next up previous contents
Next: The curl of a Up: Applications of the derivative. Previous: Curl of a plane

   
The divergence of a vector field in the plane.

Let $\overrightarrow{F} $ be a vector field in the plane, i.e. at every point (x,y) in the plane we attach a vector $\overrightarrow{F} (x,y)=M(x,y) \overrightarrow{i} + N(x,y) \overrightarrow{j} $. For example, think of the vector $\overrightarrow{F} (x,y)$ as the velocity vector at the point P(x,y) of a liquid flowing in the plane.

Consider a small rectangle, whose sides are parallel to the coordinate axes, as in Figure  9. In fact the figure is identical to Fig,  8, but we use it differently.


  
Figure 9:
\begin{figure}
\mbox{\epsfig{file=div-rectangle.eps,height=4cm} }
\end{figure}

The rate at which the liquid leaves the rectangle through each side of the rectangle is given as follows:

Top side $\overrightarrow{F} (x,y+ \Delta y) \cdot \overrightarrow{j}\Delta x = N(x,y+\Delta y) \Delta x$
Right side $\overrightarrow{F} (x+\Delta x, y) \cdot \overrightarrow{i}\Delta y = M(x + \Delta x ,y ) \Delta y$
Bottom $\overrightarrow{F} (x,y) \cdot (-\overrightarrow{j} ) \Delta x = -N(x,y) \Delta x$
Left side $\overrightarrow{F} (x,y) \cdot (-\overrightarrow{i} ) \Delta y = -M(x,y) \Delta y$

Combine opposite sides:

Top and Bottom $N(x,y+\Delta y) \Delta x - N(x,y+) \Delta x \approx \frac {\partial N}{\partial y} \Delta y \Delta x$
Right and left sides $M(x + \Delta x ,y ) \Delta y -M(x,y) \Delta y \approx \frac {\partial M}{\partial x} \Delta x \Delta y$
Add the two formulas on the right, divide by the area of the rectangle ( $\Delta x \Delta y$), then take the limit when $\Delta x$ and $\Delta y$ approach 0. We get the flux density of $\overrightarrow{F} $ at the point P(x,y).

Definition 6.32   The flux density or divergence of the vector field $\overrightarrow{F} $ at the point (x0,y0) is the number

\begin{displaymath}\text{div} \overrightarrow{F} = \frac {\partial M}{\partial x} +
\frac {\partial N}{\partial y}
\end{displaymath}

Example 6.33   Let $\overrightarrow{F} (x,y)=\underbrace{(x^2-y)}_{M(x,y)}\overrightarrow{i} +
\underbrace{(y^2-2x)}_{N(x,y)}\overrightarrow{j} $. Then $\frac {\partial M}{\partial x} =2x$ and $\frac {\partial N}{\partial y}=2y$. We have: $\text{div}\overrightarrow{F} (x,y)=2x+2y$.

Remark 6.34   Suppose that a liquid is entering a region D in the plane through the point P0(x0,y0). The lines of flow diverge at that point and $\text{div } \overrightarrow{F} (P_0)$ is positive. If the liquid leaves D at the point P0, then $\text{div } \overrightarrow{F} (P_0)$ is negative.

As an actual example of a vector field, see Figure  10;it has been released on the Internet by the U.S. National Center for Environmental Prediction (NCEP). (http://grads.iges.org/pix/hemi.jet.html).


  
Figure 10: The winds over the northern hemisphere on the 5th of March, 2000.
\begin{figure}
\mbox{\epsfig{file=hemi5.00hr.eps,height=13cm} }
\end{figure}

We quote here part of their comments (200mb Winds):


next up previous contents
Next: The curl of a Up: Applications of the derivative. Previous: Curl of a plane
Noah Dana-Picard
2001-05-30