The differential arc length
of a parametric curve is given by
(2.2). Now if we replace the parametric curve by
a curve
,
which lies on the parametric surface
, then

(3.11)

where

(3.12)

The first fundamental form is defined as

(3.13)

and
,
,
are called the first fundamental form coefficients and
play important roles in many intrinsic properties of a
surface.
The first fundamental form
can be rewritten as

(3.14)

Since
^{3.1} and
,
is positive definite, provided that the surface is regular.
That is
and
if and only if
and
.

Example 3.2.1.
Let us compute the arc length of a curve
,
for
on a hyperbolic
paraboloid
where
as shown in
Fig. 3.4 (a).
We have

and along the curve the first fundamental form coefficients are

thus,

Finally the arc length for
is given by

Figure 3.4:
Hyperbolic paraboloid: (a) arc length along
,
,
(b) area bounded by positive
and
axes and a quarter circle

The angle between two curves on a parametric surface
and
can be evaluated by taking the inner product of the tangent vectors
of
and
, yielding

(3.17)

As a result of the above equation, the orthogonality condition for the two
tangent vectors
and
is:

(3.18)

In particular when the two curves are the
and
iso-parametric curves,
(3.17) reduces to

(3.19)

Thus the iso-parametric curves are orthogonal if
.

The area of a small parallelogram with vertices
,
,
and
as illustrated in Fig. 3.5,
is approximated by

(3.20)

or in differential form

(3.21)

Figure 3.5:
Area of small surface patch

Example 3.2.2.
Let us compute the area of a region of the hyperbolic paraboloid that
is used in Example 3.2.1. The region is bounded by positive
and
axes and a
quarter circle
as shown in Fig. 3.4 (b).
Substituting
into
(3.21), we obtain

To perform the integration it is easier to change variables,
,
, so that