In the book "In pursuit of the Unknown" by Ian Stewart, page 19 of chapter "Pythagoras' Theorem" shows the equation for the distance between two points in a non-euclidian space from point $(x,y,z)$ to $(x+dx,y+dy,z+dz)$ as:
$ds^2 = X dx^2 + Y dy^2 + Z dz^2 + 2U dx\,dy + 2V dx\,dz + 2W dy\,dz$
Being $ds$ the distance between the two points, $dx$ the distance between $x$ and $x+dx$, and so on.
Could you clarify the meaning of $X, Y, Z, U, V, W$ in this formula?
Thank you!
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paul0207
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Hello and welcome to MSE! A little tip to improve readability: use MathJax everywhere mathematical symbols are found in your question, not just for equations. The more readable the question is, the more inviting it is for reviewers to provide answers. – user3733558 Mar 27 '21 at 21:59
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They're functions of $x,y,z$ that depend on what the space is. For instance in 2D hyperbolic geometry, modeled on the upper half-plane, $ds^2=\frac{1}{y^2}{dx}^2+\frac{1}{y^2}dy^2$. Really you want to look up "metric tensor." The metric describes how the real geometry of your space compares with the geometry of your coordinates you're using to model them, informally. – anon Mar 28 '21 at 00:39
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Thanks runway44. From what I read about metric tensor I understand it measures the curvature of the space. – paul0207 Mar 28 '21 at 00:51