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DEtools

 PDEchangecoords
 perform a coordinate system transformation on a partial differential equation

 Calling Sequence PDEchangecoords(pdes, c_ivar, c_name, n_ivar)

Parameters

 pdes - partial differential equation, or list or set of equations c_ivar - list of current independent variables c_name - name of new coordinate system n_ivar - (optional) list of new independent variables

Description

 • Important: The command DEtools[PDEchangecoords] has been deprecated. Use the superseding command PDEtools[dchange] instead.
 • Given an nth order partial differential equation (or list of such equations), a list of the current independent variables, and the name of a new coordinate system, PDEchangecoords applies the appropriate transformations to each of the partial differential equations.  Currently, only two or three independent variables are allowed.
 • The PDE is written in term of new partial derivatives (with respect to n_ivar).  New partial derivatives are determined by the inverse of the Jacobian matrix of the transformations.
 • pdes is a single PDE, or list or set of PDEs, to which the coordinate transformation is applied.  It is assumed that the given PDEs are defined in Cartesian (rectangular) coordinates.
 • c_ivar should only list the names of the current independent variables. The number of independent variables should match the dimension of the coordinate system transformation in use.
 • c_name indicates the coordinate system to convert to.
 For two dimensions, the available coordinate systems are bipolar, cardioid, cassinian, elliptic, hyperbolic, invcassinian, invelliptic, logarithmic, logcosh, maxwell, parabolic, polar, rose, and tangent.
 For three dimensions, the available coordinate systems are bipolarcylindrical, bispherical, cardioidal, cardioidcylindrical, casscylindrical, confocalellip, confocalparab, conical, cylindrical, ellcylindrical, ellipsoidal, hypercylindrical, invcasscylindrical, invellcylindrical, invoblatespheroidal, invprospheroidal, logcoshcylindrical, logcylindrical, maxwellcylindrical, oblatespheroidal, paraboloidal, paracylindrical, prolatespheroidal, rosecylindrical, sixsphere, spherical, tangentcylindrical, tangentsphere, and toroidal.
 • n_ivar is a list of the names of the new independent variables.  If no new independent variable names are specified, then DEtools[PDEchangecoords] uses the variables specified in the second argument.
 • Further coordinate transformations may be created and used by way of addcoords.
 • This function is part of the DEtools package, and so it can be used in the form PDEchangecoords(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[PDEchangecoords](..).

Examples

Important: The command DEtools[PDEchangecoords] has been deprecated. Use the superseding command PDEtools[dchange] instead.

 > $\mathrm{with}\left(\mathrm{DEtools}\right):$
 > $\mathrm{PDEchangecoords}\left(\frac{\partial }{\partial x}z\left(x,y\right),\left[x,y\right],\mathrm{polar}\right)$
 $\frac{{\mathrm{cos}}{}\left({y}\right){}\left(\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{z}{}\left({x}{,}{y}\right)\right){}{x}{-}{\mathrm{sin}}{}\left({y}\right){}\left(\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{z}{}\left({x}{,}{y}\right)\right)}{{x}}$ (1)
 > $\mathrm{PDEchangecoords}\left({\mathrm{D}}_{1,2}\left(w\right)\left(x,y,z\right)-{\mathrm{D}}_{1}\left(w\right)\left(x,y,z\right),\left[x,y,z\right],\mathrm{cylindrical},\left[r,\mathrm{φ},\mathrm{θ}\right]\right)$
 $\frac{{\mathrm{sin}}{}\left({\mathrm{\phi }}\right){}{\mathrm{cos}}{}\left({\mathrm{\phi }}\right){}{{\mathrm{D}}}_{{1}{,}{1}}{}\left({w}\right){}\left({r}{,}{\mathrm{\phi }}{,}{\mathrm{\theta }}\right){}{{r}}^{{2}}{-}{\mathrm{cos}}{}\left({\mathrm{\phi }}\right){}{{\mathrm{D}}}_{{1}}{}\left({w}\right){}\left({r}{,}{\mathrm{\phi }}{,}{\mathrm{\theta }}\right){}{r}{}{\mathrm{sin}}{}\left({\mathrm{\phi }}\right){-}{\mathrm{cos}}{}\left({\mathrm{\phi }}\right){}{{\mathrm{D}}}_{{1}}{}\left({w}\right){}\left({r}{,}{\mathrm{\phi }}{,}{\mathrm{\theta }}\right){}{{r}}^{{2}}{+}{2}{}{{\mathrm{D}}}_{{1}{,}{2}}{}\left({w}\right){}\left({r}{,}{\mathrm{\phi }}{,}{\mathrm{\theta }}\right){}{{\mathrm{cos}}{}\left({\mathrm{\phi }}\right)}^{{2}}{}{r}{+}{{\mathrm{D}}}_{{2}}{}\left({w}\right){}\left({r}{,}{\mathrm{\phi }}{,}{\mathrm{\theta }}\right){}{\mathrm{sin}}{}\left({\mathrm{\phi }}\right){}{r}{-}{2}{}{{\mathrm{D}}}_{{2}}{}\left({w}\right){}\left({r}{,}{\mathrm{\phi }}{,}{\mathrm{\theta }}\right){}{{\mathrm{cos}}{}\left({\mathrm{\phi }}\right)}^{{2}}{-}{\mathrm{cos}}{}\left({\mathrm{\phi }}\right){}{\mathrm{sin}}{}\left({\mathrm{\phi }}\right){}{{\mathrm{D}}}_{{2}{,}{2}}{}\left({w}\right){}\left({r}{,}{\mathrm{\phi }}{,}{\mathrm{\theta }}\right){-}{{\mathrm{D}}}_{{1}{,}{2}}{}\left({w}\right){}\left({r}{,}{\mathrm{\phi }}{,}{\mathrm{\theta }}\right){}{r}{+}{{\mathrm{D}}}_{{2}}{}\left({w}\right){}\left({r}{,}{\mathrm{\phi }}{,}{\mathrm{\theta }}\right)}{{{r}}^{{2}}}$ (2)
 > $\mathrm{PDEchangecoords}\left(\left[\frac{\partial }{\partial y}F\left(x,y\right)-y\mathrm{cos}\left(x\right)=x,\frac{\partial }{\partial y}F\left(x,y\right)=y\right],\left[x,y\right],\mathrm{maxwell}\right)$
 $\left[\frac{{\mathrm{\pi }}{}{{ⅇ}}^{{x}}{}{\mathrm{sin}}{}\left({y}\right){}\left(\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{F}{}\left({x}{,}{y}\right)\right)}{{{ⅇ}}^{{2}{}{x}}{+}{2}{}{{ⅇ}}^{{x}}{}{\mathrm{cos}}{}\left({y}\right){+}{1}}{+}\frac{{\mathrm{\pi }}{}\left({1}{+}{{ⅇ}}^{{x}}{}{\mathrm{cos}}{}\left({y}\right)\right){}\left(\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{F}{}\left({x}{,}{y}\right)\right)}{{{ⅇ}}^{{2}{}{x}}{+}{2}{}{{ⅇ}}^{{x}}{}{\mathrm{cos}}{}\left({y}\right){+}{1}}{-}\frac{\left({y}{+}{{ⅇ}}^{{x}}{}{\mathrm{sin}}{}\left({y}\right)\right){}{\mathrm{cos}}{}\left(\frac{{x}{+}{1}{+}{{ⅇ}}^{{x}}{}{\mathrm{cos}}{}\left({y}\right)}{{\mathrm{\pi }}}\right)}{{\mathrm{\pi }}}{-}\frac{{x}{+}{1}{+}{{ⅇ}}^{{x}}{}{\mathrm{cos}}{}\left({y}\right)}{{\mathrm{\pi }}}{=}{0}{,}\frac{{\mathrm{\pi }}{}{{ⅇ}}^{{x}}{}{\mathrm{sin}}{}\left({y}\right){}\left(\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{F}{}\left({x}{,}{y}\right)\right)}{{{ⅇ}}^{{2}{}{x}}{+}{2}{}{{ⅇ}}^{{x}}{}{\mathrm{cos}}{}\left({y}\right){+}{1}}{+}\frac{{\mathrm{\pi }}{}\left({1}{+}{{ⅇ}}^{{x}}{}{\mathrm{cos}}{}\left({y}\right)\right){}\left(\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{F}{}\left({x}{,}{y}\right)\right)}{{{ⅇ}}^{{2}{}{x}}{+}{2}{}{{ⅇ}}^{{x}}{}{\mathrm{cos}}{}\left({y}\right){+}{1}}{-}\frac{{y}{+}{{ⅇ}}^{{x}}{}{\mathrm{sin}}{}\left({y}\right)}{{\mathrm{\pi }}}{=}{0}\right]$ (3)

Create and then use a new coordinate system, reverse'':

 > $\mathrm{addcoords}\left(\mathrm{reverse},\left[x,y\right],\left[y-x,y-\frac{x}{4}\right]\right)$
 Warning, not an orthogonal coordinate system - no scale factors calculated.
 > $\mathrm{PDEchangecoords}\left(\frac{{\partial }^{2}}{\partial v\partial u}z\left(u,v\right)=\frac{1\left(\frac{\partial }{\partial v}z\left(u,v\right)\right)}{3}-\frac{8}{9},\left[u,v\right],\mathrm{reverse},\left[x,y\right]\right)$
 ${-}\frac{{16}{}\frac{{{\partial }}^{{2}}}{{\partial }{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{z}{}\left({x}{,}{y}\right)}{{9}}{-}\frac{{20}{}\frac{{{\partial }}^{{2}}}{{\partial }{x}{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{z}{}\left({x}{,}{y}\right)}{{9}}{-}\frac{{4}{}\frac{{{\partial }}^{{2}}}{{\partial }{{y}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{z}{}\left({x}{,}{y}\right)}{{9}}{-}\frac{{4}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{z}{}\left({x}{,}{y}\right)}{{9}}{-}\frac{{4}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{z}{}\left({x}{,}{y}\right)}{{9}}{+}\frac{{8}}{{9}}{=}{0}$ (4)
 > $-\frac{9}{4}$
 ${4}{}\frac{{{\partial }}^{{2}}}{{\partial }{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{z}{}\left({x}{,}{y}\right){+}{5}{}\frac{{{\partial }}^{{2}}}{{\partial }{x}{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{z}{}\left({x}{,}{y}\right){+}\frac{{{\partial }}^{{2}}}{{\partial }{{y}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{z}{}\left({x}{,}{y}\right){+}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{z}{}\left({x}{,}{y}\right){+}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{z}{}\left({x}{,}{y}\right){-}{2}{=}{0}$ (5)
 > $\mathrm{PDEchangecoords}\left({\mathrm{D}}_{3}\left(\mathrm{η}\right)\left(x,y,z\right),\left[x,y,z\right],\mathrm{confocalparab}\left(2,5\right)\right)$
 ${-}\frac{{2}{}\left({x}{-}{4}\right){}\left({x}{-}{25}\right){}{{\mathrm{D}}}_{{1}}{}\left({\mathrm{\eta }}\right){}\left({x}{,}{y}{,}{z}\right)}{{{x}}^{{2}}{-}{x}{}{y}{-}{x}{}{z}{+}{y}{}{z}}{+}\frac{{2}{}\left({y}{-}{4}\right){}\left({y}{-}{25}\right){}{{\mathrm{D}}}_{{2}}{}\left({\mathrm{\eta }}\right){}\left({x}{,}{y}{,}{z}\right)}{{x}{}{y}{-}{x}{}{z}{-}{{y}}^{{2}}{+}{y}{}{z}}{-}\frac{{2}{}\left({z}{-}{4}\right){}\left({z}{-}{25}\right){}{{\mathrm{D}}}_{{3}}{}\left({\mathrm{\eta }}\right){}\left({x}{,}{y}{,}{z}\right)}{{x}{}{y}{-}{x}{}{z}{-}{y}{}{z}{+}{{z}}^{{2}}}$ (6)
 > $\mathrm{PDE}≔\left({x}^{2}-1\right){\mathrm{D}}_{1,1}\left(z\right)\left(x,y\right)+2xy{\mathrm{D}}_{1,2}\left(z\right)\left(x,y\right)+\left({y}^{2}-1\right){\mathrm{D}}_{2,2}\left(z\right)\left(x,y\right)+2x{\mathrm{D}}_{1}\left(z\right)\left(x,y\right)+2y{\mathrm{D}}_{2}\left(z\right)\left(x,y\right)=0:$
 > $\mathrm{temp_ans}≔\mathrm{PDEchangecoords}\left(\mathrm{PDE},\left[x,y\right],\mathrm{polar},\left[r,\mathrm{θ}\right]\right):$

Create and then use a new coordinate system, stretch'':

 > $\mathrm{addcoords}\left(\mathrm{stretch},\left[\mathrm{σ},\mathrm{ρ}\right],\left[\frac{1}{\mathrm{cos}\left(\mathrm{σ}\right)},\mathrm{ρ}\right]\right):$
 > $\mathrm{ans}≔\mathrm{PDEchangecoords}\left(\mathrm{temp_ans},\left[r,\mathrm{θ}\right],\mathrm{stretch},\left[\mathrm{σ},\mathrm{ρ}\right]\right):$
 > $\mathrm{ans}≔\mathrm{convert}\left(\mathrm{ans},\mathrm{diff}\right)$
 ${\mathrm{ans}}{≔}{-}\left(\frac{{{\partial }}^{{2}}}{{\partial }{{\mathrm{\rho }}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{z}{}\left({\mathrm{\sigma }}{,}{\mathrm{\rho }}\right){-}\frac{{{\partial }}^{{2}}}{{\partial }{{\mathrm{\sigma }}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{z}{}\left({\mathrm{\sigma }}{,}{\mathrm{\rho }}\right)\right){}{{\mathrm{cos}}{}\left({\mathrm{\sigma }}\right)}^{{2}}{=}{0}$ (7)