Editing Function of a real variable (section)

==One-dimensional space curves in <math>\mathbb{R}</math><sup>''n''</sup>==

[[File:Space curve.svg|200px|thumb|Space curve in 3d. The [[position vector]] '''r''' is parametrized by a scalar ''t''. At '''r''' = '''a''' the red line is the tangent to the curve, and the blue plane is normal to the curve.]]

===Formulation===

Given the functions {{nowrap|''r''<sub>1</sub> {{=}} ''r''<sub>1</sub>(''t'')}}, {{nowrap|''r''<sub>2</sub> {{=}} ''r''<sub>2</sub>(''t'')}}, ..., {{nowrap|''r''<sub>''n''</sub> {{=}} ''r''<sub>''n''</sub>(''t'')}} all of a common variable ''t'', so that:

:<math>\begin{align}
r_1 : \mathbb{R} \rightarrow \mathbb{R} & \quad r_2 : \mathbb{R} \rightarrow \mathbb{R} & \cdots &  \quad r_n : \mathbb{R} \rightarrow \mathbb{R} \\
r_1 = r_1(t) &  \quad r_2 = r_2(t) & \cdots &  \quad r_n = r_n(t) \\
\end{align}</math>

or taken together:

:<math>\mathbf{r} : \mathbb{R} \rightarrow \mathbb{R}^n \,,\quad \mathbf{r} = \mathbf{r}(t) </math>

then the parametrized ''n''-tuple,

:<math>\mathbf{r}(t) = [r_1(t), r_2(t), \ldots , r_n(t)] </math>

describes a one-dimensional [[space curve]].

===Tangent line to curve===

At a point {{nowrap|'''r'''(''t'' {{=}} ''c'') {{=}} '''a''' {{=}} (''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''n''</sub>)}} for some constant ''t'' = ''c'', the equations of the one-dimensional tangent line to the curve at that point are given in terms of the [[ordinary derivative]]s of ''r''<sub>1</sub>(''t''), ''r''<sub>2</sub>(''t''), ..., ''r''<sub>''n''</sub>(''t''), and ''r'' with respect to ''t'':

:<math>\frac{r_1(t) - a_1}{dr_1(t)/dt} = \frac{r_2(t) - a_2}{dr_2(t)/dt} = \cdots = \frac{r_n(t) - a_n}{dr_n(t)/dt} </math>

===Normal plane to curve===

The equation of the ''n''-dimensional [[hyperplane]] normal to the tangent line at '''r''' = '''a''' is:

:<math>(p_1 - a_1)\frac{dr_1(t)}{dt} + (p_2 - a_2)\frac{dr_2(t)}{dt} + \cdots + (p_n - a_n)\frac{dr_n(t)}{dt} = 0</math>

or in terms of the [[dot product]]:

:<math>(\mathbf{p} - \mathbf{a})\cdot \frac{d\mathbf{r}(t)}{dt} = 0</math>

where {{nowrap|'''p''' {{=}} (''p''<sub>1</sub>, ''p''<sub>2</sub>, ..., ''p''<sub>''n''</sub>)}} are points ''in the plane'', not on the space curve.

===Relation to kinematics===

[[File:Kinematics.svg|thumb|300px|Kinematic quantities of a classical particle: mass ''m'', position '''r''', velocity '''v''', acceleration '''a'''.]]

The physical and geometric interpretation of ''d'''''r'''(''t'')/''dt'' is the "[[velocity]]" of a point-like [[particle]] moving along the path '''r'''(''t''), treating '''r''' as the spatial [[position vector]] coordinates parametrized by time ''t'', and is a vector tangent to the space curve for all ''t'' in the instantaneous direction of motion. At ''t'' = ''c'', the space curve has a tangent vector {{nowrap|''d'''''r'''(''t'')/''dt''{{!}}<sub>''t'' {{=}} ''c''</sub>}}, and the hyperplane normal to the space curve at ''t'' = ''c'' is also normal to the tangent at ''t'' = ''c''. Any vector in this plane ('''p''' − '''a''') must be normal to {{nowrap|''d'''''r'''(''t'')/''dt''{{!}}<sub>''t'' {{=}} ''c''</sub>}}.

Similarly, ''d''<sup>2</sup>'''r'''(''t'')/''dt''<sup>2</sup> is the "[[acceleration]]" of the particle, and is a vector normal to the curve directed along the [[Radius of curvature (mathematics)|radius of curvature]].