Proof of orthogonality in a coordinate system

Question

Given any coordinate transformation, how could I prove that the new coordinate system is orthogonal?

Answer 1

To prove that a new system of coordinates is orthogonal you have to find the basis of the new system, made up by the tangent vectors. For ex, if you have new coordinates $u,v$ on the plane and these are related to $x,y$ (Cartesian, orthogonal) coordinates by means of $x=f(u,v)$, $y=g(u,v)$, those new coordinates are orthogonal if the tangent vectors $$ (\partial f/\partial u,\partial g/\partial u) $$ and $$ (\partial f/\partial v,\partial g/\partial v) $$

are orthogonal. As a concrete example you can take polar coordinates on the plane $x=r\cos\theta$ and $y=r\sin\theta$. Polar coordinates are orthogonal (as can be seen geometrically as well).

Answer 2

The orthogonality does not depend on the coordinate system. Are you sure you are not talking about orthonormal/orthogonal basis ? To show that a basis of $n$ vectors$\{v_1,\cdots, v_n\}$ is orthogonal you have to show that $v_i \cdot v_j$, the dot product between $v_i$ and $v_j$ is equal to zero if and only if $i = j$