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# 3.2 First fundamental form I (metric)

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

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)

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

#### Footnotes

...3.1
Here the vector identity
 (3.15)

with the special case
 (3.16)

is used.

Next: 3.3 Second fundamental form Up: 3. Differential Geometry of Previous: 3.1 Tangent plane and   Contents   Index
December 2009