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VectorCalculus

 TangentVector
 compute the tangent vector to a curve

 Calling Sequence TangentVector(C, t, n)

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 n - (optional) equation of the form normalized=true or normalized=false, or simply normalized

Description

 • The TangentVector(C, t) command computes the tangent vector to the curve C that is parameterized by t. Note that this vector is not normalized by default, so it is a scalar multiple of the unit tangent vector to the curve C. Therefore, by default, if C is a curve in ${ℝ}^{3}$, the result is generally different from the output of TNBFrame(C, t, output=['T']).
 • If n is given as either normalized=true or normalized, then the resulting vector will be normalized before it is returned. As discussed above, the default value is false, so that the result is not normalized.
 • The curve 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 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 that 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

 > $\mathrm{with}\left(\mathrm{VectorCalculus}\right):$
 > $\mathrm{T1}≔\mathrm{TangentVector}\left(t↦⟨t,{t}^{2},{t}^{3}⟩\right):$
 > $\mathrm{T1}\left(t\right)$
 $\left[\begin{array}{c}{1}\\ {2}{}{t}\\ {3}{}{{t}}^{{2}}\end{array}\right]$ (1)
 > $\mathrm{T2}≔\mathrm{TangentVector}\left(t↦⟨t,{t}^{2},{t}^{3}⟩,\mathrm{normalized}\right):$
 > $\mathrm{T2}\left(t\right)$
 $\left[\begin{array}{c}\frac{{1}}{\sqrt{{9}{}{{t}}^{{4}}{+}{4}{}{{t}}^{{2}}{+}{1}}}\\ \frac{{2}{}{t}}{\sqrt{{9}{}{{t}}^{{4}}{+}{4}{}{{t}}^{{2}}{+}{1}}}\\ \frac{{3}{}{{t}}^{{2}}}{\sqrt{{9}{}{{t}}^{{4}}{+}{4}{}{{t}}^{{2}}{+}{1}}}\end{array}\right]$ (2)
 > $\mathrm{TangentVector}\left(\mathrm{PositionVector}\left(\left[\mathrm{cos}\left(t\right),\mathrm{sin}\left(t\right)\right]\right),t\right)$
 $\left[\begin{array}{c}{-}{\mathrm{sin}}{}\left({t}\right)\\ {\mathrm{cos}}{}\left({t}\right)\end{array}\right]$ (3)
 > $\mathrm{TangentVector}\left(⟨a\mathrm{exp}\left(-t\right),t⟩\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}a::\mathrm{constant}$
 $\left[\begin{array}{c}{-}{a}{}{{ⅇ}}^{{-}{t}}\\ {1}\end{array}\right]$ (4)
 > $\mathrm{TangentVector}\left(t↦⟨a\cdot \mathrm{cos}\left(t\right),b\cdot \mathrm{sin}\left(t\right),t⟩\right)$
 ${t}{→}{\mathrm{VectorCalculus:-VectorSpace}}{}\left({\mathrm{cartesian}}{,}\left[{a}{}{\mathrm{cos}}{}\left({t}\right){,}{b}{}{\mathrm{sin}}{}\left({t}\right){,}{t}\right]\right){:-}{\mathrm{Vector}}{}\left(\left[{-}{a}{}{\mathrm{sin}}{}\left({t}\right){,}{b}{}{\mathrm{cos}}{}\left({t}\right){,}{1}\right]\right)$ (5)
 > $\mathrm{SetCoordinates}\left('\mathrm{polar}'\right)$
 ${\mathrm{polar}}$ (6)
 > $\mathrm{TangentVector}\left(⟨1,t⟩,t\right)$
 $\left[\begin{array}{c}{0}\\ {1}\end{array}\right]$ (7)
 > $\mathrm{TangentVector}\left(⟨\mathrm{exp}\left(-t\right),t⟩,t\right)$
 $\left[\begin{array}{c}{-}\sqrt{{{ⅇ}}^{{-}{2}{}{t}}}\\ \sqrt{{{ⅇ}}^{{-}{2}{}{t}}}\end{array}\right]$ (8)