Let $R$ and $S$ be type $\left(\genfrac{}{}{0ex}{}{r}{0}\right)$ and type $\left(\genfrac{}{}{0ex}{}{s}{0}\right)$ contravariant tensor fields on a manifold $M$, respectively. The Schouten bracket $T\=\left[R\,S\right]$ is a contravariant tensor field of type $r\+s-1$ which generalizes the Lie bracket of two vector fields. The Schouten bracket enjoys the following properties:

Let $R$and $S$ be type $\left(\genfrac{}{}{0ex}{}{1}{r}\right)$ and type $\left(\genfrac{}{}{0ex}{}{1}{s}\right)$ tensor fields on a manifold $M$, each of which is skew-symmetric in its covariant indices. Such tensors are often referred to as vector-valued differential forms (of degrees $r$ and $s$). The Frolicher-Nijenhuis bracket $T\=\left[R\,S\right]$ is a vector-valued differential-form of degree $r\+s$. The Frolicher-Nijenhuis bracket enjoys the following properties:

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

with(DifferentialGeometry):with(Tensor):

DGsetup([x, y, z, w], M):

X := evalDG(x*D_y - z*D_w);

$X:=x\mathrm{D\_y}-z\mathrm{D\_w}$

T1 := convert(X, DGtensor);

$\mathrm{T1}:=x\mathrm{D\_y}-z\mathrm{D\_w}$

T2 := evalDG(w*D_x &s D_y + y*D_z &s D_w);

$\mathrm{T2}:=\frac{w}{2}\mathrm{D\_x}\mathrm{D\_y}+\frac{w}{2}\mathrm{D\_y}\mathrm{D\_x}+\frac{y}{2}\mathrm{D\_z}\mathrm{D\_w}+\frac{y}{2}\mathrm{D\_w}\mathrm{D\_z}$

TensorBrackets(T1, T2, "Schouten");

$-\frac{1}{2}z\mathrm{D\_x}\mathrm{D\_y}-\frac{1}{2}z\mathrm{D\_y}\mathrm{D\_x}-w\mathrm{D\_y}\mathrm{D\_y}\+\frac{1}{2}x\mathrm{D\_z}\mathrm{D\_w}\+\frac{1}{2}x\mathrm{D\_w}\mathrm{D\_z}\+y\mathrm{D\_w}\mathrm{D\_w}$

LieDerivative(X, T2);

$-\frac{1}{2}z\mathrm{D\_x}\mathrm{D\_y}-\frac{1}{2}z\mathrm{D\_y}\mathrm{D\_x}-w\mathrm{D\_y}\mathrm{D\_y}\+\frac{1}{2}x\mathrm{D\_z}\mathrm{D\_w}\+\frac{1}{2}x\mathrm{D\_w}\mathrm{D\_z}\+y\mathrm{D\_w}\mathrm{D\_w}$

W0 := evalDG(D_x &w D_y + z*D_y &w D_z + f(x, y, z, w)*D_z &w D_w);

$\mathrm{W0}:=\mathrm{D\_x}\bigwedge \mathrm{D\_y}+z\mathrm{D\_y}\bigwedge \mathrm{D\_z}+f\left(x\,y\,z\,w\right)\mathrm{D\_z}\bigwedge \mathrm{D\_w}$

W := convert(W0, DGtensor);

$W:=\mathrm{D\_x}\mathrm{D\_y}-\mathrm{D\_y}\mathrm{D\_x}+z\mathrm{D\_y}\mathrm{D\_z}-z\mathrm{D\_z}\mathrm{D\_y}+f\left(x\,y\,z\,w\right)\mathrm{D\_z}\mathrm{D\_w}-f\left(x\,y\,z\,w\right)\mathrm{D\_w}\mathrm{D\_z}$

S := TensorBrackets(W, W, "Schouten"):

pde:=Tools:-DGinfo(S, "CoefficientSet");

$\mathrm{pde}:=\left\{-\frac{4}{3}\frac{\partial }{\partial y}f\left(x\,y\,z\,w\right)\,\frac{4}{3}\frac{\partial }{\partial y}f\left(x\,y\,z\,w\right)\,\frac{4}{3}z\left(\frac{\partial }{\partial z}f\left(x\,y\,z\,w\right)\right)-\frac{4}{3}\frac{\partial }{\partial x}f\left(x\,y\,z\,w\right)-\frac{4}{3}f\left(x\,y\,z\,w\right)\,\frac{4}{3}f\left(x\,y\,z\,w\right)-\frac{4}{3}z\left(\frac{\partial }{\partial z}f\left(x\,y\,z\,w\right)\right)\+\frac{4}{3}\frac{\partial }{\partial x}f\left(x\,y\,z\,w\right)\right\}$

pdsolve(pde);

$\left\{f\left(x\,y\,z\,w\right)\=\mathrm{\_F1}\left(z{\ⅇ}^x\,w\right){\ⅇ}^{-x}\right\}$

T3 := evalDG(y^2*D_x &s D_y + x*z *D_y &s D_w);

$\mathrm{T3}:=\frac{y^2}{2}\mathrm{D\_x}\mathrm{D\_y}+\frac{y^2}{2}\mathrm{D\_y}\mathrm{D\_x}+\frac{xz}{2}\mathrm{D\_y}\mathrm{D\_w}+\frac{xz}{2}\mathrm{D\_w}\mathrm{D\_y}$

T4 := evalDG(D_y &s D_w);

$\mathrm{T4}:=\frac{1}{2}\mathrm{D\_y}\mathrm{D\_w}+\frac{1}{2}\mathrm{D\_w}\mathrm{D\_y}$

T5 := evalDG(x*z*w* &s (D_y, D_y, D_z));

$\mathrm{T5}:=\frac{xzw}{3}\mathrm{D\_y}\mathrm{D\_y}\mathrm{D\_z}+\frac{xzw}{3}\mathrm{D\_y}\mathrm{D\_z}\mathrm{D\_y}+\frac{xzw}{3}\mathrm{D\_z}\mathrm{D\_y}\mathrm{D\_y}$

F :=(X, Y, Z) -> TensorBrackets(X, TensorBrackets(Y, Z, "Schouten"), "Schouten");

$F:=\left(X\,Y\,Z\right)\→\mathrm{DifferentialGeometry:-Tensor}:-\mathrm{TensorBrackets}\left(X\,\mathrm{DifferentialGeometry:-Tensor}:-\mathrm{TensorBrackets}\left(Y\,Z\,''Schouten''\right)\,''Schouten''\right)$

F(T3, T4, T5) &plus F(T5, T3, T4) &plus F(T4, T5, T3);

$0\mathrm{D\_x}\mathrm{D\_x}\mathrm{D\_x}\mathrm{D\_x}\mathrm{D\_x}$

X := evalDG(z*D_y + x*D_w);

$X:=z\mathrm{D\_y}\+x\mathrm{D\_w}$

T6 := convert(X, DGtensor);

$\mathrm{T6}:=z\mathrm{D\_y}\+x\mathrm{D\_w}$

T7 := evalDG(w^2* D_x &t (dx &w dy &w dw) + y^2* D_y &t(dx &w dz &w dw));

$\mathrm{T7}:=w^2\mathrm{D\_x}\mathrm{dx}\mathrm{dy}\mathrm{dw}-w^2\mathrm{D\_x}\mathrm{dx}\mathrm{dw}\mathrm{dy}-w^2\mathrm{D\_x}\mathrm{dy}\mathrm{dx}\mathrm{dw}\+w^2\mathrm{D\_x}\mathrm{dy}\mathrm{dw}\mathrm{dx}\+w^2\mathrm{D\_x}\mathrm{dw}\mathrm{dx}\mathrm{dy}-w^2\mathrm{D\_x}\mathrm{dw}\mathrm{dy}\mathrm{dx}\+y^2\mathrm{D\_y}\mathrm{dx}\mathrm{dz}\mathrm{dw}-y^2\mathrm{D\_y}\mathrm{dx}\mathrm{dw}\mathrm{dz}-y^2\mathrm{D\_y}\mathrm{dz}\mathrm{dx}\mathrm{dw}\+y^2\mathrm{D\_y}\mathrm{dz}\mathrm{dw}\mathrm{dx}\+y^2\mathrm{D\_y}\mathrm{dw}\mathrm{dx}\mathrm{dz}-y^2\mathrm{D\_y}\mathrm{dw}\mathrm{dz}\mathrm{dx}$

S1 := TensorBrackets(T6, T7, "Frolicher-Nijenhuis");

$\mathrm{S1}:=2xw\mathrm{D\_x}\mathrm{dx}\mathrm{dy}\mathrm{dw}\+w^2\mathrm{D\_x}\mathrm{dx}\mathrm{dz}\mathrm{dw}-2xw\mathrm{D\_x}\mathrm{dx}\mathrm{dw}\mathrm{dy}-w^2\mathrm{D\_x}\mathrm{dx}\mathrm{dw}\mathrm{dz}-2xw\mathrm{D\_x}\mathrm{dy}\mathrm{dx}\mathrm{dw}\+2xw\mathrm{D\_x}\mathrm{dy}\mathrm{dw}\mathrm{dx}-w^2\mathrm{D\_x}\mathrm{dz}\mathrm{dx}\mathrm{dw}\+w^2\mathrm{D\_x}\mathrm{dz}\mathrm{dw}\mathrm{dx}\+2xw\mathrm{D\_x}\mathrm{dw}\mathrm{dx}\mathrm{dy}\+w^2\mathrm{D\_x}\mathrm{dw}\mathrm{dx}\mathrm{dz}-2xw\mathrm{D\_x}\mathrm{dw}\mathrm{dy}\mathrm{dx}-w^2\mathrm{D\_x}\mathrm{dw}\mathrm{dz}\mathrm{dx}\+2zy\mathrm{D\_y}\mathrm{dx}\mathrm{dz}\mathrm{dw}-2zy\mathrm{D\_y}\mathrm{dx}\mathrm{dw}\mathrm{dz}-2zy\mathrm{D\_y}\mathrm{dz}\mathrm{dx}\mathrm{dw}\+2zy\mathrm{D\_y}\mathrm{dz}\mathrm{dw}\mathrm{dx}\+2zy\mathrm{D\_y}\mathrm{dw}\mathrm{dx}\mathrm{dz}-2zy\mathrm{D\_y}\mathrm{dw}\mathrm{dz}\mathrm{dx}-w^2\mathrm{D\_w}\mathrm{dx}\mathrm{dy}\mathrm{dw}\+w^2\mathrm{D\_w}\mathrm{dx}\mathrm{dw}\mathrm{dy}\+w^2\mathrm{D\_w}\mathrm{dy}\mathrm{dx}\mathrm{dw}-w^2\mathrm{D\_w}\mathrm{dy}\mathrm{dw}\mathrm{dx}-w^2\mathrm{D\_w}\mathrm{dw}\mathrm{dx}\mathrm{dy}\+w^2\mathrm{D\_w}\mathrm{dw}\mathrm{dy}\mathrm{dx}$

S1 &minus LieDerivative(X, T7);

$0\mathrm{D\_x}\mathrm{dx}\mathrm{dx}\mathrm{dx}$

S1 &plus TensorBrackets(T7, T6, "Frolicher-Nijenhuis");

$0\mathrm{D\_x}\mathrm{dx}\mathrm{dx}\mathrm{dx}$

T8 := evalDG(w^2* z*D_x &t (dx &w dy) + y^2*x *D_y &t(dx &w dz));

$\mathrm{T8}:=w^{2}z\mathrm{D\_x}\mathrm{dx}\mathrm{dy}-w^{2}z\mathrm{D\_x}\mathrm{dy}\mathrm{dx}\+y^{2}x\mathrm{D\_y}\mathrm{dx}\mathrm{dz}-y^{2}x\mathrm{D\_y}\mathrm{dz}\mathrm{dx}$

TensorBrackets(T8, T8, "Frolicher-Nijenhuis");

$0\mathrm{D\_x}\mathrm{dx}\mathrm{dx}\mathrm{dx}\mathrm{dx}$

T9 := evalDG(w^2* z^2*D_x &t (dy &w dw) + y^2*x^2* D_y &t(dx &w dw));

$\mathrm{T9}:=w^2{z}^2\mathrm{D\_x}\mathrm{dy}\mathrm{dw}-w^2{z}^2\mathrm{D\_x}\mathrm{dw}\mathrm{dy}\+y^2{x}^2\mathrm{D\_y}\mathrm{dx}\mathrm{dw}-y^2{x}^2\mathrm{D\_y}\mathrm{dw}\mathrm{dx}$

C := Connection(SymmetrizeIndices(z^2*D_y &t dx &t dz + x*y*D_y &t dx &t dy, [2, 3], "Symmetric"));

$C:=\frac{1}{2}yx\mathrm{D\_y}\mathrm{dx}\mathrm{dy}\+\frac{1}{2}{z}^2\mathrm{D\_y}\mathrm{dx}\mathrm{dz}\+\frac{1}{2}yx\mathrm{D\_y}\mathrm{dy}\mathrm{dx}\+\frac{1}{2}{z}^2\mathrm{D\_y}\mathrm{dz}\mathrm{dx}$

TensorBrackets(T8, T9, "Frolicher-Nijenhuis") &minus TensorBrackets(T8, T9, C, "Frolicher-Nijenhuis");

$0\mathrm{D\_x}\mathrm{dx}\mathrm{dx}\mathrm{dx}\mathrm{dx}$

T10 := evalDG(D_x &t dy + z*D_y &t dz +f(x, y, z, w)*D_z &t dw);

$\mathrm{T10}:=\mathrm{D\_x}\mathrm{dy}\+z\mathrm{D\_y}\mathrm{dz}\+f\left(x\,y\,z\,w\right)\mathrm{D\_z}\mathrm{dw}$

PDEtools[declare](f(x, y, z, w));

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

S2 := TensorBrackets(T10, T10, "Frolicher-Nijenhuis");

$\mathrm{S2}:=-2z{f}_x\mathrm{D\_y}\mathrm{dx}\mathrm{dw}-2z{f}_y\mathrm{D\_y}\mathrm{dy}\mathrm{dw}-\left(2f\+2z{f}_z\right)\mathrm{D\_y}\mathrm{dz}\mathrm{dw}\+2z{f}_x\mathrm{D\_y}\mathrm{dw}\mathrm{dx}\+2z{f}_y\mathrm{D\_y}\mathrm{dw}\mathrm{dy}\+\left(2f\+2z{f}_z\right)\mathrm{D\_y}\mathrm{dw}\mathrm{dz}\+2f_x\mathrm{D\_z}\mathrm{dy}\mathrm{dw}\+2z{f}_y\mathrm{D\_z}\mathrm{dz}\mathrm{dw}-2f_x\mathrm{D\_z}\mathrm{dw}\mathrm{dy}-2z{f}_y\mathrm{D\_z}\mathrm{dw}\mathrm{dz}$

pde := Tools:-DGinfo(S2, "CoefficientSet");

$\mathrm{pde}:=\left\{-2f_x\,2f_x\,-2z{f}_x\,2z{f}_x\,-2z{f}_y\,2z{f}_y\,2f\+2z{f}_z\,-2z{f}_z-2f\right\}$

pdsolve(pde);

$\left\{f\=\frac{\mathrm{\_F1}\left(w\right)}{z}\right\}$