The most familiar example is that of elementary Euclidean geometry: the two-dimensional Euclidean metric tensor. In the usual 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 :

So
by trigonometric identities.

In general, in a Cartesian coordinate system on a Euclidean space, the partial derivatives 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 is given by

Induced metric
Metric as a section of a bundle
The round metric on a sphere
Euclidean metric