I'm trying to compute the following differential form

$\omega = x(dy\wedge dz) + y(dx \wedge dz) + z(dx \wedge dy)$

but using a change of coordinates into spherical coords. So far, this is my code:

w[x_, y_, z_, dx_, dy_, dz_] := x*(dy\[TensorWedge]dz) + y*(dz\[TensorWedge]dz) + z*(dz\[TensorWedge]dy)

Spherical[r_, p_, t_] := {r*Cos[p]*   Cos[t], r*Cos[p]*Sin[t], r*Sin[p]}

dx = Dot[Grad[Spherical[r, p, t][[1]], {r, p, t}], {Dt[r], Dt[p], Dt[t]}]
dy = Dot[Grad[Spherical[r, p, t][[2]], {r, p, t}], {Dt[r], Dt[p], Dt[t]}]
dz = Dot[Grad[Spherical[r, p, t][[3]], {r, p, t}], {Dt[r], Dt[p], Dt[t]}]

$Assumptions = (r | t | p) \[Element] Reals && (Dt[r] | Dt[t] | Dt[p]) \[Element] Vectors[{d}];

My problem comes from the following line:

dy\[TensorWedge] dz // TensorExpand

(* r Cos[t] Cos[p] Sin[p] Dt[t]\[TensorWedge]Dt[r] + r^2 Cos[t] Cos[p]^2 Dt[t]\[TensorWedge]Dt[p] + r Cos[p]^2 Sin[t] Dt[r]\[TensorWedge]Dt[p] - r Sin[t] Sin[p]^2 Dt[p]\[TensorWedge]Dt[r] *)

As you can see, the last two summands are $d\rho\wedge d\theta$ and $d\theta\wedge d\rho$. I guess that the problem is either that I'm not defining antisymmetry in TensorWedge (which I think it should have by default, since $d\rho \wedge d\rho$ is not there) or, the coefficients before the wedge products are not letting mathematica find the symmetries (but this does not make sense to me either, since I've defined before what is a real number and what is a vector).

Am I doing something wrong?

Thank you very much.

I'm on Mathematica 10.2, by the way.


1 Answer 1


TensorExpand doesn't make use of TensorWedge antisymmetry, but TensorReduce does:

TensorReduce[dy \[TensorWedge] dz]

-r Cos[p]^2 Sin[t] Dt[p] \[TensorWedge] Dt[r] - r Sin[p]^2 Sin[t] Dt[p] \[TensorWedge] Dt[r] - r^2 Cos[p]^2 Cos[t] Dt[p] \[TensorWedge] Dt[t] - r Cos[p] Cos[t] Sin[p] Dt[r] \[TensorWedge] Dt[t]


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