91影视

We need to find an example of a field with negative divergence at the origin.

If $\mathbf{F}(x,y,z)=F_1(x,y,z)\mathbf{i}+F_2(x,y,z)\mathbf{j}+F_3(x,y,z)\mathbf{k}$is a vector field, then the divergence of the vector field is defined as follows:

$\mathrm{鈭�}鈰�\mathbf{F}=\frac{\mathrm{鈭�}{F}_1}{\mathrm{鈭�}x}+\frac{\mathrm{鈭�}{F}_2}{\mathrm{鈭�}y}+\frac{\mathrm{鈭�}{F}_3}{\mathrm{鈭�}z}$

Notice that, the value of 鈭団媴F is a scalar.

Consider the following vector field $\mathbf{F}(x,y,z)=\sin 鈦�x\mathbf{i}鈭�3\sin 鈦�y\mathbf{j}+\sin 鈦�z\mathbf{k}$

The divergence of a vector field will be

localid="1654152522225" $\begin{array}{rcl}\begin{array}{r}\mathrm{鈭�}鈰�\mathbf{F}\end{array}& =& \frac{\mathrm{鈭�}}{\mathrm{鈭�}x}(\sin 鈦�x)+\frac{\mathrm{鈭�}}{\mathrm{鈭�}y}(鈭�3\sin 鈦�y)+\frac{\mathrm{鈭�}}{\mathrm{鈭�}z}(\sin 鈦�z)\\ & =& \cos 鈦�x鈭�3\cos 鈦�y+\cos 鈦�z\end{array}$

At the origin, that is, at $\left(x,y,z\right)=\left(0,0,0\right)$, the divergence of this vector field will be,

localid="1654152584029" $\begin{array}{rcl}\begin{array}{r}\mathrm{鈭�}鈰�\mathbf{F}\end{array}& =& \cos 鈦�0鈭�3\cos 鈦�0+\cos 鈦�0\\ & =& \begin{array}{r}1鈭�3鈰�1+1\end{array}\\ & =& \begin{array}{r}1鈭�3+1\end{array}\\ & =& 鈭�1\end{array}$

Which is negative.

Therefore, an example of a field with negative divergence at the origin is,

$\mathbf{F}(x,y,z)=\sin 鈦�x\mathbf{i}鈭�3\sin 鈦�y\mathbf{j}+\sin 鈦�z\mathbf{k}$.