Give an example of a field with negative divergence at the origin.

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:

$\mathrm{\nabla }\cdot \mathbf{F}=\frac{\mathrm{\partial }{F}_1}{\mathrm{\partial }x}+\frac{\mathrm{\partial }{F}_2}{\mathrm{\partial }y}+\frac{\mathrm{\partial }{F}_3}{\mathrm{\partial }z}$

Notice that, the value of ∇⋅F is a scalar.

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

The divergence of a vector field will be

localid="1654152522225" $\begin{array}{rcl}\begin{array}{r}\mathrm{\nabla }\cdot \mathbf{F}\end{array}& =& \frac{\mathrm{\partial }}{\mathrm{\partial }x}(\sin x)+\frac{\mathrm{\partial }}{\mathrm{\partial }y}(-3\sin y)+\frac{\mathrm{\partial }}{\mathrm{\partial }z}(\sin z)\\ & =& \cos x-3\cos y+\cos z\end{array}$

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" $\begin{array}{rcl}\begin{array}{r}\mathrm{\nabla }\cdot \mathbf{F}\end{array}& =& \cos 0-3\cos 0+\cos 0\\ & =& \begin{array}{r}1-3\cdot 1+1\end{array}\\ & =& \begin{array}{r}1-3+1\end{array}\\ & =& -1\end{array}$

Which is negative.

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

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