[Dune] Basic question on coordinate transformation

[Aleksejs Fomins]

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Dear Dune,

I have always been puzzled by Jacobian transformations. And now, I might have a bug in their implementation in curvilinear geometry, so I thought it's a good time to sort out the unclear steps.

When I look at, e.g., affine geometry, I see that the Jacobian Transposed is defined in such a way that
(1) global() = origin + J * local
So "J" is a matrix that transforms from local to global coordinates.

However, if I look at the implementation of a surface normal, e.g. in GeometryGrid->Intersection->outerNormal I see that
refNormal = refElement.integrationOuterNormal( indexInInside() );
jit.mv( refNormal, normal );
so in order to transform the local (reference) normal to global coordinates, one multiplies it by the inverse of the Jacobian.

I am somewhat unable to convince myself why this works. My simplest counter-proof is to take the above equation (1) at 2 different locations and subtract from one another

p_global = g2 - g1 = J * l2 - J * l1 = J (l2 - l1) = J * p_local

I would be grateful if somebody could hint me on why the transformation of the normal is correct, or perhaps the nicest place to read about this.

Thank you,
Aleksejs
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