Let be a regular surface with points in the tangent space of . Then the first fundamental form is the inner product of tangent vectors,
(1)

The first fundamental form satisfies
(2)

The first fundamental form (or line element) is given explicitly by the Riemannian metric
(3)

It determines the arc length of a curve on a surface. The coefficients are given by
(4)
 
(5)
 
(6)

The coefficients are also denoted , , and . In curvilinear coordinates (where ), the quantities
(7)
 
(8)

are called scale factors.