Derive: The First Fundamental Quadratic Form is

where:

Derivation. The differential of arc length ds on surface S is related to dx, dy and dz by

1)        ds² = dx² + dy² + dz²

and dx, dy and dz are related to du and dv by

Substituting 2) into 1) we have

Expanding and rearranging 3) gives the First Fundamental Quadratic Form.