ChangeofVariables - Maple Help

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Student[MultivariateCalculus]

 ChangeOfVariables
 return the change of variables of a multivariate integral function or algebraic expression
 Revert

 Calling Sequence ChangeOfVariables(f, g) ChangeOfVariables(f, h) Revert(m)

Parameters

 f - algebraic expression or inert multiple integral g - list of equations h - list of indexed names m - inert integral with unevaluated region of integration

Description

 • The ChangeOfVariables command returns a new integral or algebraic expression specified by the change of variables equations.
 • The second parameter of the command can be either a list of equations representing a suitable transformation or a list with two entries containing coordinate systems.
 In the first form, the transformation equations must specify new variable names for any coordinate that is to be changed; unchanged variables can be omitted from the transformation or given in the form of an identity equation, such as $x=x$. For example, ChangeOfVariables(Int(Int(a*x,x),y), [u=a*x]) is valid, as is ChangeOfVariables(Int(Int(a*x,x),y), [u=a*x,y=y]), but ChangeOfVariables(Int(Int(a*x,x),y), [y=a*x,x=y]) is not.
 • The coordinate systems available are cartesian and polar for two dimensions, and cartesian, cylindrical, and spherical for three dimensions.
 The coordinate systems must be specified in the form [OldCoordinateSystem[OldVariables],NewCoordinateSystem[NewVariables]]. Examples:
 * [polar[r,theta], cartesian[x,y]]
 * [cylindrical[r,theta,u], cartesian[x,y,z]]
 * [spherical[R,theta1,phi], cylindrical[r,theta2,z]]
 In the polar, cylindrical, and spherical coordinate systems, the first variable is assumed to be the radial component.
 • In general, when working with definite integrals, it is not always possible to find a new region of integration after applying a change of variables. Thus, if ChangeOfVariables cannot find the new region of integration, it retains the old region and indicates that the region has not been converted. If this occurs, use the Revert function to return to the previous integral before ChangeOfVariables was applied. Revert only works on integrals where the region of integration cannot be found. Otherwise, the same integral is returned.

Examples

 > $\mathrm{with}\left({\mathrm{Student}}_{\mathrm{MultivariateCalculus}}\right):$
 > $a≔{{∫}}_{2}^{5}{{∫}}_{1}^{4}x\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}y$
 ${a}{≔}{{\int }}_{{2}}^{{5}}\left({{\int }}_{{1}}^{{4}}{x}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}$ (1)
 > $b≔\mathrm{ChangeOfVariables}\left(a,\left[{\mathrm{cartesian}}_{x,y},{\mathrm{polar}}_{r,\mathrm{θ}}\right]\right)$
 ${b}{≔}{{\int }}_{{y}{=}{2}}^{{y}{=}{5}}\left({{\int }}_{{x}{=}{1}}^{{x}{=}{4}}{{r}}^{{2}}{}{\mathrm{cos}}{}\left({\mathrm{\theta }}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{r}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{\theta }}$ (2)
 > $c≔\mathrm{ChangeOfVariables}\left(b,\left[r=u+v,\mathrm{θ}=v-2\right]\right)$
 ${c}{≔}{{\int }}_{{y}{=}{2}}^{{y}{=}{5}}\left({{\int }}_{{x}{=}{1}}^{{x}{=}{4}}{\left({u}{+}{v}\right)}^{{2}}{}{\mathrm{cos}}{}\left({v}{-}{2}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{u}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{v}$ (3)
 > $\mathrm{Revert}\left(b\right)$
 ${{\int }}_{{2}}^{{5}}\left({{\int }}_{{1}}^{{4}}{x}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}$ (4)
 > $\mathrm{Revert}\left(c\right)$
 ${{\int }}_{{y}{=}{2}}^{{y}{=}{5}}\left({{\int }}_{{x}{=}{1}}^{{x}{=}{4}}{{r}}^{{2}}{}{\mathrm{cos}}{}\left({\mathrm{\theta }}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{r}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{\theta }}$ (5)
 > $\mathrm{Revert}\left(\mathrm{Revert}\left(c\right)\right)$
 ${{\int }}_{{2}}^{{5}}\left({{\int }}_{{1}}^{{4}}{x}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}$ (6)
 > $a≔{∫}{∫}{∫}xy\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}y\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}z$
 ${a}{≔}{\int }\left({\int }\left({\int }{x}{}{y}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{z}$ (7)
 > $\mathrm{ChangeOfVariables}\left(a,\left[{\mathrm{cartesian}}_{x,y,z},{\mathrm{cylindrical}}_{r,\mathrm{θ},u}\right]\right)$
 ${\int }\left({\int }\left({\int }{{r}}^{{3}}{}{\mathrm{cos}}{}\left({\mathrm{\theta }}\right){}{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{r}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{\theta }}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{u}$ (8)
 > $\mathrm{ChangeOfVariables}\left(a,\left[{\mathrm{cartesian}}_{x,y,z},{\mathrm{cylindrical}}_{r,\mathrm{θ},z}\right]\right)$
 ${\int }\left({\int }\left({\int }{{r}}^{{3}}{}{\mathrm{cos}}{}\left({\mathrm{\theta }}\right){}{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{r}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{\theta }}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{z}$ (9)
 > $\mathrm{ChangeOfVariables}\left(a,\left[x=u+1,y=y,z=w-2\right]\right)$
 ${\int }\left({\int }\left({\int }\left({u}{+}{1}\right){}{y}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{u}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{w}$ (10)
 > $\mathrm{ChangeOfVariables}\left(x+z,\left[x=u+1,z=w-2\right]\right)$
 ${u}{-}{1}{+}{w}$ (11)