Differential - Maple Help

LieAlgebrasOfVectorFields

 Differential
 construct the differential of an expression

 Calling Sequence Differential( f, space = vars)

Parameters

 f - a scalar expression, or a list, set or rtable of scalar expressions vars - (optional) list of names giving the coordinates of the space

Description

 • The differential command computes the differential of an algebraic expression, and returns a OneForm object
 • The differential of a function f on a space with coordinates $\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)$ is $\mathrm{df}=\sum _{i=1}^{n}$ $\frac{\partial }{\partial {x}_{i}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}f\left({x}_{1},\mathrm{..},{x}_{n}\right)$ $\cdot {\mathrm{dx}}_{i}$. See differential.
 • The space on which f lives can be specified using the space= option. It may be necessary to specify this option if the expression f contains names that are not coordinates, or if f is independent of a name that is a coordinate.
 • The Differential command maps itself over indexed structures such as lists and vectors.
 • This command is part of the LieAlgebrasOfVectorFields package. It can be called using the form Differential(...) after executing the command with(LieAlgebrasOfVectorFields). It can also be directly called using the form :-LieAlgebrasOfVectorFields[Differential](...). The Differential command returns a OneForm object.  For more detail see Overview of the OneForm object.

Examples

 > $\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right):$

 > $\mathrm{Differential}\left(f\left(x,y\right)\right)$
 $\left(\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}{,}{y}\right)\right){}{\mathrm{dx}}{+}\left(\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}{,}{y}\right)\right){}{\mathrm{dy}}$ (1)

 > $\mathrm{Differential}\left({a}^{2}{y}^{2},\mathrm{space}=\left[x,y\right]\right)$
 ${2}{}{{a}}^{{2}}{}{y}{}{\mathrm{dy}}$ (2)

 > $\mathrm{Differential}\left(\left[{x}^{2},{y}^{2},2xy\right],\mathrm{space}=\left[x,y\right]\right)$
 $\left[{2}{}{x}{}{\mathrm{dx}}{,}{2}{}{y}{}{\mathrm{dy}}{,}{2}{}{y}{}{\mathrm{dx}}{+}{2}{}{x}{}{\mathrm{dy}}\right]$ (3)

Compatibility

 • The LieAlgebrasOfVectorFields[Differential] command was introduced in Maple 2020.