### Divergence and Curl

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.
How to use   ||   Examples   ||   Other Notes

How to Use
• 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 fx+gy, and enter it into the text input field marked "div F ="
• Compute the curl gx-fy, 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
 Positive Divergence:    F (x,y)= Positive Curl: F (x,y)=<-y,x>

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.