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)\:$

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

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

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

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

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

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

$a_x\mathrm{dx}+a_y\mathrm{dy}+a_z\mathrm{dz}$

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

$\mathrm{\ω1}≔a\mathrm{dx}+b\mathrm{dy}+c\mathrm{dz}$

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

$-\left(-b_x+a_y\right)\mathrm{dx}\bigwedge \mathrm{dy}-\left(-c_x+a_z\right)\mathrm{dx}\bigwedge \mathrm{dz}-\left(-c_y+b_z\right)\mathrm{dy}\bigwedge \mathrm{dz}$

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

$\mathrm{\ω2}≔c\mathrm{dx}\bigwedge \mathrm{dy}-b\mathrm{dx}\bigwedge \mathrm{dz}+a\mathrm{dy}\bigwedge \mathrm{dz}$

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

$\left(a_x+b_y+c_z\right)\mathrm{dx}\bigwedge \mathrm{dy}\bigwedge \mathrm{dz}$

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

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

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

$\mathrm{\ω4}≔-{\ⅇ}^y\cos \left(z\right)\mathrm{dx}\bigwedge \mathrm{dy}+\frac{\left({\ⅇ}^y\sin \left(z\right)x+\sin \left(y\right)\right)\mathrm{dx}}{x}\bigwedge \mathrm{dz}+\ln \left(x\right)\cos \left(y\right)\mathrm{dy}\bigwedge \mathrm{dz}$

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

$0\mathrm{dx}\bigwedge \mathrm{dy}\bigwedge \mathrm{dz}$

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

$\left[-\mathrm{dx}\bigwedge \mathrm{dy}\,\mathrm{dx}\bigwedge \mathrm{dy}\bigwedge \mathrm{dz}\right]$

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

$A≔\left[\begin{array}{cc}y\mathrm{dx}& z\mathrm{dy}\\ 0\mathrm{dx}& \mathrm{dz}\end{array}\right]$

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

$\left[\begin{array}{cc}-\mathrm{dx}\bigwedge \mathrm{dy}& -\mathrm{dy}\bigwedge \mathrm{dz}\\ 0\mathrm{dx}\bigwedge \mathrm{dy}& 0\mathrm{dx}\bigwedge \mathrm{dy}\end{array}\right]$

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

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

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

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

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

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

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

$\frac{\mathrm{\Θ3}}{x}$

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

$\frac{y\left(y+1\right)\mathrm{\Θ1}}{x^2}\bigwedge \mathrm{\Θ2}+\frac{\left(-y^3+xz-y^2\right)\mathrm{\Θ1}}{x^3}\bigwedge \mathrm{\Θ3}+\frac{y^2\left(3x+z\right)\mathrm{\Θ2}}{x^3}\bigwedge \mathrm{\Θ3}$

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

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

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

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

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

$-\mathrm{\θ1}\bigwedge \mathrm{\θ3}-\mathrm{\θ2}\bigwedge \mathrm{\θ4}$

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

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

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

$\mathrm{\beta }f$

$\mathrm{ExteriorDerivative}\left(\left(\mathrm{\alpha }\&w\mathrm{\beta }\right)\&w\mathrm{\beta }\right)$

$-2\mathrm{\alpha }\bigwedge \mathrm{\beta }\bigwedge d\mathrm{\beta }+\mathrm{\beta }f\bigwedge \mathrm{\beta }\bigwedge \mathrm{\beta }$