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I need to integrate the product of two polynomial functions defined on an arbitrary (convex) planar quadrilateral defined by 4 points in $\mathbb{R}^3$.

I was trying to firstly rotate the system of reference so to make the problem 2D and then transform the quadrilateral into the unit square and perform the integration there with the correct change of coordinates. This is the reason I opened this question.

However some users suggested different strategies, so I move the question to a higher level.

Is there any theorem or formula to perform such integration?

Thank you very much

Luca
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  • Do the four points lie in the same plane? If so, it makes sense to first rotate the coordinate system so the problem becomes 2-dimensional – Yuriy S Apr 18 '19 at 09:23
  • You are right. Forgot to specify that. I'll add it to the problem – Luca Apr 18 '19 at 09:24
  • I'm pretty sure my answer to the linked question essentially answers this one. You can work out the area of the infinitesimal quadrilaterals determined by $dt$ and $ds$. – Ethan Bolker Apr 18 '19 at 10:08
  • @EthanBolker I obtained the function for (x,y) = f(s,t), but I'm having a bit of troubles inverting it to find (s,t) = g(x,y) as the function is (as expected) non-linear – Luca Apr 18 '19 at 10:35
  • @LucaAmerio You may not need to. I think its Jacobian when multiplied by $dsdt$ gives you the area of the infinitesimal quadrilateral at $f(s,t)$. https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant – Ethan Bolker Apr 18 '19 at 11:47
  • @EthanBolker shouldn't I express the function i want to integrate $f(x,y)$ as $f(s,t) = f(g(x,y))$ where $g(x,y)$ is the coordinates transformation from $x,y$ to $s,t$? – Luca Apr 18 '19 at 12:02
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    Yes, more or less. Search double integral change of variable . Here's one link: http://math.mit.edu/~jorloff/suppnotes/suppnotes02/cv.pdf . If when work out the integral you seem to need to invert the transformation you may need just the inverse of the determinant rather than the inverse of the transformation. – Ethan Bolker Apr 18 '19 at 12:16
  • The Jacobian of the map in your original question does vanish along a line, but that line generally (always?) lies outside of the unit square. – amd Apr 18 '19 at 20:40

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