- #1
WalkingInMud
- 5
- 0
A parametric equation, say r(t), is smoothly parametrized if:
1. its derivative is continuous, and
2. its derivative does not equal zero for all t in the domain of r.
Now that sounds simple enough. Now let's say we have the tractrix:
r(t) = (t-tanht)i + sechtj, ...
then r'(t) = [ 1/(1+x^2) ]i + [ tantsect ]j, right?
Now, without reverting to MatLab or Maple to view the graph, how do we mathematically explain that r'(t) is continuous (or not)?
Do I just state that is is/isn't -by inspection, or ...?
1. its derivative is continuous, and
2. its derivative does not equal zero for all t in the domain of r.
Now that sounds simple enough. Now let's say we have the tractrix:
r(t) = (t-tanht)i + sechtj, ...
then r'(t) = [ 1/(1+x^2) ]i + [ tantsect ]j, right?
Now, without reverting to MatLab or Maple to view the graph, how do we mathematically explain that r'(t) is continuous (or not)?
Do I just state that is is/isn't -by inspection, or ...?