Let $g_{\mathrm{ab}}$ be a metric (of any signature) on the tangent bundle of a manifold $M$ of dimension$$$n\>2.$ The metric determines: the covariant derivative ${\nabla }_a$, the Schouten tensor $P_{\mathrm{ab}}$, the Weyl tensor $W_{\mathrm{abcd}}$ and the Cotton tensor $C_{\mathrm{abc}}\.$ The Bach tensor is defined as

The first calling sequence computes $B_{\mathrm{ab}}$ directly from the given metric using the formula above. The second calling sequence computes $B_{\mathrm{ab}}$ from the given metric and Christoffel connection. The third calling sequence computes $B_{\mathrm{ab}}$ directly from the given metric Christoffel connection, curvature and Cotton tensors.

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

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

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

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

$g≔\mathrm{evalDG}\left(-\mathrm{du}\&s\mathrm{dv}\+\mathrm{dx}\&t\mathrm{dx}\+\mathrm{dy}\&t\mathrm{dy}\+{\ⅇ}^{xy}\mathrm{du}\&t\mathrm{du}\right)$

$g:=\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,{\ⅇ}^{xy}\right]\,\left[\left[1\,2\right]\,-\frac{1}{2}\right]\,\left[\left[2\,1\right]\,-\frac{1}{2}\right]\,\left[\left[3\,3\right]\,1\right]\,\left[\left[4\,4\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,{\ⅇ}^{xy}\right]\,\left[\left[1\,2\right]\,-\frac{1}{2}\right]\,\left[\left[2\,1\right]\,-\frac{1}{2}\right]\,\left[\left[3\,3\right]\,1\right]\,\left[\left[4\,4\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,{\ⅇ}^{xy}\right]\,\left[\left[1\,2\right]\,-\frac{1}{2}\right]\,\left[\left[2\,1\right]\,-\frac{1}{2}\right]\,\left[\left[3\,3\right]\,1\right]\,\left[\left[4\,4\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,{\ⅇ}^{xy}\right]\,\left[\left[1\,2\right]\,-\frac{1}{2}\right]\,\left[\left[2\,1\right]\,-\frac{1}{2}\right]\,\left[\left[3\,3\right]\,1\right]\,\left[\left[4\,4\right]\,1\right]\right]\right]\right)$

$B≔\mathrm{BachTensor}\left(g\right)$

$B:=\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,-\frac{{\ⅇ}^{xy}\left(x^4+2x^2{y}^2+y^4+8xy+4\right)}{4}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,-\frac{{\ⅇ}^{xy}\left(x^4+2x^2{y}^2+y^4+8xy+4\right)}{4}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,-\frac{{\ⅇ}^{xy}\left(x^4+2x^2{y}^2+y^4+8xy+4\right)}{4}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,-\frac{{\ⅇ}^{xy}\left(x^4+2x^2{y}^2+y^4+8xy+4\right)}{4}\right]\right]\right]\right)$

$\mathrm{TensorInnerProduct}\left(g\,g\,B\right)$

$0$

$\mathrm{Gamma}≔\mathrm{Christoffel}\left(g\right)$

$\mathrm{\Γ}:=\mathrm{\_DG}\left(\left[\left[''connection''\,M\,\left[\left[''con\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[2\,1\,3\right]\,-y{\ⅇ}^{xy}\right]\,\left[\left[2\,1\,4\right]\,-x{\ⅇ}^{xy}\right]\,\left[\left[2\,3\,1\right]\,-y{\ⅇ}^{xy}\right]\,\left[\left[2\,4\,1\right]\,-x{\ⅇ}^{xy}\right]\,\left[\left[3\,1\,1\right]\,-\frac{y{\ⅇ}^{xy}}{2}\right]\,\left[\left[4\,1\,1\right]\,-\frac{x{\ⅇ}^{xy}}{2}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''connection''\,M\,\left[\left[''con\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[2\,1\,3\right]\,-y{\ⅇ}^{xy}\right]\,\left[\left[2\,1\,4\right]\,-x{\ⅇ}^{xy}\right]\,\left[\left[2\,3\,1\right]\,-y{\ⅇ}^{xy}\right]\,\left[\left[2\,4\,1\right]\,-x{\ⅇ}^{xy}\right]\,\left[\left[3\,1\,1\right]\,-\frac{y{\ⅇ}^{xy}}{2}\right]\,\left[\left[4\,1\,1\right]\,-\frac{x{\ⅇ}^{xy}}{2}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''connection''\,M\,\left[\left[''con\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[2\,1\,3\right]\,-y{\ⅇ}^{xy}\right]\,\left[\left[2\,1\,4\right]\,-x{\ⅇ}^{xy}\right]\,\left[\left[2\,3\,1\right]\,-y{\ⅇ}^{xy}\right]\,\left[\left[2\,4\,1\right]\,-x{\ⅇ}^{xy}\right]\,\left[\left[3\,1\,1\right]\,-\frac{y{\ⅇ}^{xy}}{2}\right]\,\left[\left[4\,1\,1\right]\,-\frac{x{\ⅇ}^{xy}}{2}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''connection''\,M\,\left[\left[''con\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[2\,1\,3\right]\,-y{\ⅇ}^{xy}\right]\,\left[\left[2\,1\,4\right]\,-x{\ⅇ}^{xy}\right]\,\left[\left[2\,3\,1\right]\,-y{\ⅇ}^{xy}\right]\,\left[\left[2\,4\,1\right]\,-x{\ⅇ}^{xy}\right]\,\left[\left[3\,1\,1\right]\,-\frac{y{\ⅇ}^{xy}}{2}\right]\,\left[\left[4\,1\,1\right]\,-\frac{x{\ⅇ}^{xy}}{2}\right]\right]\right]\right)$

$\mathrm{BachTensor}\left(g\,\mathrm{Gamma}\right)$

$\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,-\frac{{\ⅇ}^{xy}\left(x^4+2x^2{y}^2+y^4+8xy+4\right)}{4}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,-\frac{{\ⅇ}^{xy}\left(x^4+2x^2{y}^2+y^4+8xy+4\right)}{4}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,-\frac{{\ⅇ}^{xy}\left(x^4+2x^2{y}^2+y^4+8xy+4\right)}{4}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,-\frac{{\ⅇ}^{xy}\left(x^4+2x^2{y}^2+y^4+8xy+4\right)}{4}\right]\right]\right]\right)$

$R≔\mathrm{CurvatureTensor}\left(g\right)$

$R:=\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_bas''\,''cov\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[2\,3\,1\,3\right]\,y^2{\ⅇ}^{xy}\right]\,\left[\left[2\,3\,1\,4\right]\,{\ⅇ}^{xy}\left(xy+1\right)\right]\,\left[\left[2\,3\,3\,1\right]\,-y^2{\ⅇ}^{xy}\right]\,\left[\left[2\,3\,4\,1\right]\,-{\ⅇ}^{xy}\left(xy+1\right)\right]\,\left[\left[2\,4\,1\,3\right]\,{\ⅇ}^{xy}\left(xy+1\right)\right]\,\left[\left[2\,4\,1\,4\right]\,x^2{\ⅇ}^{xy}\right]\,\left[\left[2\,4\,3\,1\right]\,-{\ⅇ}^{xy}\left(xy+1\right)\right]\,\left[\left[2\,4\,4\,1\right]\,-x^2{\ⅇ}^{xy}\right]\,\left[\left[3\,1\,1\,3\right]\,\frac{y^2{\ⅇ}^{xy}}{2}\right]\,\left[\left[3\,1\,1\,4\right]\,\frac{{\ⅇ}^{xy}\left(xy+1\right)}{2}\right]\,\left[\left[3\,1\,3\,1\right]\,-\frac{y^2{\ⅇ}^{xy}}{2}\right]\,\left[\left[3\,1\,4\,1\right]\,-\frac{{\ⅇ}^{xy}\left(xy+1\right)}{2}\right]\,\left[\left[4\,1\,1\,3\right]\,\frac{{\ⅇ}^{xy}\left(xy+1\right)}{2}\right]\,\left[\left[4\,1\,1\,4\right]\,\frac{x^2{\ⅇ}^{xy}}{2}\right]\,\left[\left[4\,1\,3\,1\right]\,-\frac{{\ⅇ}^{xy}\left(xy+1\right)}{2}\right]\,\left[\left[4\,1\,4\,1\right]\,-\frac{x^2{\ⅇ}^{xy}}{2}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_bas''\,''cov\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[2\,3\,1\,3\right]\,y^2{\ⅇ}^{xy}\right]\,\left[\left[2\,3\,1\,4\right]\,{\ⅇ}^{xy}\left(xy+1\right)\right]\,\left[\left[2\,3\,3\,1\right]\,-y^2{\ⅇ}^{xy}\right]\,\left[\left[2\,3\,4\,1\right]\,-{\ⅇ}^{xy}\left(xy+1\right)\right]\,\left[\left[2\,4\,1\,3\right]\,{\ⅇ}^{xy}\left(xy+1\right)\right]\,\left[\left[2\,4\,1\,4\right]\,x^2{\ⅇ}^{xy}\right]\,\left[\left[2\,4\,3\,1\right]\,-{\ⅇ}^{xy}\left(xy+1\right)\right]\,\left[\left[2\,4\,4\,1\right]\,-x^2{\ⅇ}^{xy}\right]\,\left[\left[3\,1\,1\,3\right]\,\frac{y^2{\ⅇ}^{xy}}{2}\right]\,\left[\left[3\,1\,1\,4\right]\,\frac{{\ⅇ}^{xy}\left(xy+1\right)}{2}\right]\,\left[\left[3\,1\,3\,1\right]\,-\frac{y^2{\ⅇ}^{xy}}{2}\right]\,\left[\left[3\,1\,4\,1\right]\,-\frac{{\ⅇ}^{xy}\left(xy+1\right)}{2}\right]\,\left[\left[4\,1\,1\,3\right]\,\frac{{\ⅇ}^{xy}\left(xy+1\right)}{2}\right]\,\left[\left[4\,1\,1\,4\right]\,\frac{x^2{\ⅇ}^{xy}}{2}\right]\,\left[\left[4\,1\,3\,1\right]\,-\frac{{\ⅇ}^{xy}\left(xy+1\right)}{2}\right]\,\left[\left[4\,1\,4\,1\right]\,-\frac{x^2{\ⅇ}^{xy}}{2}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_bas''\,''cov\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[2\,3\,1\,3\right]\,y^2{\ⅇ}^{xy}\right]\,\left[\left[2\,3\,1\,4\right]\,{\ⅇ}^{xy}\left(xy+1\right)\right]\,\left[\left[2\,3\,3\,1\right]\,-y^2{\ⅇ}^{xy}\right]\,\left[\left[2\,3\,4\,1\right]\,-{\ⅇ}^{xy}\left(xy+1\right)\right]\,\left[\left[2\,4\,1\,3\right]\,{\ⅇ}^{xy}\left(xy+1\right)\right]\,\left[\left[2\,4\,1\,4\right]\,x^2{\ⅇ}^{xy}\right]\,\left[\left[2\,4\,3\,1\right]\,-{\ⅇ}^{xy}\left(xy+1\right)\right]\,\left[\left[2\,4\,4\,1\right]\,-x^2{\ⅇ}^{xy}\right]\,\left[\left[3\,1\,1\,3\right]\,\frac{y^2{\ⅇ}^{xy}}{2}\right]\,\left[\left[3\,1\,1\,4\right]\,\frac{{\ⅇ}^{xy}\left(xy+1\right)}{2}\right]\,\left[\left[3\,1\,3\,1\right]\,-\frac{y^2{\ⅇ}^{xy}}{2}\right]\,\left[\left[3\,1\,4\,1\right]\,-\frac{{\ⅇ}^{xy}\left(xy+1\right)}{2}\right]\,\left[\left[4\,1\,1\,3\right]\,\frac{{\ⅇ}^{xy}\left(xy+1\right)}{2}\right]\,\left[\left[4\,1\,1\,4\right]\,\frac{x^2{\ⅇ}^{xy}}{2}\right]\,\left[\left[4\,1\,3\,1\right]\,-\frac{{\ⅇ}^{xy}\left(xy+1\right)}{2}\right]\,\left[\left[4\,1\,4\,1\right]\,-\frac{x^2{\ⅇ}^{xy}}{2}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_bas''\,''cov\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[2\,3\,1\,3\right]\,y^2{\ⅇ}^{xy}\right]\,\left[\left[2\,3\,1\,4\right]\,{\ⅇ}^{xy}\left(xy+1\right)\right]\,\left[\left[2\,3\,3\,1\right]\,-y^2{\ⅇ}^{xy}\right]\,\left[\left[2\,3\,4\,1\right]\,-{\ⅇ}^{xy}\left(xy+1\right)\right]\,\left[\left[2\,4\,1\,3\right]\,{\ⅇ}^{xy}\left(xy+1\right)\right]\,\left[\left[2\,4\,1\,4\right]\,x^2{\ⅇ}^{xy}\right]\,\left[\left[2\,4\,3\,1\right]\,-{\ⅇ}^{xy}\left(xy+1\right)\right]\,\left[\left[2\,4\,4\,1\right]\,-x^2{\ⅇ}^{xy}\right]\,\left[\left[3\,1\,1\,3\right]\,\frac{y^2{\ⅇ}^{xy}}{2}\right]\,\left[\left[3\,1\,1\,4\right]\,\frac{{\ⅇ}^{xy}\left(xy+1\right)}{2}\right]\,\left[\left[3\,1\,3\,1\right]\,-\frac{y^2{\ⅇ}^{xy}}{2}\right]\,\left[\left[3\,1\,4\,1\right]\,-\frac{{\ⅇ}^{xy}\left(xy+1\right)}{2}\right]\,\left[\left[4\,1\,1\,3\right]\,\frac{{\ⅇ}^{xy}\left(xy+1\right)}{2}\right]\,\left[\left[4\,1\,1\,4\right]\,\frac{x^2{\ⅇ}^{xy}}{2}\right]\,\left[\left[4\,1\,3\,1\right]\,-\frac{{\ⅇ}^{xy}\left(xy+1\right)}{2}\right]\,\left[\left[4\,1\,4\,1\right]\,-\frac{x^2{\ⅇ}^{xy}}{2}\right]\right]\right]\right)$

$C≔\mathrm{CottonTensor}\left(g\right)$

$C:=\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\,3\right]\,\frac{{\ⅇ}^{xy}\left(x^{2}y+y^3+2x\right)}{4}\right]\,\left[\left[1\,1\,4\right]\,\frac{{\ⅇ}^{xy}\left(x^3+x{y}^2+2y\right)}{4}\right]\,\left[\left[1\,3\,1\right]\,-\frac{{\ⅇ}^{xy}\left(x^{2}y+y^3+2x\right)}{4}\right]\,\left[\left[1\,4\,1\right]\,-\frac{{\ⅇ}^{xy}\left(x^3+x{y}^2+2y\right)}{4}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\,3\right]\,\frac{{\ⅇ}^{xy}\left(x^{2}y+y^3+2x\right)}{4}\right]\,\left[\left[1\,1\,4\right]\,\frac{{\ⅇ}^{xy}\left(x^3+x{y}^2+2y\right)}{4}\right]\,\left[\left[1\,3\,1\right]\,-\frac{{\ⅇ}^{xy}\left(x^{2}y+y^3+2x\right)}{4}\right]\,\left[\left[1\,4\,1\right]\,-\frac{{\ⅇ}^{xy}\left(x^3+x{y}^2+2y\right)}{4}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\,3\right]\,\frac{{\ⅇ}^{xy}\left(x^{2}y+y^3+2x\right)}{4}\right]\,\left[\left[1\,1\,4\right]\,\frac{{\ⅇ}^{xy}\left(x^3+x{y}^2+2y\right)}{4}\right]\,\left[\left[1\,3\,1\right]\,-\frac{{\ⅇ}^{xy}\left(x^{2}y+y^3+2x\right)}{4}\right]\,\left[\left[1\,4\,1\right]\,-\frac{{\ⅇ}^{xy}\left(x^3+x{y}^2+2y\right)}{4}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\,3\right]\,\frac{{\ⅇ}^{xy}\left(x^{2}y+y^3+2x\right)}{4}\right]\,\left[\left[1\,1\,4\right]\,\frac{{\ⅇ}^{xy}\left(x^3+x{y}^2+2y\right)}{4}\right]\,\left[\left[1\,3\,1\right]\,-\frac{{\ⅇ}^{xy}\left(x^{2}y+y^3+2x\right)}{4}\right]\,\left[\left[1\,4\,1\right]\,-\frac{{\ⅇ}^{xy}\left(x^3+x{y}^2+2y\right)}{4}\right]\right]\right]\right)$

$\mathrm{BachTensor}\left(g\,\mathrm{Gamma}\,R\,C\right)$

$\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,-\frac{{\ⅇ}^{xy}\left(x^4+2x^2{y}^2+y^4+8xy+4\right)}{4}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,-\frac{{\ⅇ}^{xy}\left(x^4+2x^2{y}^2+y^4+8xy+4\right)}{4}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,-\frac{{\ⅇ}^{xy}\left(x^4+2x^2{y}^2+y^4+8xy+4\right)}{4}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,-\frac{{\ⅇ}^{xy}\left(x^4+2x^2{y}^2+y^4+8xy+4\right)}{4}\right]\right]\right]\right)$

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

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

$\mathrm{g0}≔\mathrm{evalDG}\left(-\mathrm{du}\&s\mathrm{dv}\+\mathrm{dx}\&t\mathrm{dx}\+\mathrm{dy}\&t\mathrm{dy}\+xy\mathrm{du}\&t\mathrm{du}\right)$

$\mathrm{g0}:=\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,xy\right]\,\left[\left[1\,2\right]\,-\frac{1}{2}\right]\,\left[\left[2\,1\right]\,-\frac{1}{2}\right]\,\left[\left[3\,3\right]\,1\right]\,\left[\left[4\,4\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,xy\right]\,\left[\left[1\,2\right]\,-\frac{1}{2}\right]\,\left[\left[2\,1\right]\,-\frac{1}{2}\right]\,\left[\left[3\,3\right]\,1\right]\,\left[\left[4\,4\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,xy\right]\,\left[\left[1\,2\right]\,-\frac{1}{2}\right]\,\left[\left[2\,1\right]\,-\frac{1}{2}\right]\,\left[\left[3\,3\right]\,1\right]\,\left[\left[4\,4\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,xy\right]\,\left[\left[1\,2\right]\,-\frac{1}{2}\right]\,\left[\left[2\,1\right]\,-\frac{1}{2}\right]\,\left[\left[3\,3\right]\,1\right]\,\left[\left[4\,4\right]\,1\right]\right]\right]\right)$

$\mathrm{RicciTensor}\left(\mathrm{g0}\right)$

$\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,0\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,0\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,0\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,0\right]\right]\right]\right)$

$\mathrm{g1}≔\mathrm{evalDG}\left({\ⅇ}^{f\left(u\,v\,x\,y\right)}\mathrm{g0}\right)$

$\mathrm{g1}:=\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,{\ⅇ}^{f\left(u\,v\,x\,y\right)}xy\right]\,\left[\left[1\,2\right]\,-\frac{{\ⅇ}^{f\left(u\,v\,x\,y\right)}}{2}\right]\,\left[\left[2\,1\right]\,-\frac{{\ⅇ}^{f\left(u\,v\,x\,y\right)}}{2}\right]\,\left[\left[3\,3\right]\,{\ⅇ}^{f\left(u\,v\,x\,y\right)}\right]\,\left[\left[4\,4\right]\,{\ⅇ}^{f\left(u\,v\,x\,y\right)}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,{\ⅇ}^{f\left(u\,v\,x\,y\right)}xy\right]\,\left[\left[1\,2\right]\,-\frac{{\ⅇ}^{f\left(u\,v\,x\,y\right)}}{2}\right]\,\left[\left[2\,1\right]\,-\frac{{\ⅇ}^{f\left(u\,v\,x\,y\right)}}{2}\right]\,\left[\left[3\,3\right]\,{\ⅇ}^{f\left(u\,v\,x\,y\right)}\right]\,\left[\left[4\,4\right]\,{\ⅇ}^{f\left(u\,v\,x\,y\right)}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,{\ⅇ}^{f\left(u\,v\,x\,y\right)}xy\right]\,\left[\left[1\,2\right]\,-\frac{{\ⅇ}^{f\left(u\,v\,x\,y\right)}}{2}\right]\,\left[\left[2\,1\right]\,-\frac{{\ⅇ}^{f\left(u\,v\,x\,y\right)}}{2}\right]\,\left[\left[3\,3\right]\,{\ⅇ}^{f\left(u\,v\,x\,y\right)}\right]\,\left[\left[4\,4\right]\,{\ⅇ}^{f\left(u\,v\,x\,y\right)}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,{\ⅇ}^{f\left(u\,v\,x\,y\right)}xy\right]\,\left[\left[1\,2\right]\,-\frac{{\ⅇ}^{f\left(u\,v\,x\,y\right)}}{2}\right]\,\left[\left[2\,1\right]\,-\frac{{\ⅇ}^{f\left(u\,v\,x\,y\right)}}{2}\right]\,\left[\left[3\,3\right]\,{\ⅇ}^{f\left(u\,v\,x\,y\right)}\right]\,\left[\left[4\,4\right]\,{\ⅇ}^{f\left(u\,v\,x\,y\right)}\right]\right]\right]\right)$

$\mathrm{BachTensor}\left(\mathrm{g1}\right)$

$\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,0\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,0\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,0\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,0\right]\right]\right]\right)$