Is there an efficient way to prove orthogonality of a coordinate system?

Question

Suppose we define a new orthogonal coordinate system, such as spherical coordinates defined by $$x = r \sin \theta \cos \phi, y = r \sin \theta \sin \phi, z = r \cos \theta.$$ Is there an efficient and general way to prove that the coordinate system is orthogonal, other than by writing out the off-diagonal metric coefficients and finding that they are all zero? Of course it's not that bad for spherical coordinates, but for (say) toroidal or bispherical it gets messy.

Answer 1

Let's say you have a system of coordinates $u, v, w$ with tangent vectors $e_u, e_v, e_w$. Given the relationship between these coordintaes and Cartesian coordinates, you can express $e_u$ as a linear combination of $e_x, e_y, e_z$, and the same for the others.

You can use a wedge product to determine the size of the parallelepiped formed from these vectors, and you should be able to express that wedge product in terms of the Cartesian basis vectors. That is, you should be able to find

$$e_u \wedge e_v \wedge e_w = \alpha (e_x \wedge e_y \wedge e_z)$$

for some number $\alpha$.

In the general case, the magnitude of $\alpha$ is bounded as follows:

$$|\alpha| \leq |e_u||e_v||e_w|$$

The bound is saturated when the vectors are orthogonal. This allows you to compute only the diagonal metric components and a wedge product (equivalent to computing a determinant). Debatable whether this is "easier" than computing all the metric components, though.