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Next: The divergence of a Up: Applications of the derivative. Previous: Lagrange's multipliers.

   
Curl of a plane vector field.

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} $. We will define the circulation density of $\overrightarrow{F} (x,y)$ at a point in the plane.

Consider a small rectangle, whose sides are parallel to the coordinate axes, as in Figure  8.


  
Figure 8:
\begin{figure}
\mbox{\epsfig{file=div-rectangle.eps,height=4cm} }
\end{figure}

On each side of the rectangle, we have the following flow rates:

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

Combine opposite sides:

Top and Bottom $-M(x,y+\Delta y) \Delta x - M(x,y+) \Delta x \approx - \frac {\partial M}{\partial y} \Delta y \Delta x$
Right and left sides $N(x + \Delta x ,y ) \Delta y -N(x,y) \Delta y \approx \frac {\partial N}{\partial x} \Delta x \Delta y$
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 circulation density of $\overrightarrow{F} $ at the point P(x,y).

Definition 6.30   The circulation density or curl of the vector field $\overrightarrow{F} $ at the point (x0,y0) is the number

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

Example 6.31   Let $\overrightarrow{F} (x,y)=\underbrace{(x^2-y)}_{M(x,y)}\overrightarrow{i} +
\underbrace{(y^2-2x)}_{N(x,y)}\overrightarrow{j} $. Then $\frac {\partial M}{\partial y} =-1$ and $\frac {\partial N}{\partial x}=-2$. We have: $\text{curl}\overrightarrow{F} (x,y)=-1$.


next up previous contents
Next: The divergence of a Up: Applications of the derivative. Previous: Lagrange's multipliers.
Noah Dana-Picard
2001-05-30