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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])
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  • $\begingroup$ Unfortunately, Wedge has no built-in meaning in Mathematica. $\endgroup$ – march Oct 2 '17 at 21:48
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    $\begingroup$ Yes, that's the first detail in Wedge's documentation page: "Wedge[x,y,[Ellipsis]] has no built-in meaning." Wedge is there to allow you define your own favorite wedge product, of which there are many. TensorWedge is a tensorial operation. $\endgroup$ – Itai Seggev Oct 3 '17 at 5:55
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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]
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