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.

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

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

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

$\mathrm{Curvature}\left(\mathrm{PositionVector}\left(\left[\cos \left(t\right)\,\sin \left(t\right)\right]\right)\right)$

$\sqrt{{\sin \left(t\right)}^2+{\cos \left(t\right)}^2}$

$c≔\mathrm{Curvature}\left(t\mapsto ⟨t\,t^2\,t^3⟩\right)\:$

$\mathrm{simplify}\left(c\left(t\right)\right)\mathbf{assuming}t::\mathrm{real}$

$\frac{2\sqrt{9t^4+9t^2+1}}{{\left(9t^4+4t^2+1\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$

$\mathrm{Curvature}\left(⟨a\cos \left(t\right)\,a\sin \left(t\right)\,t⟩\right)\mathbf{assuming}a::\mathrm{constant}$

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

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

${\mathrm{polar}}_{r,\mathrm{\theta }}$

$\mathrm{Curvature}\left(⟨\exp \left(-t\right)\,t⟩\right)$

$\frac{\sqrt{{\left(\frac{\sqrt{2}\left(-{\ⅇ}^{-t}\cos \left(t\right)-{\ⅇ}^{-t}\sin \left(t\right)\right)}{2\sqrt{{\ⅇ}^{-2t}}}+\frac{\sqrt{2}{\ⅇ}^{-t}\sin \left(t\right)}{\sqrt{{\ⅇ}^{-2t}}}\right)}^2+{\left(\frac{\sqrt{2}\left(-{\ⅇ}^{-t}\sin \left(t\right)+{\ⅇ}^{-t}\cos \left(t\right)\right)}{2\sqrt{{\ⅇ}^{-2t}}}-\frac{\sqrt{2}{\ⅇ}^{-t}\cos \left(t\right)}{\sqrt{{\ⅇ}^{-2t}}}\right)}^2}\sqrt{2}}{2\sqrt{{\ⅇ}^{-2t}}}$

$\mathrm{simplify}\left(\right)\mathbf{assuming}t::\mathrm{real}$

$\frac{\sqrt{2}{\ⅇ}^t}{2}$