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DifferentialAlgebra[Tools]

 ToJet
 rewrite in jet notation a mathematical expression written in function notation
 FromJet
 rewrite in function notation a mathematical expression written in jet notation

 Calling Sequence ToJet(expr, DepVars, notation = ...) FromJet(expr, DepVars, differentiationnotation = ...)

Parameters

 expr - algebraic expression or equation DepVars - function or list of functions indicating the dependent variables of the problem differentiationnotation = ... - (optional) can be diff (default), Diff, or D; specifies the derivative notation to return notation = ... - (optional) can be jet (default), tjet, diff, or Diff; specifies the jet notation to return

Description

 • The ToJet and FromJet commands rewrite mathematical expressions back and forth using jet and function notation respectively.
 • For ToJet, the option notation can be used to select the jet notation. The possible values are jet and tjet, respectively corresponding to the notations jetvariables and jetvariableswithbrackets explained in the help page of the equivalent PDEtools[ToJet] command.
 • For FromJet, the option differentiationnotation can be used to select the notation used for derivatives.  The possible values are diff, Diff and D, explained in the help page of the equivalent PDEtools[FromJet] command.

Examples

 > $\mathrm{with}\left(\mathrm{DifferentialAlgebra}:-\mathrm{Tools},\mathrm{ToJet},\mathrm{FromJet}\right)$
 $\left[{\mathrm{ToJet}}{,}{\mathrm{FromJet}}\right]$ (1)
 > $\mathrm{DepVars}≔\left[u\left(x,t\right),v\left(t\right),w\left(y\right)\right]$
 ${\mathrm{DepVars}}{≔}\left[{u}{}\left({x}{,}{t}\right){,}{v}{}\left({t}\right){,}{w}{}\left({y}\right)\right]$ (2)
 > $\mathrm{PDE}≔\frac{\partial }{\partial x}\left(u\left(x,t\right)v\left(t\right)w\left(y\right)\right)+\frac{{\partial }^{2}}{\partial y\partial t}\left(u\left(x,t\right)v\left(t\right)w\left(y\right)\right)$
 ${\mathrm{PDE}}{≔}\left(\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}{,}{t}\right)\right){}{v}{}\left({t}\right){}{w}{}\left({y}\right){+}\left(\frac{{\partial }}{{\partial }{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}{,}{t}\right)\right){}{v}{}\left({t}\right){}\left(\frac{{ⅆ}}{{ⅆ}{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{w}{}\left({y}\right)\right){+}{u}{}\left({x}{,}{t}\right){}\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{v}{}\left({t}\right)\right){}\left(\frac{{ⅆ}}{{ⅆ}{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{w}{}\left({y}\right)\right)$ (3)

The following is $\mathrm{PDE}$ in jet notation

 > $\mathrm{jet_PDE}≔\mathrm{ToJet}\left(\mathrm{PDE},\mathrm{DepVars}\right)$
 ${\mathrm{jet_PDE}}{≔}{u}{}{{v}}_{{t}}{}{{w}}_{{y}}{+}{v}{}{w}{}{{u}}_{{x}}{+}{v}{}{{u}}_{{t}}{}{{w}}_{{y}}$ (4)

This is $\mathrm{PDE}$ in tjet notation, or jet notation with brackets

 > $\mathrm{tjet_PDE}≔\mathrm{ToJet}\left(\mathrm{PDE},\mathrm{DepVars},\mathrm{notation}=\mathrm{tjet}\right)$
 ${\mathrm{tjet_PDE}}{≔}{u}\left[\right]{}{{v}}_{{t}}{}{{w}}_{{y}}{+}{{u}}_{{t}}{}{v}\left[\right]{}{{w}}_{{y}}{+}{{u}}_{{x}}{}{v}\left[\right]{}{w}\left[\right]$ (5)

You can express jet_PDE and tjet_PDE in function notation, using diff, Diff, or D for expression the derivatives. By default diff notation is used.

 > $\mathrm{FromJet}\left(\mathrm{jet_PDE},\mathrm{DepVars}\right)$
 $\left(\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}{,}{t}\right)\right){}{v}{}\left({t}\right){}{w}{}\left({y}\right){+}\left(\frac{{\partial }}{{\partial }{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}{,}{t}\right)\right){}{v}{}\left({t}\right){}\left(\frac{{ⅆ}}{{ⅆ}{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{w}{}\left({y}\right)\right){+}{u}{}\left({x}{,}{t}\right){}\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{v}{}\left({t}\right)\right){}\left(\frac{{ⅆ}}{{ⅆ}{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{w}{}\left({y}\right)\right)$ (6)
 > $\mathrm{FromJet}\left(\mathrm{jet_PDE},\mathrm{DepVars},\mathrm{differentiationnotation}='\mathrm{Diff}'\right)$
 $\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}{,}{t}\right){}{v}{}\left({t}\right){}{w}{}\left({y}\right){+}\frac{{\partial }}{{\partial }{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({x}{,}{t}\right){}{v}{}\left({t}\right){}\frac{{ⅆ}}{{ⅆ}{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{w}{}\left({y}\right){+}{u}{}\left({x}{,}{t}\right){}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{v}{}\left({t}\right){}\frac{{ⅆ}}{{ⅆ}{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{w}{}\left({y}\right)$ (7)
 > $\mathrm{FromJet}\left(\mathrm{jet_PDE},\mathrm{DepVars},\mathrm{differentiationnotation}='\mathrm{D}'\right)$
 ${{\mathrm{D}}}_{{1}}{}\left({u}\right){}\left({x}{,}{t}\right){}{v}{}\left({t}\right){}{w}{}\left({y}\right){+}{{\mathrm{D}}}_{{2}}{}\left({u}\right){}\left({x}{,}{t}\right){}{v}{}\left({t}\right){}{\mathrm{D}}{}\left({w}\right){}\left({y}\right){+}{u}{}\left({x}{,}{t}\right){}{\mathrm{D}}{}\left({v}\right){}\left({t}\right){}{\mathrm{D}}{}\left({w}\right){}\left({y}\right)$ (8)