A Polar Functions Applet

This applet looks at functions written in polar form.
In particular, it looks at the function f(r,theta)=cos(m*theta)*r^n, where m and n can be modified.

Functions of this form are particularly useful when considering what can go wrong with continuity and differentiability.  In all cases the key point will be the origin.  

Consider when m=3.
If  n=0 the function is not continuous.
If n=1/2 the partials don't exist  since the slices have cusps.
If n=1, we have tangent lines in all directions, but no tangent plane.
If n=3/2, the function is differentiable.
If n=2, we have the standard monkey saddle.

To convert the functions to the more typical x-y format, recall that cos(m*theta) is the real part of (cos(theta)+i*sin(theta))^m,
and that r^2=(x^2+y^2).

This applet was designed as by modifying an applet from a demo by Tom Banchoff at Brown University.  It is used with permission.   Go to the Banchoff Applet Help page.

Return to the Banchoff Applet page.
Return to the SLU Calculus Applet page.

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Last updated By Mike May, S.J. ,  October 6, 2006.