How can I force mathematica to expand for example this expression
$$(\cos (\theta ) dr-r d\theta \sin (\theta ))\wedge (\sin (\theta ) dr+r d\theta \cos (\theta ))$$
into what is should be, that is
$$r dr\wedge d\theta \, ?$$
For your convenience:
(Cos[θ] Dt[r] - r Dt[θ] Sin[θ])\[Wedge](r Cos[θ] Dt[θ] + Dt[r] Sin[θ])
One idea is to use TensorReduce. I will assume that r is real, and that Dt[r] and Dt[θ] are symbolic vectors:
$Assumptions = r ∈ Reals && (Dt[r]|Dt[θ]) ∈ Vectors[{d}];
Then, we can use Simplify + TensorReduce:
Simplify @ TensorReduce[
TensorWedge[Cos[θ] Dt[r]-r Dt[θ] Sin[θ], r Cos[θ] Dt[θ]+Dt[r] Sin[θ]]
]
r Dt[r] \[TensorWedge] Dt[θ]
Wedgehas no built-in meaning in Mathematica. – march Oct 02 '17 at 21:48Wedge's documentation page: "Wedge[x,y,[Ellipsis]] has no built-in meaning."Wedgeis there to allow you define your own favorite wedge product, of which there are many.TensorWedgeis a tensorial operation. – Itai Seggev Oct 03 '17 at 05:55