Gradient of Divergence/Examples

Examples of Use of Gradient of Divergence

Fluid Density Increase

Let $\mathbf V$ be the velocity of a fluid at a point in a region of space $R$.

We have that $\operatorname {div} \mathbf V$ is the scalar rate of change of density with respect to time.

Hence $\grad \operatorname {div} \mathbf V$ gives the magnitude and direction in space of the greatest rate of increase of the space of the density.

Increase of Electric Charge

Let $\mathbf V$ be an electric force in a given electric field at a point in a region of space $R$.

We have that $\operatorname {div} \mathbf V$ is the space charge density.

Hence $\grad \operatorname {div} \mathbf V$ gives the magnitude and direction in space of the greatest rate of increase of the space of the electric charge at a given point.