How do I verify a vector identity using Mathematica?

Question

I am trying to verify well known Frenet–Serret formulas in general setting using Mathematica. I need to consider a general space curve $r(s)=(x(s),y(s),z(s))$ in $\mathbb{R}^3$ and define its unit tangent $T$ and unit normal $N$ vectors by $$T=\dfrac{dr}{ds}$$ and $$\dfrac{dT}{ds}=\kappa N.$$ Then I define the unit binomial vector $B$ in such a way that $$\dfrac{dN}{ds}=-\kappa T+\tau B$$ and now I need to find an expression for $B$ only interms of derivative of $r(s)$ using this formula as the definition. The difficulty to do this computation by hand arise as formulas for $\kappa, \tau$ are too long.

However I do not know how I can do this symbolic computations through Mathematica. I tried to find a reference or any computation of this kind that I can use as an example, but couldn't succeed. Any help that you can do is highly appreciated.

Answer 1

You can use FrenetSerretSystem:

FrenetSerretSystem[{x[s], y[s], z[s]}, s][[-1, -1]] //TeXForm

$\left\{\frac{y'(s) z''(s)-y''(s) z'(s)}{\sqrt{\left(x'(s) y''(s)-x''(s) y'(s)\right)^2+\left(x''(s) z'(s)-x'(s) z''(s)\right)^2+\left(y'(s) z''(s)-y''(s) z'(s)\right)^2}},\frac{x''(s) z'(s)-x'(s) z''(s)}{\sqrt{\left(x'(s) y''(s)-x''(s) y'(s)\right)^2+\left(x''(s) z'(s)-x'(s) z''(s)\right)^2+\left(y'(s) z''(s)-y''(s) z'(s)\right)^2}},\frac{x'(s) y''(s)-x''(s) y'(s)}{\sqrt{\left(x'(s) y''(s)-x''(s) y'(s)\right)^2+\left(x''(s) z'(s)-x'(s) z''(s)\right)^2+\left(y'(s) z''(s)-y''(s) z'(s)\right)^2}}\right\}$

Answer 2

Since N is a built-in symbol, we will use n instead of N. The required formulas are

r = {x[s], y[s], z[s]};
T = D[r, s]
\[Kappa] = Norm[D[T, s]]
n = D[T, s]/\[Kappa]
\[Tau] = Norm[(\[Kappa]*T + D[n, s])]
B = (\[Kappa]*T + D[n, s])/\[Tau]