Editing Frenet–Serret formulas (section)

==Proof of the Frenet–Serret formulas ==

The first Frenet–Serret formula holds by the definition of the normal {{math|'''N'''}} and the curvature {{mvar|κ}}, and the third Frenet–Serret formula holds by the definition of the torsion {{mvar|τ}}. Thus what is needed is to show the second Frenet–Serret formula.

Since {{math|'''T''', '''N''', '''B'''}} are orthogonal unit vectors with {{math|1='''B''' = '''T''' × '''N'''}}, one also has {{math|1='''T''' = '''N''' × '''B'''}} and {{math|1='''N''' = '''B''' × '''T'''}}. Differentiating the last equation with respect to {{mvar|s}} gives

<math display=block>
  \frac{\partial \mathbf N}{\partial s} = \left( \frac{\partial \mathbf B}{\partial s} \right) \times \mathbf T + \mathbf B \times \left(\frac{\partial \mathbf T}{\partial s} \right)
</math>

Using that <math>\tfrac{\partial \mathbf B}{\partial s} = -\tau \mathbf N</math> and <math>\tfrac{\partial \mathbf T}{\partial s} = \kappa \mathbf N, </math> this becomes

<math display=block>\begin{align}
  \frac{\partial \mathbf N}{\partial s} &= -\tau (\mathbf N \times \mathbf T) + \kappa (\mathbf B \times \mathbf N) \\
  &= \tau \mathbf B - \kappa \mathbf T
\end{align}</math>

This is exactly the second Frenet–Serret formula.