Expand wedge product

Question

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])

Answer 1

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]