Torsion - Maple Help

Student[VectorCalculus]

 Torsion
 compute the torsion of a curve in R^3

 Calling Sequence Torsion(C, t)

Parameters

 C - free or position Vector or Vector-valued procedure; specify the components of the curve t - (optional) name; specify the parameter of the curve

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{Student}\left[\mathrm{VectorCalculus}\right]\right):$
 > $\mathrm{Torsion}\left(⟨at+b,ct+d,et+f⟩,t\right)$
 ${0}$ (1)
 > $\mathrm{simplify}\left(\mathrm{Torsion}\left(⟨t,{t}^{2},{t}^{3}⟩\right)\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}t::\mathrm{real}$
 $\frac{{3}}{{9}{}{{t}}^{{4}}{+}{9}{}{{t}}^{{2}}{+}{1}}$ (2)
 > $\mathrm{Torsion}\left(t↦⟨\mathrm{cos}\left(t\right),\mathrm{sin}\left(t\right),t⟩\right)$
 ${t}{↦}\frac{\sqrt{{2}}{\cdot }{{\mathrm{sin}}{}\left({t}\right)}^{{2}}}{{2}{\cdot }\sqrt{{2}{\cdot }{{\mathrm{sin}}{}\left({t}\right)}^{{2}}{+}{2}{\cdot }{{\mathrm{cos}}{}\left({t}\right)}^{{2}}}{\cdot }\sqrt{\left(\frac{{1}}{{4}}{+}\frac{{{\mathrm{sin}}{}\left({t}\right)}^{{2}}}{{4}}{+}\frac{{{\mathrm{cos}}{}\left({t}\right)}^{{2}}}{{4}}\right){\cdot }\left({1}{+}{{\mathrm{sin}}{}\left({t}\right)}^{{2}}{+}{{\mathrm{cos}}{}\left({t}\right)}^{{2}}\right)}}{+}\frac{\sqrt{{2}}{\cdot }{{\mathrm{cos}}{}\left({t}\right)}^{{2}}}{{2}{\cdot }\sqrt{{2}{\cdot }{{\mathrm{sin}}{}\left({t}\right)}^{{2}}{+}{2}{\cdot }{{\mathrm{cos}}{}\left({t}\right)}^{{2}}}{\cdot }\sqrt{\left(\frac{{1}}{{4}}{+}\frac{{{\mathrm{sin}}{}\left({t}\right)}^{{2}}}{{4}}{+}\frac{{{\mathrm{cos}}{}\left({t}\right)}^{{2}}}{{4}}\right){\cdot }\left({1}{+}{{\mathrm{sin}}{}\left({t}\right)}^{{2}}{+}{{\mathrm{cos}}{}\left({t}\right)}^{{2}}\right)}}$ (3)
 > $\mathrm{SetCoordinates}\left(\mathrm{cylindrical}\right)$
 ${{\mathrm{cylindrical}}}_{{r}{,}{\mathrm{\theta }}{,}{z}}$ (4)
 > $\mathrm{simplify}\left(\mathrm{Torsion}\left(⟨\mathrm{exp}\left(-at\right),t,t⟩,t\right)\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{real}$
 $\frac{{{ⅇ}}^{{2}{}{a}{}{t}}}{{1}{+}{{ⅇ}}^{{2}{}{a}{}{t}}}$ (5)