# 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.

Compute the divergence of the the velocity field \(\vec V=\left\langle 3xy^2, 3x^2y, z^3 \right\rangle\). At which points is it expanding or contracting?

\(\vec\nabla\cdot\vec V=3y^2+3x^2+3z^2\)

The fluid is expanding everywhere.

We use the definition of the divergence: \[\begin{aligned} \vec\nabla\cdot\vec V &=\dfrac{\partial }{\partial x}(3xy^2) +\dfrac{\partial }{\partial y}(3x^2y) +\dfrac{\partial }{\partial z}(z^3) \\ &=3y^2+3x^2+3z^2 \end{aligned}\] Since this is always positive, the fluid is expanding everywhere.

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