Editing Differentiable curve (section)

=== Tangent vector ===

If a curve {{math|'''γ'''}} represents the path of a particle over time, then the instantaneous [[velocity]] of the particle at a given position {{math|''P''}} is expressed by a [[Vector (geometric)|vector]], called the ''[[tangent vector]]'' to the curve at {{math|''P''}}. Mathematically, given a parametrized {{math|''C''<sup>1</sup>}} curve {{math|1='''''γ''''' = '''''γ'''''(''t'')}}, for every value {{math|''t'' {{=}} ''t''<sub>0</sub>}} of the time parameter, the vector
<math display="block"> \boldsymbol{\gamma}'(t_0) = \left.\frac{\mathrm{d}}{\mathrm{d}t}\boldsymbol{\gamma}(t)\right|_{t=t_0} </math>
is the tangent vector at the point {{math|''P'' {{=}} '''γ'''(''t''<sub>0</sub>)}}. Generally speaking, the tangent vector may be [[zero vector|zero]]. The tangent vector's magnitude
<math display="block">\left\|\boldsymbol{\gamma}'(t_0)\right\|</math>
is the speed at the time {{math|''t''<sub>0</sub>}}.

The first Frenet vector {{math|'''e'''<sub>1</sub>(''t'')}} is the unit tangent vector in the same direction, called simply the tangent direction, defined at each regular point of {{math|'''γ'''}}:
<math display="block">\mathbf{e}_{1}(t) = \frac{ \boldsymbol{\gamma}'(t) }{ \left\| \boldsymbol{\gamma}'(t) \right\|}.</math>
If the time parameter is replaced by the arc length, {{math|''t'' {{=}} ''s''}}, then the tangent vector has unit length and the formula simplifies:
<math display="block">\mathbf{e}_{1}(s) = \boldsymbol{\gamma}'(s).</math>
However, then it is no longer applicable the interpretation in terms of the particle's velocity (with [[dimension (physics)|dimension]] of length per time).
The tangent direction determines the orientation of the curve, or the forward direction, corresponding to the increasing values of the parameter. The tangent direction taken as a curve traces the [[spherical image]] of the original curve.