Torsion - Maple Help
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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) command computes the torsion of the curve C, which must have exactly three components, that is, the curve that this Vector represents is in ${ℝ}^{3}$.
 • The curve C can be specified as a free or position Vector or as a Vector valued procedure.  This determines the returned object type.
 • 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 which 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 two are not compatible, an error is raised.

Examples

 > $\mathrm{with}\left(\mathrm{VectorCalculus}\right):$
 > $\mathrm{Torsion}\left(⟨at+b,ct+d,et+f⟩,t\right)$
 ${0}$ (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)
 > $\mathrm{SetCoordinates}\left('\mathrm{cylindrical}'\right)$
 ${\mathrm{cylindrical}}$ (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)