I have already ask this question on stackoverflow, but since it concerns as mathematics than computer science, I ask it here too.
I would like to make a classical computation using maple. I would like define an abstract moving frame (e_1,e_2), then
1) get the dual frame omega_1, omega_2 (always abstract) omega_i(e_j)=delta_i^j
2) define omega_12 such that d omega_1= omega_12 wedge omega_2
3) define Gauss curvature K such that d omega_12=K omega_1\wedge omega_2
4) Finally I would like to get an abstract formula for \overline{K} the curvature of the a new frame (\overline{e}_1,\overline{e}_2)=e^f (e_1,e_2) , where f is a abstract function, with respect to K and f.
I should use the package Differential geometry but even I don't succeed to define the frame. The exmpale in the description of the library is
DGsetup([x,y,z,w],M):
Example 1. Define a 3-dimensional subspace of vectors by the span of S and compute a simpler base for this subspace relative to the coordinate basis T for the tangent space of M.
S1:=evalDG([D_x−D_y,D_x+D_y,D_x+D_y+D_w])
But S1 is an explicit frame I want to define an abstract one as I can do for functions. For instance if I want an abstract Liebnitz rule,
diff(f(x)*g(x), x);
/ d \ / d \
|--- f(x)| g(x) + f(x) |--- g(x)|
\ dx / \ dx /
Here I have tryied the following:
with(DifferentialGeometry); DGsetup([x, y], M); S1 := evalDG([e1, e2]); M > B1 := DGbasis(S1); Error, (in DifferentialGeometry:-DGbasis) expected 1st argument to be a list of biforms, forms, vectors tensors, matrices, vectors. Received [e1, e2]
EDIT:
Here is what I have start to do with Atlas. As you see, it doesn't use the symmetries of the Lie derivative. And I don't know how to make my 4)
@Paul, I must confess that it is not clear to me exactly what you are looking for in regards to 4.), but I think that what you can do with Atlas combined with a little exterior algebra works quite nicely. (I have attached a screen shot of my Maple file below and I am having some issues with some of the simplifications as well. I seem to remember having these problems before, but of late I have been using Mathematica and a newer version of Atlas. The attached screen shot is from what I believe to be the original version of Atlas and Maple 15 so it is possibly outdated. I apologize for any confusion that this causes.)
Notation: In keeping with the notation of Atlas, I use $\iota$ for the interior product of a vector field with a differential form.
Key Theorems:
Now, assuming that we are on a two dimensional manifold and that $\omega^{1}$ and $\omega^{2}$ form an orthonormal coframe for a metric tensor $g$, i.e., $g = \omega^{1} \otimes \omega^{1} + \omega^{2} \otimes \omega^{2}$, one calculates $\omega^{1}_{2}$ to be
$$ \omega^{1}_{2} = \frac{1}{2}\left(\,\iota_{{E_{{2}}}} \left( \mathrm{d} \left( \omega^{{1}} \right) \right) + \,\iota_{{E_{{1}}}} \left( \iota_{{E_{{2}}}} \left( \mathrm{d} \left( \omega^{{1}} \right) \right) \right) \omega^{{1}}-\,\iota_{{E_{{1}}}} \left( \mathrm{d} \left( \omega^{{2}} \right) \right) + \,\iota_{{E_{{1}}}} \left( \iota_{{E_{{2}}}} \left( \mathrm{d} \left( \omega^{{2}} \right) \right) \right) \omega^{{2}}\right). $$
Going term-by-term through $\omega^{1}_{2}$ we have
$\iota_{{E_{{2}}}} \left( \mathrm{d} \left( \omega^{{1}} \right) \right)= \mathcal{L}_{E_{2}}\omega^{1} = -C_{2k}^{1}\omega^{k} = C_{12}^{1}\omega^{1}$
$\iota_{{E_{{1}}}} \left( \iota_{{E_{{2}}}} \left( \mathrm{d} \left( \omega^{{1}} \right) \right) \right) \omega^{{1}} = -\omega^{1}\left(\left[E_{2}, E_{1}\right]\right)\omega^{1}= C_{12}^{1}\omega^{1}$
We thus have $$ \omega^{1}_{2} = C_{12}^{1}\omega^{1} + C_{12}^{2}\omega^{2}, $$ and one can see that $$\mathrm{d}\omega^{1} = -\omega^{1}_{2} \wedge \omega^2 = -C_{12}^{1}\omega^{1} \wedge \omega^{2}.$$ Furthermore, we have $$\mathrm{d}\omega^{2} = -\omega^{2}_{1} \wedge \omega^1 = \omega^{1}_{2}\wedge \omega^{1} = C_{12}^{2} \omega^{2}\wedge \omega^{1} = -C_{12}^{2}\omega^{1}\wedge\omega^{2} $$
To calculate the curvature, we now have \begin{align*} \mathrm{d}\omega^{1}_{2} &= \mathrm{d}\left(C_{12}^{1}\omega^{1} + C_{12}^{2}\omega^{2}\right)\\ &= \mathrm{d}C_{12}^{1}\wedge \omega^{1} + C_{12}^{1}\mathrm{d}\omega^{1} +\mathrm{d}C_{12}^{2}\wedge\omega^{2} +C_{12}^{2}\mathrm{d}\omega^{2}\\ &= \mathrm{d}C_{12}^{1}\wedge \omega^{1} + C_{12}^{1}\left(-C_{12}^{1}\omega^{1}\wedge \omega^{2}\right) +\mathrm{d}C_{12}^{2}\wedge\omega^{2} +C_{12}^{2}\left(-C_{12}^{2}\omega^{1}\wedge\omega^{2}\right)\\ &= \left(E_{1}C_{12}^{1}\omega^{1} + E_{2}C_{12}^{2}\omega^{2}\right) \wedge \omega^{1} - \left(C_{12}^{1}\right)^2\omega^{1}\wedge \omega^{2} +\left(E_{1}C_{12}^{2}\omega^{1} +E_{2}C_{12}^{2}\omega^{2}\right)\wedge\omega^{2} -\left(C_{12}^{2}\right)^2\omega^{1}\wedge\omega^{2}\\ &=E_{2}C_{12}^{2}\,\omega^{2} \wedge \omega^{1} -\left(C_{12}^{1}\right)^2\left(\omega^{1}\wedge \omega^{2}\right) + E_{1}C_{12}^{2}\omega^{1}\wedge\omega^{2} -\left(C_{12}^{2}\right)^2\omega^{1}\wedge\omega^{2}\\ &= \underbrace{\left(-E_{2}C_{12}^{2} - \left(C_{12}^{1}\right)^2 + E_{1}C_{12}^{2} - \left(C_{12}^{2}\right)^2\right)}_{K}\omega^{1}\wedge \omega^{2} \end{align*}
Note: In the Maple file created with Atlas, I wrote $\omega^{1}_{2}$ as $$\omega^{1}_{2} = \phi = A \omega^{1} + B \omega^{2}, $$ where $A= C_{12}^{1}$ and $B = C_{12}^{2}$. I also did not spell out explicitly where I used the three key theorems in the substitutions. Let me know if you have any additional questions and/or spot any errors.
I initially thought that there would be a more direct solution using Atlas, but this is the best that I see at the moment.