ATAR Notes: Forum

VCE Stuff => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Mathematics => Topic started by: /0 on February 18, 2010, 05:17:16 pm

Title: Smoothness in parametric equations
Post by: /0 on February 18, 2010, 05:17:16 pm: Is there a test of some kind to determine whether a parametric equation is smooth?

Say you have a parametric equation such as $\gamma (t) = (\cos^3t,\sin^3t)$

So $\dot{\gamma}(t)= 3\sin t\cos t (-\cos t,\sin t)$ , and obviously the derivative is defined for all t.

However, if you graph $\gamma$ you will get cusps at $t = n\frac{\pi}{2}$
Title: Re: Smoothness in parametric equations
Post by: TrueTears on February 18, 2010, 05:20:51 pm: For r(t) to be smooth r'(t) must be continuous and never equal to the 0 vector.

But r'(t) = 0 when t = npi/2

It is explained in Stewarts pg 831 on the bottom.
Title: Re: Smoothness in parametric equations
Post by: /0 on February 18, 2010, 05:27:26 pm: ah thanks TT, didn't know that