For my Calc III class, I need to find $T(t)$, $N(t)$, and $k(t)$ using Mathematica, given $r(t) = {sqrt(t), (1/2)t^2, t^3 - (5/2)t^2 - 17}$. While the former and latter have been found, attempting to use the same method used to get $T(t)$ in order to find $N(t)$ has had... messy results, as seen in the following image.

(If the image doesn't show, it is basically an eight line mess full of Abs.

The code used is:

N1[t_] = Simplify[T'[t] / Norm[T'[t]], t ∈ Reals]

By hand, it starts getting ridiculously nasty around $||T'(t)||$, and I can't even begin to figure out $N(t)$, so I couldn't tell you if the result given is even remotely correct. What Mathematica gave me for $T(t)$ matched up with what I got by hand, so I know that, at the very least, Mathematica got that right or otherwise as wrong as I have.

I've tried doing a Google search for anything that could help compute $N(t)$, but nothing has proven useful.

For $T(t)$, I have gotten the following:

T(t) = {1 / sqrt(t) sqrt((1/t)(4t^3((3t - 5)^2 + 1) + 1)), 2t / sqrt((1/t)(4t^3((3t - 5)^2 + 1) + 1)), 2t(3t - 5) / sqrt((1/t)(4t^3((3t - 5)^2 + 1) + 1))}

(I apologize if the formatting is nasty; I've never really done this before, so I don't know how to properly clean it up.)

  • 1
    $\begingroup$ Probably better off avoiding Norm[] altogether: Simplify[T'[t]/Sqrt[T'[t].T'[t]]]. You might be interested in looking at FrenetSerretSystem[], too. $\endgroup$
    – J. M.'s torpor
    Oct 7 '16 at 2:11
  • $\begingroup$ It helped... some. Insofar as it reduced it to a two-line mess of fractions full of Abs. $\endgroup$
    – krourou2
    Oct 7 '16 at 2:17
  • $\begingroup$ I mean, you should use something similar to get your tangent vector, too: Simplify[r'[t]/Sqrt[r'[t].r'[t]]]. Make sure to clear all variables before trying all of these. $\endgroup$
    – J. M.'s torpor
    Oct 7 '16 at 2:19
  • $\begingroup$ ... Wow. Oh wow, that's beautiful. Thank you so much. $\endgroup$
    – krourou2
    Oct 7 '16 at 2:48
  • $\begingroup$ Is it only closely related Finding unit tangent, normal, and binormal vectors for a given r(t) or simply a duplicate? If not try to explain why? $\endgroup$
    – Artes
    Oct 7 '16 at 9:04

All you have to do is replace the assumption for Simplify with

t > 0

and it will work. The reason is that your function contains a square root of t which precludes evaluation of the Euclidean norm for $t<0$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.