Line Element

Also known as the first Fundamental Form

$$ds^2=g_{ab}\,dx^a\,dx^b.$$

In the principal axis frame for 3-D,

$$ds^2=g_{aa}(dx^a)^2+g_{bb}(dx^b)^2+g_{cc}(dx^c)^2.$$

At Ordinary Points on a surface, the line element is positive definite.

See also Area Element, Fundamental Forms, Volume Element