MATH 251 Lecture 30

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Divergence

Start with a vector field ${\vec {F}}(x,y)=a(x,y){\hat {\imath }}+b(x,y){\hat {\jmath }}$

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

$\mathrm {div} {\vec {F}}=\nabla \cdot {\vec {F}}={\frac {\partial a}{\partial x}}+{\frac {\partial b}{\partial y}}$

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: $\nabla \cdot {\vec {F}}>0$

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

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

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 $a{\hat {\imath }}$ 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

$\mathrm {APB} ={\frac {a(x_{0}+h,y_{0})-a(x_{0}-h,y_{0})}{2h}}+{\frac {b(x_{0},y_{0}+h)-b(x_{0},y_{0}-h)}{2h}}$

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

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

Example

${\begin{aligned}{\vec {F}}&=3x^{2}{\hat {\imath }}-(x+4y){\hat {\imath }}\\\nabla \cdot {\vec {F}}&=6x-4\end{aligned}}$

Explanation of Notation

$\nabla ={\frac {\partial }{\partial x}}{\hat {\imath }}+{\frac {\partial }{\partial y}}{\hat {\jmath }}$ is a "differential operator":

gradient

$\nabla f={\frac {\partial f}{\partial x}}{\hat {\imath }}+{\frac {\partial f}{\partial y}}{\hat {\jmath }}$

divergence

${\begin{aligned}\nabla \cdot {\vec {F}}&=\left({\frac {\partial }{\partial x}}{\hat {\imath }}+{\frac {\partial }{\partial y}}{\hat {\jmath }}\right)\cdot \left(a{\hat {\imath }}+b{\hat {\jmath }}\right)\\&={\frac {\partial a}{\partial x}}+{\frac {\partial b}{\partial y}}\end{aligned}}$

Curl

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

Let $\partial _{x}={\tfrac {\partial }{\partial x}}$

${\begin{aligned}\nabla \times {\vec {F}}&=\left(\partial _{x}{\hat {\imath }}+\partial _{y}{\hat {\jmath }}+\partial _{k}{\hat {k}}\right)\times \left(a{\hat {\imath }}+b{\hat {\jmath }}+c{\hat {k}}\right)\\&={\begin{vmatrix}{\hat {\imath }}&{\hat {\jmath }}&{\hat {k}}\\\partial _{x}&\partial _{y}&\partial _{z}\\a&b&c\end{vmatrix}}\end{aligned}}$

Special Case

When ${\vec {F}}=a(x,y){\hat {\imath }}+b(x,y){\hat {\jmath }}$, the curl is equal to:

${\begin{vmatrix}{\hat {\imath }}&{\hat {\jmath }}&{\hat {k}}\\\partial _{x}&\partial _{y}&\partial _{z}\\a&b&0\end{vmatrix}}=\left({\frac {\partial b}{\partial x}}-{\frac {\partial x}{\partial y}}\right){\hat {k}}$