17. Divergence, Curl and Potentials

a. The Divergence Operator

1. Algebraic Definition of Divergence

When we compute the dot product of two vectors, we multiply corresponding components and add them up. When the first of those two vectors is a vector differential operator and the second is a vector field, multiplication is replaced by differentiation, i.e. each component of the differential operator differentiates the corresponding component of the vector field and we add them up.

The divergence of a vector field \(\vec F=\left\langle F_1, F_2, F_3 \right\rangle\), is the function \[\begin{aligned} \text{div} \vec F &=\vec\nabla\cdot\vec F =\left\langle \dfrac{\partial}{\partial x}, \dfrac{\partial}{\partial y}, \dfrac{\partial}{\partial z}\right\rangle \cdot\left\langle F_1, F_2, F_3 \right\rangle \\[2pt] &=\dfrac{\partial F_1}{\partial x} +\dfrac{\partial F_2}{\partial y} +\dfrac{\partial F_3}{\partial z} =\partial_x F_1+\partial_y F_2+\partial_z F_3 \end{aligned}\]

For \(\vec F=\left\langle x^2y,y^2+z^2,z^3-x^2z\right\rangle\), find the divergence \(\vec\nabla\cdot\vec F\).

We use the definition of the divergence: \[\begin{aligned} \vec\nabla\cdot\vec F &=\dfrac{\partial }{\partial x}(x^2y) +\dfrac{\partial }{\partial y}(y^2+z^2) +\dfrac{\partial }{\partial z}(z^3-x^2z) \\ &=2xy+2y+3z^2-x^2 \end{aligned}\]

Fluid Velocity Interpretation

If the vector field is the velocity field of a fluid, \(\vec V\), then the divergence of the velocity field, \(\vec\nabla\cdot\vec V\), measures how much the fluid is spreading out at each point. (See the next page.) If \(\vec\nabla\cdot\vec V=0\), we say vector field is divergence-free and the fluid is incompressible. At a point where \(\vec\nabla\cdot\vec V \gt 0\), we say the fluid is diverging or expanding and the point is a source. At a point where \(\vec\nabla\cdot\vec V \lt 0\), we say the fluid is converging or contracting and the point is a sink.