structural:summarization, information:classification
Euclidean metric

The most familiar example is that of elementary Euclidean geometry: the two-dimensional Euclidean metric tensor. In the usual Cartesian (x, y) coordinates, we can write

The length of a curve reduces to the formula:

The Euclidean metric in some other common coordinate systems can be written as follows.

Polar coordinates (r, θ):

So

by trigonometric identities.

In general, in a Cartesian coordinate system x on a Euclidean space, the partial derivatives ∂ / ∂ x are orthonormal with respect to the Euclidean metric. Thus the metric tensor is the Kronecker delta δ in this coordinate system. The metric tensor with respect to arbitrary (possibly curvilinear) coordinates q is given by