∂x}& \frac{∂}{∂y}& \frac{∂}{∂z}\\ F_1(x,y,z)& F_2(x,y,z)& F_3(x,y,z)\end{array}\right|$

$=\left[\frac{∂F_3}{∂y}-\frac{∂F_2}{∂z}\right]\mathbf{i}-\left[\frac{∂F_3}{∂x}-\frac{∂F_1}{∂z}\right]\mathbf{j}+\left[\frac{∂F_2}{∂x}-\frac{∂F_1}{∂y}\right]\mathbf{k}$

The curl of vectoris defined as follows:

$CurlF(x,y,z)=\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ \frac{∂}{∂x}& \frac{∂}{∂y}& \frac{∂}{∂z}\\ 2xyz& x^{2}z& x^{2}y\end{array}\right|$

$=\left[\frac{∂\left(x^{2}y\right)}{∂y}-\frac{∂\left(x^{2}z\right)}{∂z}\right]\mathbf{i}-\left[\frac{∂\left(x^{2}y\right)}{∂x}-\frac{∂\left(2xyz\right)}{∂z}\right]\mathbf{j}+\left[\frac{∂\left(x^{2}z\right)}{∂x}-\frac{∂\left(2xyz\right)}{∂y}\right]\mathbf{k}$

$=\left[x^2-x^2\right]\mathbf{i}-[2xy-2xy]\mathbf{j}+[2xz-2xz]\mathbf{k}$

$$\begin{gathered} =0\mathbf{i}+0\mathbf{j}+0\mathbf{k} \\ \mathbf{=}\mathbf{0} \end{gathered}$$

Note that the vector field:;

$CurlF=0$

Hence, It is proved that the Vector field$\mathbf{F}(x,y,z)$is rotational at the point $P=(1,1,1)$.