Uses selected flow curve segments in a vector field
F (x,y)=
<f (x,y),g (x,y)>
to demonstrate the effects of the divergence and curl of the
vector field.
Enter the functions
f (x,y) and
g (x,y) in the text input fields marked
"f (x,y)=" and
"g(x,y)="
(Example:
f (x,y)=-y,
g (x,y)=x)
Using partial derivatives, compute and enter the divergence and
curl into the corresponding text input fields:
Compute the divergence f_{x}+g_{y},
and enter it into the text input field marked
"div F ="
Compute the curl g_{x}-f_{y},
and enter it into the text input field marked
"curl F ="
(Example: for
F (x,y)=<-y,x>,
div F=0 and curl F=1)
Click the "Graph" button
(this button also refreshes the graph)
Choose the initial point
(x,y)
(Example:
(x,y)=(0,1))
This can be done either of two ways:
Enter the values directly into the text fields marked
"x=" and
"y="
and click the
"Graph" button to refresh.
The text input fields can accept any decimal number input.
Use the mouse to click and drag the
red point on the graph.
The point (x,y) represents the center of a square, drawn
in blue.
To change the size of the square, enter a value into the text
input field marked "s=" -- this value represents
the length of a side of the square.
Four flow curve segments are drawn in yellow,
one from each corner of the
square, parameterized from 0 to t --
to change the lengths of the flow curves, enter a value into the text
input field marked "t=".
The opposite endpoints of the curve segments are connected to form another
quadrilateral (not necessarily square), drawn in blue.
To erase the graph and all input fields, click the
"Clear" button
The text input fields marked
"f(x,y)=",
"g(x,y)=",
"div F=", and
"curl F="
can accept
a wide variety of expressions to represent functions, and the
buttons under the graph allow various manipulations of
the graph coordinates.
Examples
Other Notes
The starting square and the resulting quadrilateral (both drawn in
blue) give the basis for the geometric
interpretations of divergence and curl:
If divergence is positive, then the verteces move apart,
resulting in a quadrilateral with area larger than the original square;
if divergence is negative, the verteces move together.
If curl is positive, then the verteces rotate counterclockwise;
if curl is negative, the rotation is clockwise.
The flow curve segments are graphed by determining
numerical approximations using the classical (order four)
Runge-Kutta Method.