If the variables and are taken to depend on a third variable, , taking values in an interval , then will trace out a parametric curve in parametric surface . The arc length of that curve is given by the integral

where formula_3 represents the Euclidean norm. Here the chain rule has been applied, and the subscripts denote partial derivatives:

The integrand is the restriction to the curve of the square root of the (দ্বিঘাত সমীকরণ)ের বর্গমূল differential

where

The quantity in () is called the line element, while is called the first fundamental form of . Intuitively, it represents the principal part of the square of the displacement undergone by when is increased by units, and is increased by units.

Using matrix notation, the first fundamental form becomes

Induced metric
Tangent–cotangent isomorphism
বৃত্তের পরিধির চাপের দৈর্ঘ
Canonical measure and volume form