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.
