The exterior derivative of a differential p-form omega is a differential form d(omega) of degree p + 1.  There are two standard ways to intrinsically define the exterior derivative d.

The exterior derivative can be defined directly in terms of the Lie bracket.  For a 1-form alpha and a 2-form beta this definition is:

Alternatively, d can be defined uniquely as that linear operator acting on differential forms such that:

The ExteriorDerivative command can also be applied to a list of differential forms.

This command is part of the DifferentialGeometry package, and so can be used in the form ExteriorDerivative(...) only after executing the command with(DifferentialGeometry).  It can always be used in the long form DifferentialGeometry:-ExteriorDerivative.

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{LieAlgebras}\right)\:$

$\mathrm{DGsetup}\left(\left[x\,y\,z\right]\,M\right)\:$

${\mathrm{PDEtools}}_{\mathrm{declare}}\left(a\left(x\,y\,z\right)\,b\left(x\,y\,z\right)\,c\left(x\,y\,z\right)\right)$

$a\left(x\,y\,z\right)\mathrm{will\; now\; be\; displayed\; as}a$

$b\left(x\,y\,z\right)\mathrm{will\; now\; be\; displayed\; as}b$

$c\left(x\,y\,z\right)\mathrm{will\; now\; be\; displayed\; as}c$

$\mathrm{ExteriorDerivative}\left(a\left(x\,y\,z\right)\right)$

$\mathrm{\_DG}\left(\left[\left[''form''\,M\,1\right]\,\left[\left[\left[1\right]\,\mathrm{diff}\left(a\left(x\,y\,z\right)\,x\right)\right]\,\left[\left[2\right]\,\mathrm{diff}\left(a\left(x\,y\,z\right)\,y\right)\right]\,\left[\left[3\right]\,\mathrm{diff}\left(a\left(x\,y\,z\right)\,z\right)\right]\right]\right]\right)$

$\mathrm{\ω1}≔\mathrm{evalDG}\left(a\left(x\,y\,z\right)\mathrm{dx}\+b\left(x\,y\,z\right)\mathrm{dy}\+c\left(x\,y\,z\right)\mathrm{dz}\right)$

$\mathrm{\_DG}\left(\left[\left[''form''\,M\,1\right]\,\left[\left[\left[1\right]\,a\left(x\,y\,z\right)\right]\,\left[\left[2\right]\,b\left(x\,y\,z\right)\right]\,\left[\left[3\right]\,c\left(x\,y\,z\right)\right]\right]\right]\right)$

$\mathrm{ExteriorDerivative}\left(\mathrm{\ω1}\right)$

$\mathrm{\_DG}\left(\left[\left[''form''\,M\,2\right]\,\left[\left[\left[1\,2\right]\,\mathrm{diff}\left(b\left(x\,y\,z\right)\,x\right)-\left(\mathrm{diff}\left(a\left(x\,y\,z\right)\,y\right)\right)\right]\,\left[\left[1\,3\right]\,\mathrm{diff}\left(c\left(x\,y\,z\right)\,x\right)-\left(\mathrm{diff}\left(a\left(x\,y\,z\right)\,z\right)\right)\right]\,\left[\left[2\,3\right]\,\mathrm{diff}\left(c\left(x\,y\,z\right)\,y\right)-\left(\mathrm{diff}\left(b\left(x\,y\,z\right)\,z\right)\right)\right]\right]\right]\right)$

$\mathrm{\ω2}≔\mathrm{evalDG}\left(c\left(x\,y\,z\right)\mathrm{dx}\&w\mathrm{dy}-b\left(x\,y\,z\right)\mathrm{dx}\&w\mathrm{dz}\+a\left(x\,y\,z\right)\mathrm{dy}\&w\mathrm{dz}\right)$

$\mathrm{\_DG}\left(\left[\left[''form''\,M\,2\right]\,\left[\left[\left[1\,2\right]\,c\left(x\,y\,z\right)\right]\,\left[\left[1\,3\right]\,-b\left(x\,y\,z\right)\right]\,\left[\left[2\,3\right]\,a\left(x\,y\,z\right)\right]\right]\right]\right)$

$\mathrm{ExteriorDerivative}\left(\mathrm{\ω2}\right)$

$\mathrm{\_DG}\left(\left[\left[''form''\,M\,3\right]\,\left[\left[\left[1\,2\,3\right]\,\mathrm{diff}\left(a\left(x\,y\,z\right)\,x\right)\+\mathrm{diff}\left(b\left(x\,y\,z\right)\,y\right)\+\mathrm{diff}\left(c\left(x\,y\,z\right)\,z\right)\right]\right]\right]\right)$

$\mathrm{\ω3}≔\mathrm{evalDG}\left({\ⅇ}^y\cos \left(z\right)\mathrm{dx}\+\ln \left(x\right)\sin \left(y\right)\mathrm{dz}\right)$

$\mathrm{\_DG}\left(\left[\left[''form''\,M\,1\right]\,\left[\left[\left[1\right]\,\exp \left(y\right)\cos \left(z\right)\right]\,\left[\left[3\right]\,\ln \left(x\right)\sin \left(y\right)\right]\right]\right]\right)$

$\mathrm{\ω4}≔\mathrm{ExteriorDerivative}\left(\mathrm{\ω3}\right)$

$\mathrm{\_DG}\left(\left[\left[''form''\,M\,2\right]\,\left[\left[\left[1\,2\right]\,-\exp \left(y\right)\cos \left(z\right)\right]\,\left[\left[1\,3\right]\,\frac{\exp \left(y\right)\sin \left(z\right)x\+\sin \left(y\right)}{x}\right]\,\left[\left[2\,3\right]\,\ln \left(x\right)\cos \left(y\right)\right]\right]\right]\right)$

$\mathrm{ExteriorDerivative}\left(\mathrm{\ω4}\right)$

$\mathrm{\_DG}\left(\left[\left[''form''\,M\,3\right]\,\left[\left[\left[1\,2\,3\right]\,0\right]\right]\right]\right)$

$\mathrm{ExteriorDerivative}\left(\left[y\mathrm{dx}\,z\mathrm{dx}\&w\mathrm{dy}\right]\right)$

$\left[\mathrm{\_DG}\left(\left[\left[''form''\,M\,2\right]\,\left[\left[\left[1\,2\right]\,-1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,M\,3\right]\,\left[\left[\left[1\,2\,3\right]\,1\right]\right]\right]\right)\right]$

$A≔\mathrm{Matrix}\left(\left[\left[y\&mult\mathrm{dx}\,z\&mult\mathrm{dy}\right]\,\left[0\&mult\mathrm{dx}\,\mathrm{dz}\right]\right]\right)$

$\left[\begin{array}{cc}\mathrm{\_DG}\left(\left[\left[''form''\,M\,1\right]\,\left[\left[\left[1\right]\,y\right]\right]\right]\right)& \mathrm{\_DG}\left(\left[\left[''form''\,M\,1\right]\,\left[\left[\left[2\right]\,z\right]\right]\right]\right)\\ \mathrm{\_DG}\left(\left[\left[''form''\,M\,1\right]\,\left[\left[\left[1\right]\,0\right]\right]\right]\right)& \mathrm{\_DG}\left(\left[\left[''form''\,M\,1\right]\,\left[\left[\left[3\right]\,1\right]\right]\right]\right)\end{array}\right]$

$\mathrm{ExteriorDerivative}\left(A\right)$

$\left[\begin{array}{cc}\mathrm{\_DG}\left(\left[\left[''form''\,M\,2\right]\,\left[\left[\left[1\,2\right]\,-1\right]\right]\right]\right)& \mathrm{\_DG}\left(\left[\left[''form''\,M\,2\right]\,\left[\left[\left[2\,3\right]\,-1\right]\right]\right]\right)\\ \mathrm{\_DG}\left(\left[\left[''form''\,M\,2\right]\,\left[\left[\left[1\,2\right]\,0\right]\right]\right]\right)& \mathrm{\_DG}\left(\left[\left[''form''\,M\,2\right]\,\left[\left[\left[1\,2\right]\,0\right]\right]\right]\right)\end{array}\right]$

$\mathrm{Fr}≔\mathrm{evalDG}\left(\left[x\mathrm{dx}\+y\mathrm{dy}\+z\mathrm{dz}\,x\mathrm{dy}\+y\mathrm{dz}\,x\mathrm{dz}\right]\right)$

$\left[\mathrm{\_DG}\left(\left[\left[''form''\,M\,1\right]\,\left[\left[\left[1\right]\,x\right]\,\left[\left[2\right]\,y\right]\,\left[\left[3\right]\,z\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,M\,1\right]\,\left[\left[\left[2\right]\,x\right]\,\left[\left[3\right]\,y\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,M\,1\right]\,\left[\left[\left[3\right]\,x\right]\right]\right]\right)\right]$

$\mathrm{FrData}≔\mathrm{FrameData}\left(\mathrm{Fr}\,P\right)$

$\mathrm{FrData}≔\left[d\mathrm{\Θ1}=0\,d\mathrm{\Θ2}=\frac{\mathrm{\Θ1}\bigwedge \mathrm{\Θ2}}{x^2}-\frac{y\mathrm{\Θ1}\bigwedge \mathrm{\Θ3}}{x^3}+\frac{\left(x+z\right)\mathrm{\Θ2}\bigwedge \mathrm{\Θ3}}{x^3}\,d\mathrm{\Θ3}=\frac{\mathrm{\Θ1}\bigwedge \mathrm{\Θ3}}{x^2}-\frac{y\mathrm{\Θ2}\bigwedge \mathrm{\Θ3}}{x^3}\right]$

$\mathrm{DGsetup}\left(\mathrm{FrData}\right)$

$\mathrm{frame\; name:\; P}$

$\mathrm{ExteriorDerivative}\left(z\right)$

$\mathrm{\_DG}\left(\left[\left[''form''\,P\,1\right]\,\left[\left[\left[3\right]\,\frac{1}{x}\right]\right]\right]\right)$

$\mathrm{ExteriorDerivative}\left(x\mathrm{\Θ1}\+y^2\mathrm{\Θ2}\right)$

$\mathrm{\_DG}\left(\left[\left[''form''\,P\,2\right]\,\left[\left[\left[1\,2\right]\,\frac{y\left(y\+1\right)}{x^2}\right]\,\left[\left[1\,3\right]\,\frac{-y^3\+xz-y^2}{x^3}\right]\,\left[\left[2\,3\right]\,\frac{y^2\left(3x\+z\right)}{x^3}\right]\right]\right]\right)$

$\mathrm{LD}≔\mathrm{LieAlgebraData}\left(\left[\left[\mathrm{x1}\,\mathrm{x3}\right]\=\mathrm{x1}\,\left[\mathrm{x1}\,\mathrm{x4}\right]\=-\mathrm{x2}\,\left[\mathrm{x2}\,\mathrm{x3}\right]\=\mathrm{x2}\,\left[\mathrm{x2}\,\mathrm{x4}\right]\=\mathrm{x1}\right]\,\left[\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\right]\,\mathrm{Alg1}\right)$

$\mathrm{LD}≔\left[\left[\mathrm{e1}\,\mathrm{e3}\right]=\mathrm{e1}\,\left[\mathrm{e1}\,\mathrm{e4}\right]=-\mathrm{e2}\,\left[\mathrm{e2}\,\mathrm{e3}\right]=\mathrm{e2}\,\left[\mathrm{e2}\,\mathrm{e4}\right]=\mathrm{e1}\right]$

$\mathrm{DGsetup}\left(\mathrm{LD}\right)$

$\mathrm{Lie\; algebra:\; Alg1}$

$\mathrm{ExteriorDerivative}\left(\mathrm{\θ1}\right)$

$\mathrm{\_DG}\left(\left[\left[''form''\,\mathrm{Alg1}\,2\right]\,\left[\left[\left[1\,3\right]\,-1\right]\,\left[\left[2\,4\right]\,-1\right]\right]\right]\right)$

$\mathrm{DGsetup}\left(\left[f\=\mathrm{dgform}\left(0\right)\,\mathrm{\α}\=\mathrm{dgform}\left(1\right)\,\mathrm{\β}\=\mathrm{dgform}\left(2\right)\right]\,\left[d\left(\mathrm{\α}\right)\=f\mathrm{\β}\right]\,\mathrm{M11}\right)$

$\mathrm{frame\; name:\; M11}$

$\mathrm{ExteriorDerivative}\left(\mathrm{\α}\right)$

$\mathrm{\_DG}\left(\left[\left[''form''\,\mathrm{M11}\,2\right]\,\left[\left[\left[2\right]\,f\right]\right]\right]\right)$

$\mathrm{ExteriorDerivative}\left(\left(\mathrm{\α}\&w\mathrm{\β}\right)\&w\mathrm{\β}\right)$

$\mathrm{\_DG}\left(\left[\left[''form''\,\mathrm{M11}\,6\right]\,\left[\left[\left[1\,2\,3\right]\,-2\right]\,\left[\left[2\,2\,2\right]\,f\right]\right]\right]\right)$