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Student[VectorCalculus]

 Curvature
 compute the curvature of a curve

 Calling Sequence Curvature(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 Curvature(C, t) calling sequence computes the curvature of the curve C.
 • The curve C can be specified as a free or position Vector or a Vector-valued procedure.  This determines the returned object type.
 • If t is not specified, the function tries to determine a suitable variable name by using 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, that 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, it 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

 > $\mathrm{with}\left({\mathrm{Student}}_{\mathrm{VectorCalculus}}\right):$
 > $\mathrm{Curvature}\left(⟨\mathrm{cos}\left(t\right),\mathrm{sin}\left(t\right),t⟩,t\right)$
 $\frac{\sqrt{{2}{}{{\mathrm{cos}}{}\left({t}\right)}^{{2}}{+}{2}{}{{\mathrm{sin}}{}\left({t}\right)}^{{2}}}{}\sqrt{{2}}}{{4}}$ (1)
 > $\mathrm{Curvature}\left(\mathrm{PositionVector}\left(\left[\mathrm{cos}\left(t\right),\mathrm{sin}\left(t\right)\right]\right)\right)$
 $\sqrt{{{\mathrm{sin}}{}\left({t}\right)}^{{2}}{+}{{\mathrm{cos}}{}\left({t}\right)}^{{2}}}$ (2)
 > $c≔\mathrm{Curvature}\left(t→⟨t,{t}^{2},{t}^{3}⟩\right):$
 > $\mathrm{simplify}\left(c\left(t\right)\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}t::\mathrm{real}$
 $\frac{{2}{}\sqrt{{9}{}{{t}}^{{4}}{+}{9}{}{{t}}^{{2}}{+}{1}}}{{\left({9}{}{{t}}^{{4}}{+}{4}{}{{t}}^{{2}}{+}{1}\right)}^{{3}}{{2}}}}$ (3)
 > $\mathrm{Curvature}\left(⟨a\mathrm{cos}\left(t\right),a\mathrm{sin}\left(t\right),t⟩\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}a::\mathrm{constant}$
 $\frac{\sqrt{\frac{{{a}}^{{2}}{}{{\mathrm{cos}}{}\left({t}\right)}^{{2}}}{{{a}}^{{2}}{+}{1}}{+}\frac{{{a}}^{{2}}{}{{\mathrm{sin}}{}\left({t}\right)}^{{2}}}{{{a}}^{{2}}{+}{1}}}}{\sqrt{{{a}}^{{2}}{+}{1}}}$ (4)
 > $\mathrm{SetCoordinates}\left(\mathrm{polar}\right)$
 ${{\mathrm{polar}}}_{{r}{,}{\mathrm{\theta }}}$ (5)
 > $\mathrm{Curvature}\left(⟨{ⅇ}^{-t},t⟩\right)$
 $\frac{\sqrt{{\left(\frac{\sqrt{{2}}{}\left({-}{{ⅇ}}^{{-}{t}}{}{\mathrm{cos}}{}\left({t}\right){-}{{ⅇ}}^{{-}{t}}{}{\mathrm{sin}}{}\left({t}\right)\right)}{{2}{}\sqrt{{{ⅇ}}^{{-}{2}{}{t}}}}{+}\frac{\sqrt{{2}}{}{{ⅇ}}^{{-}{t}}{}{\mathrm{sin}}{}\left({t}\right)}{\sqrt{{{ⅇ}}^{{-}{2}{}{t}}}}\right)}^{{2}}{+}{\left(\frac{\sqrt{{2}}{}\left({-}{{ⅇ}}^{{-}{t}}{}{\mathrm{sin}}{}\left({t}\right){+}{{ⅇ}}^{{-}{t}}{}{\mathrm{cos}}{}\left({t}\right)\right)}{{2}{}\sqrt{{{ⅇ}}^{{-}{2}{}{t}}}}{-}\frac{\sqrt{{2}}{}{{ⅇ}}^{{-}{t}}{}{\mathrm{cos}}{}\left({t}\right)}{\sqrt{{{ⅇ}}^{{-}{2}{}{t}}}}\right)}^{{2}}}{}\sqrt{{2}}}{{2}{}\sqrt{{{ⅇ}}^{{-}{2}{}{t}}}}$ (6)
 > $\mathrm{simplify}\left(\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}t::\mathrm{real}$
 $\frac{\sqrt{{2}}{}{{ⅇ}}^{{t}}}{{2}}$ (7)