Euclidean metric and non-Cartesian systems

[rkaminski]

OK. A problem is simple and probably very stupid... In many (most of) maths books things like Euclidean metric, norm, scalar product etc. are defined in Cartesian coordinate system (that is one which is orthonormal). But what would happen if one were to define all these quantities in non-Cartesian system? For example in many cases (books) the dot product is defined as sum of coordinates for two vectors. Such expression is definitely not true for non-orthogonal systems. Can anyone comment on such issues, perhaps some can propose some detailed reading?

 

[pasmith]

Riemannian manifolds have a concept of a metric tensor $g_{ij}$, so that the element of arclength $ds$ is given by $$ds^2 = g_{ij}dx^i dx^j$$ in terms of generalized coordinates $x_1, \dots, x_n$, and the distance between two points is then the infimum of $\int_C \frac{ds}{dt}\,dt = \int_C \sqrt{g_{ij}\dot x^i \dot x^j}\,dt$ over all continuous curves $C$ between those two points.

Thus in spherical polar coordinates with $x^1 = r$, $x^2 = \theta$ and $x^3 = \phi$ the euclidean metric is given by $$\begin{array}{ccc} g_{11} = 1 & g_{12} = 0 & g_{13} = 0 \\ g_{21} = 0 & g_{22} = r^2 & g_{23} = 0 \\ g_{31} = 0 & g_{32} = 0 & g_{33} = r^2 \sin^2 \theta \end{array}$$ so that the arclength element is given by $$ds^2 = dr^2 + r^2 \sin^2 \theta d\phi^2 + r^2 d\theta^2.$$ Introductory texts on general relativity (such as Foster & Nightingale) should discuss this for the case of the (pseudo-)Riemannian geometry of space-time.

 

[rkaminski]

And does any author discuss these issues at the elementary level, not pointing out to the formalism of differential forms?