MATH 251 Lecture 30

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Divergence

Start with a vector field $\overrightarrow{F}(x,y)=a(x,y)\hat{ı}+b(x,y)\hat{ȷ}$

Divergence represents the net flow out of a point (x,y). This is a scalar field, which is a regular old function.

$\mathrm{d}\mathrm{i}\mathrm{v}\overrightarrow{F}=\mathrm{\nabla }\cdot \overrightarrow{F}=\frac{\partial a}{\partial x}+\frac{\partial b}{\partial y}$

Think of the vector field as the flow of a fluid.

A source point is a point at which there is more fluid moving away than moving toward: $\mathrm{\nabla }\cdot \overrightarrow{F}>0$

A sink point is a point at which there is more fluid moving toward than away: $\mathrm{\nabla }\cdot \overrightarrow{F}<0$

An incompressible point is a point at which there is an equal amount of fluid moving toward and away: $\mathrm{\nabla }\cdot \overrightarrow{F}=0$

Derivation

Measure fluid produced in a small box around point $(x_0,y_0)$. The box is a square from $(x_0-h,y_0-h)$ to $(x_0+h,y_0+h)$.

Let $a\hat{ı}$ represent the difference between fluid out and fluid in (out − in) in the x-direction.

• Right: $a(x_0+h,y_0)\cdot (2h)$
• Top: $b(x_0,y_0+h)\cdot (2h)$
• Left: $-a(x_0-h,y_0)\cdot (2h)$
• Bottom: $-b(x_0,y_0-h)\cdot (2h)$

The sum of all of these equals the area of the box ($4h^2$) times average production in box (APB). Solve for average production to get

$\mathrm{A}\mathrm{P}\mathrm{B}=\frac{a(x_0+h,y_0)-a(x_0-h,y_0)}{2h}+\frac{b(x_0,y_0+h)-b(x_0,y_0-h)}{2h}$

Take the limit as $h\to 0$, and this turns into a derivative:

$\frac{\partial a}{\partial x}+\frac{\partial b}{\partial y}$

Example

$\begin{array}{rl}\overrightarrow{F}& =3x^2\hat{ı}-(x+4y)\hat{ı}\\ \mathrm{\nabla }\cdot \overrightarrow{F}& =6x-4\end{array}$

Explanation of Notation

$\mathrm{\nabla }=\frac{\partial }{\partial x}\hat{ı}+\frac{\partial }{\partial y}\hat{ȷ}$ is a "differential operator":

gradient

$\mathrm{\nabla }f=\frac{\partial f}{\partial x}\hat{ı}+\frac{\partial f}{\partial y}\hat{ȷ}$

divergence

$\begin{array}{rl}\mathrm{\nabla }\cdot \overrightarrow{F}& =(\frac{\partial }{\partial x}\hat{ı}+\frac{\partial }{\partial y}\hat{ȷ})\cdot (a\hat{ı}+b\hat{ȷ})\\ & =\frac{\partial a}{\partial x}+\frac{\partial b}{\partial y}\end{array}$

Curl

Measures the circulation (imagine a vortex) of $\overrightarrow{F}$ around a point (by the right hand rule). In 3D space, $(\mathrm{\nabla }\times \overrightarrow{F})\cdot \hat{n}$ gives the circulation of $\overrightarrow{F}$ around $\hat{n}$ axis.

Let ${\partial }_x=\frac{\partial }{\partial x}$

$\begin{array}{rl}\mathrm{\nabla }\times \overrightarrow{F}& =({\partial }_x\hat{ı}+{\partial }_y\hat{ȷ}+{\partial }_k\hat{k})\times (a\hat{ı}+b\hat{ȷ}+c\hat{k})\\ & =\left|\begin{array}{ccc}\hat{ı}& \hat{ȷ}& \hat{k}\\ {\partial }_x& {\partial }_y& {\partial }_z\\ a& b& c\end{array}\right|\end{array}$

Special Case

When $\overrightarrow{F}=a(x,y)\hat{ı}+b(x,y)\hat{ȷ}$, the curl is equal to:

$\left|\begin{array}{ccc}\hat{ı}& \hat{ȷ}& \hat{k}\\ {\partial }_x& {\partial }_y& {\partial }_z\\ a& b& 0\end{array}\right|=(\frac{\partial b}{\partial x}-\frac{\partial x}{\partial y})\hat{k}$