Given a curve on a surface parametrized by two parameters , we first compute the element of arc length of the curve . For this, we need to compute the square norm of the tangent vector . The square norm of the tangent vector to the curve at is

,

where is the inner product in , and thus,

.

Following common usage, we let

,

and

.

Euler already obtained this formula in 1760. Thus, the map

is a quadratic form on , and since it is equal to , it is positive definite. This quadratric form plays a major role in the theory of surfaces, and deserves an official definition.

**Definition**. Given a surface , for any point on , letting

,

the positive definite quadratic form

is called the first fundamental form of at . It is often denoted as , and in matrix form, we have

.

The symmetric bilinear form associated with is an inner product on the tangent space at , such that

.

This inner product is also denoted as . The inner product can be used to determine the angle of two curves passing through , i.e., the angle of the tangent vectors to these two curves at . We have

.

For example, the angle between the -curve and the -curve passing through (where or is constant) is given by

.

Thus, the -curves and the -curves are orthogonal iff on .

**Example 1**. The helicoid is the surface defined over such that

.

This is the surface generated by a line parallel to the plane, touching the axis, and also touching an helix of axis . It is easily verified that . The figure below shows a portion of helicoid corresponding to and .

**Example 2**. The catenoid is the surface of revolution defined over such that

.

It is the surface obtained by rotating a catenary around the -axis. It is easily verified that . The figure below shows a portion of catenoid corresponding to and .

We will see how the first fundamental form relates to the curvature of curves on a surface.

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