The Torsion(C, t) calling sequence computes the torsion of the curve $C$, which must have exactly three components, that is, the curve that this Vector represents must be in ${ℝ}^3$.

The curve $C$ can be specified as a Vector or as a Vector-valued procedure. If $C$ is a procedure, the returned object is a procedure. Otherwise, the returned object is an expression.

If $t$ is not specified, the function tries to determine a suitable variable name from the components of $C$.  To do this, it checks all of the indeterminates of type name in the components of $C$ and removes the ones that are determined to be constants.

If the resulting set has a single entry, the single entry is the variable name.  If it has more than one entry, an error is raised.

If a coordinate system attribute is specified on $C$, $C$ is interpreted in this coordinate system.  Otherwise, the curve is interpreted as a curve in the current default coordinate system. If the curve and the coordinate system are incompatible, an error is returned.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{VectorCalculus}\right]\right)\:$

$\mathrm{Torsion}\left(⟨at+b\,ct+d\,et+f⟩\,t\right)$

$0$

$\mathrm{simplify}\left(\mathrm{Torsion}\left(⟨t\,t^2\,t^3⟩\right)\right)\mathbf{assuming}t::\mathrm{real}$

$\frac{3}{9t^4+9t^2+1}$

$\mathrm{Torsion}\left(t\mapsto ⟨\cos \left(t\right)\,\sin \left(t\right)\,t⟩\right)$

$t\mapsto \frac{\sqrt{2}\cdot {\sin \left(t\right)}^2}{2\cdot \sqrt{2\cdot {\sin \left(t\right)}^2+2\cdot {\cos \left(t\right)}^2}\cdot \sqrt{\left(\frac{1}{4}+\frac{{\sin \left(t\right)}^2}{4}+\frac{{\cos \left(t\right)}^2}{4}\right)\cdot \left(1+{\sin \left(t\right)}^2+{\cos \left(t\right)}^2\right)}}+\frac{\sqrt{2}\cdot {\cos \left(t\right)}^2}{2\cdot \sqrt{2\cdot {\sin \left(t\right)}^2+2\cdot {\cos \left(t\right)}^2}\cdot \sqrt{\left(\frac{1}{4}+\frac{{\sin \left(t\right)}^2}{4}+\frac{{\cos \left(t\right)}^2}{4}\right)\cdot \left(1+{\sin \left(t\right)}^2+{\cos \left(t\right)}^2\right)}}$

$\mathrm{SetCoordinates}\left(\mathrm{cylindrical}\right)$

${\mathrm{cylindrical}}_{r,\mathrm{\theta },z}$

$\mathrm{simplify}\left(\mathrm{Torsion}\left(⟨\exp \left(-at\right)\,t\,t⟩\,t\right)\right)\mathbf{assuming}\mathrm{real}$

$\frac{1}{{\ⅇ}^{-2at}+1}$