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VectorCalculus

 Binormal
 compute a Vector in the direction of the binormal vector to a curve in R^3

 Calling Sequence Binormal(C, t, n)

Parameters

 C - free or position Vector or Vector valued procedure; specify the components of the curve in R^3 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 Binormal(C, t) command computes a Vector in the direction of the binormal vector to a curve in R^3.  Note that this Vector is not normalized by default, so it is a scalar multiple of the unit binormal vector to the curve C. Therefore, by default, the result is generally different from the output of $\mathrm{TNBFrame}\left(C,t,\mathrm{output}=\left['B'\right]\right)$.
 • 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. However, it must have exactly three components, that is, the curve that the Vector or Vector valued procedure represents is in R^3.
 • 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, that 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

 > $\mathrm{with}\left(\mathrm{VectorCalculus}\right):$
 > $\mathrm{Binormal}\left(⟨\mathrm{cos}\left(t\right),\mathrm{sin}\left(t\right),t⟩,t\right)$
 $\left[\begin{array}{c}\frac{{\mathrm{sin}}{}\left({t}\right)}{{2}}\\ {-}\frac{{\mathrm{cos}}{}\left({t}\right)}{{2}}\\ \frac{{1}}{{2}}\end{array}\right]$ (1)
 > $\mathrm{Binormal}\left(⟨\mathrm{exp}\left(-t\right)\mathrm{cos}\left(t\right),\mathrm{exp}\left(-t\right)\mathrm{sin}\left(t\right),t⟩\right)$
 $\left[\begin{array}{c}\frac{{2}{}{{ⅇ}}^{{-}{t}}{}{\mathrm{cos}}{}\left({t}\right)}{{1}{+}{2}{}{{ⅇ}}^{{-}{2}{}{t}}}\\ \frac{{2}{}{{ⅇ}}^{{-}{t}}{}{\mathrm{sin}}{}\left({t}\right)}{{1}{+}{2}{}{{ⅇ}}^{{-}{2}{}{t}}}\\ \frac{{2}{}{{ⅇ}}^{{-}{2}{}{t}}}{{1}{+}{2}{}{{ⅇ}}^{{-}{2}{}{t}}}\end{array}\right]$ (2)
 > $\mathrm{B1}≔\mathrm{Binormal}\left(t↦⟨t,{t}^{2},{t}^{3}⟩\right):$
 > $\mathrm{B1}\left(t\right)$
 $\left[\begin{array}{c}\frac{{6}{}{{t}}^{{2}}}{{9}{}{{t}}^{{4}}{+}{4}{}{{t}}^{{2}}{+}{1}}\\ {-}\frac{{6}{}{t}}{{9}{}{{t}}^{{4}}{+}{4}{}{{t}}^{{2}}{+}{1}}\\ \frac{{2}}{{9}{}{{t}}^{{4}}{+}{4}{}{{t}}^{{2}}{+}{1}}\end{array}\right]$ (3)
 > $\mathrm{B2}≔\mathrm{Binormal}\left(t↦⟨t,{t}^{2},{t}^{3}⟩,\mathrm{normalized}\right):$
 > $\mathrm{B2}\left(t\right)$
 $\left[\begin{array}{c}\frac{{3}{}{{t}}^{{2}}}{\sqrt{\frac{{9}{}{{t}}^{{4}}{+}{9}{}{{t}}^{{2}}{+}{1}}{{\left({9}{}{{t}}^{{4}}{+}{4}{}{{t}}^{{2}}{+}{1}\right)}^{{2}}}}{}\left({9}{}{{t}}^{{4}}{+}{4}{}{{t}}^{{2}}{+}{1}\right)}\\ {-}\frac{{3}{}{t}}{\sqrt{\frac{{9}{}{{t}}^{{4}}{+}{9}{}{{t}}^{{2}}{+}{1}}{{\left({9}{}{{t}}^{{4}}{+}{4}{}{{t}}^{{2}}{+}{1}\right)}^{{2}}}}{}\left({9}{}{{t}}^{{4}}{+}{4}{}{{t}}^{{2}}{+}{1}\right)}\\ \frac{{1}}{\sqrt{\frac{{9}{}{{t}}^{{4}}{+}{9}{}{{t}}^{{2}}{+}{1}}{{\left({9}{}{{t}}^{{4}}{+}{4}{}{{t}}^{{2}}{+}{1}\right)}^{{2}}}}{}\left({9}{}{{t}}^{{4}}{+}{4}{}{{t}}^{{2}}{+}{1}\right)}\end{array}\right]$ (4)
 > $\mathrm{SetCoordinates}\left('\mathrm{cylindrical}'\right)$
 ${\mathrm{cylindrical}}$ (5)
 > $\mathrm{Binormal}\left(⟨a,t,t⟩\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}a::\left(\mathrm{positive}\wedge \mathrm{constant}\right)$
 $\left[\begin{array}{c}{0}\\ {-}\frac{{a}}{{{a}}^{{2}}{+}{1}}\\ \frac{{{a}}^{{2}}}{{{a}}^{{2}}{+}{1}}\end{array}\right]$ (6)