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.
