The command DGGramSchmidt applies the usual Gram-Schmidt procedure to a list $S\=\left[X_1\,X_2\,..\.\,X_m\right]$, of $m$ independent vectors, forms or tensors (the forms or tensors must all be of the same type). The procedure returns a list orthogonal vectors, forms or tensors $\left[Y_1\,Y_2\,..\.\,Y_m\right]$such that

The inner products of differential forms and tensors are calculated using TensorInnerProduct.

The default assumption is that the metric $g$ is positive-definite, that is, the signature is $\left(1\,1\,..\.\,1\right)$. In this case the length of each $Y_k$ is normalized to 1.  With method = "un-normalized" the $Y_k$ are left un-normalized.

If, for example, the signature of $g$ is $\left(1\,-1\,-1\,-1\right)$, use the keyword signature = [+1, -1, -1, -1]. The length of $Y_1$ is normalized to 1 and the lengths of $Y_2\,Y_3\,Y_4$ are normalized to $-1$.

If, for example, the signature of $g$ is $\left(1\,-1\,-1\,-1\right)$ and the vectors $X_1\,X_2$ are both null, then the usual Gram-Schmidt procedure will fail. In this case set signature = [[1,-1], 1, 1] and DGGramSchmidt will take $Y_1$ to be a vector of non-zero length in the span of $X_1\,X_2\.$

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

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

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

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

$g≔\mathrm{evalDG}\left(\mathrm{dx}\&t\mathrm{dx}+\mathrm{dy}\&t\mathrm{dy}+\mathrm{du}\&t\mathrm{du}+\mathrm{dv}\&t\mathrm{dv}\right)$

$g:=\mathrm{dx}\mathrm{dx}+\mathrm{dy}\mathrm{dy}+\mathrm{du}\mathrm{du}+\mathrm{dv}\mathrm{dv}$

$\mathrm{S1}≔\left[\mathrm{D\_x}+\mathrm{D\_y}\,\mathrm{D\_y}+2\mathrm{D\_u}\,\mathrm{D\_u}-\mathrm{D\_v}\,\mathrm{D\_v}\right]$

$\mathrm{S1}:=\left[\mathrm{D\_x}\+\mathrm{D\_y}\,\mathrm{D\_y}\+2\mathrm{D\_u}\,\mathrm{D\_u}-\mathrm{D\_v}\,\mathrm{D\_v}\right]$

$\mathrm{T1}≔\mathrm{DGGramSchmidt}\left(\mathrm{S1}\,g\right)$

$\mathrm{T1}:=\left[\frac{\sqrt{2}}{2}\mathrm{D\_x}+\frac{\sqrt{2}}{2}\mathrm{D\_y}\,-\frac{\sqrt{2}}{6}\mathrm{D\_x}+\frac{\sqrt{2}}{6}\mathrm{D\_y}+\frac{2\sqrt{2}}{3}\mathrm{D\_u}\,\frac{\sqrt{10}}{15}\mathrm{D\_x}-\frac{\sqrt{10}}{15}\mathrm{D\_y}+\frac{\sqrt{10}}{30}\mathrm{D\_u}-\frac{3\sqrt{10}}{10}\mathrm{D\_v}\,\frac{\sqrt{10}}{5}\mathrm{D\_x}-\frac{\sqrt{10}}{5}\mathrm{D\_y}+\frac{\sqrt{10}}{10}\mathrm{D\_u}+\frac{\sqrt{10}}{10}\mathrm{D\_v}\right]$

$\mathrm{TensorInnerProduct}\left(g\,\mathrm{T1}\,\mathrm{T1}\right)$

$\mathrm{T1u}≔\mathrm{DGGramSchmidt}\left(\mathrm{S1}\,g\,\mathrm{method}=''unnormalized''\right)$

$\mathrm{T1u}:=\left[\mathrm{D\_x}+\mathrm{D\_y}\,-\frac{1}{2}\mathrm{D\_x}+\frac{1}{2}\mathrm{D\_y}+2\mathrm{D\_u}\,\frac{2}{9}\mathrm{D\_x}-\frac{2}{9}\mathrm{D\_y}+\frac{1}{9}\mathrm{D\_u}-\mathrm{D\_v}\,\frac{1}{5}\mathrm{D\_x}-\frac{1}{5}\mathrm{D\_y}+\frac{1}{10}\mathrm{D\_u}+\frac{1}{10}\mathrm{D\_v}\right]$

$\mathrm{TensorInnerProduct}\left(g\,\mathrm{T1u}\,\mathrm{T1u}\right)$

$\mathrm{S2}≔\mathrm{evalDG}\left(\left[\mathrm{dx}\&w\mathrm{dy}-\mathrm{du}\&w\mathrm{dv}\,\mathrm{dx}\&w\mathrm{dy}\,\mathrm{dx}\&w\mathrm{dv}-\mathrm{dy}\&w\mathrm{dv}\,\mathrm{dx}\&w\mathrm{dv}\,\mathrm{dx}\&w\mathrm{du}+\mathrm{dy}\&w\mathrm{dv}\right]\right)$

$\mathrm{S2}:=\left[\mathrm{dx}\bigwedge \mathrm{dy}-\mathrm{du}\bigwedge \mathrm{dv}\,\mathrm{dx}\bigwedge \mathrm{dy}\,\mathrm{dx}\bigwedge \mathrm{dv}-\mathrm{dy}\bigwedge \mathrm{dv}\,\mathrm{dx}\bigwedge \mathrm{dv}\,\mathrm{dx}\bigwedge \mathrm{du}+\mathrm{dy}\bigwedge \mathrm{dv}\right]$

$\mathrm{T2}≔\mathrm{DGGramSchmidt}\left(\mathrm{S2}\,g\right)$

$\mathrm{T2}:=\left[\frac{1}{2}\mathrm{dx}\bigwedge \mathrm{dy}-\frac{1}{2}\mathrm{du}\bigwedge \mathrm{dv}\,\frac{1}{2}\mathrm{dx}\bigwedge \mathrm{dy}+\frac{1}{2}\mathrm{du}\bigwedge \mathrm{dv}\,\frac{1}{2}\mathrm{dx}\bigwedge \mathrm{dv}-\frac{1}{2}\mathrm{dy}\bigwedge \mathrm{dv}\,\frac{1}{2}\mathrm{dx}\bigwedge \mathrm{dv}+\frac{1}{2}\mathrm{dy}\bigwedge \mathrm{dv}\,\frac{\sqrt{2}}{2}\mathrm{dx}\bigwedge \mathrm{du}\right]$

$\mathrm{TensorInnerProduct}\left(g\,\mathrm{T2}\,\mathrm{T2}\right)$

$\mathrm{g3}≔\mathrm{evalDG}\left(\mathrm{dx}\&t\mathrm{dx}+\mathrm{dy}\&t\mathrm{dy}-\mathrm{du}\&t\mathrm{du}-\mathrm{dv}\&t\mathrm{dv}\right)$

$\mathrm{g3}:=\mathrm{dx}\mathrm{dx}+\mathrm{dy}\mathrm{dy}-\mathrm{du}\mathrm{du}-\mathrm{dv}\mathrm{dv}$

$\mathrm{S3}≔\left[\mathrm{dx}\,\mathrm{dy}\,\mathrm{du}\,\mathrm{dv}\right]$

$\mathrm{S3}:=\left[\mathrm{dx}\,\mathrm{dy}\,\mathrm{du}\,\mathrm{dv}\right]$

$\mathrm{DGGramSchmidt}\left(\mathrm{S3}\,\mathrm{g3}\right)$

$\left[\mathrm{dx}\,\mathrm{dy}\,-\mathrm{I}\mathrm{du}\,-\mathrm{I}\mathrm{dv}\right]$

$\mathrm{DGGramSchmidt}\left(\mathrm{S3}\,\mathrm{g3}\,\mathrm{signature}=\left[1\,1\,-1\,-1\right]\right)$

$\left[\mathrm{dx}\,\mathrm{dy}\,\mathrm{du}\,\mathrm{dv}\right]$

$\mathrm{g4}≔\mathrm{evalDG}\left(\mathrm{dx}\&s\mathrm{dy}+\mathrm{du}\&t\mathrm{du}+\mathrm{dv}\&t\mathrm{dv}\right)$

$\mathrm{g4}:=\frac{1}{2}\mathrm{dx}\mathrm{dy}+\frac{1}{2}\mathrm{dy}\mathrm{dx}+\mathrm{du}\mathrm{du}+\mathrm{dv}\mathrm{dv}$

$\mathrm{S4}≔\left[\mathrm{D\_x}\,\mathrm{D\_y}\,\mathrm{D\_u}\,\mathrm{D\_v}\right]$

$\mathrm{S4}:=\left[\mathrm{D\_x}\,\mathrm{D\_y}\,\mathrm{D\_u}\,\mathrm{D\_v}\right]$

$\mathrm{S4a}≔\mathrm{evalDG}\left(\left[\mathrm{D\_x}+\mathrm{D\_u}\,\mathrm{D\_y}\,\mathrm{D\_u}\,\mathrm{D\_v}\right]\right)$

$\mathrm{S4a}:=\left[\mathrm{D\_x}+\mathrm{D\_u}\,\mathrm{D\_y}\,\mathrm{D\_u}\,\mathrm{D\_v}\right]$

$\mathrm{DGGramSchmidt}\left(\mathrm{S4a}\,\mathrm{g4}\,\mathrm{signature}=\left[1\,-1\,1\,1\right]\right)$

$\left[\mathrm{D\_x}+\mathrm{D\_u}\,-\mathrm{D\_x}+2\mathrm{D\_y}-\mathrm{D\_u}\,-2\mathrm{D\_y}+\mathrm{D\_u}\,\mathrm{D\_v}\right]$

$\mathrm{DGGramSchmidt}\left(\mathrm{S4}\,\mathrm{g4}\,\mathrm{signature}=\left[\left[1\,-1\right]\,1\,1\right]\right)$

$\left[\mathrm{D\_x}+\mathrm{D\_y}\,\mathrm{D\_x}-\mathrm{D\_y}\,\mathrm{D\_u}\,\mathrm{D\_v}\right]$