Question

What is the first fundamental form of a surface?

Answer 1

The first fundamental form of a surface is a mathematical tool used in differential geometry to describe the geometry of a surface in terms of its metric properties—like distances and angles—without needing to refer to an explicit embedding in 3D space.

In simple terms, the first fundamental form helps us measure things like the length of curves and angles between curves on a surface, which are crucial for understanding how the surface behaves locally.

Here’s a breakdown of the components:

1. Surface and Parametrization: Imagine a surface (like a sphere or a plane), which can be described using a coordinate system. We usually represent it using two parameters, say \( u \) and \( v \), so we have a position vector \( \mathbf{r}(u,v) \) that describes points on the surface.
   

1. Tangent Vectors: The tangent vectors to the surface are the partial derivatives of the position vector \( \mathbf{r}(u,v) \) with respect to the parameters \( u \) and \( v \):
   

   \[
   \mathbf{r}_u = \frac{\partial \mathbf{r}}{\partial u}, \quad \mathbf{r}_v = \frac{\partial \mathbf{r}}{\partial v}
   \]
   These vectors span the tangent plane to the surface at each point.

1. Metric of the Surface: The first fundamental form encodes how lengths and angles are computed on the surface. It involves the dot products of the tangent vectors:
   

   \[
   E = \mathbf{r}_u \cdot \mathbf{r}_u, \quad F = \mathbf{r}_u \cdot \mathbf{r}_v, \quad G = \mathbf{r}_v \cdot \mathbf{r}_v
   \]
   Here:
   - \( E \) is the square of the length of the tangent vector in the \( u \)-direction,
   - \( F \) is the dot product between the tangent vectors in the \( u \) and \( v \)-directions (related to how much they "twist" relative to each other),
   - \( G \) is the square of the length of the tangent vector in the \( v \)-direction.

1. First Fundamental Form Expression: These quantities are then combined into a matrix that defines the first fundamental form:
   

   \[
   I = E\, du^2 + 2F\, du\, dv + G\, dv^2
   \]
   This form allows you to compute the length of curves on the surface and the angle between curves.

Key Points to Remember:

1. The first fundamental form is all about measuring lengths and angles on a surface, based on how the surface is parametrized.
   

1. It involves the coefficients \( E \), \( F \), and \( G \), which depend on how the surface is locally "curved."
   

1. It’s used in various fields of geometry, physics, and engineering to study the properties of curved surfaces, like how light behaves on curved surfaces or how stress distributes over a curved object.
   

Let me know if you need more examples or details on how this form is used!