Editing Polar coordinate system (section)

===Vector calculus===
[[Vector calculus]] can also be applied to polar coordinates. For a planar motion, let <math>\mathbf{r}</math> be the position vector {{math|(''r'' cos(''φ''), ''r'' sin(''φ''))}}, with ''r'' and {{math|''φ''}} depending on time ''t''.

{{anchor|Radial, transverse, normal}}We define an [[orthonormal basis]] with three unit vectors: ''radial, transverse, and normal directions''.
The ''radial direction'' is defined by normalizing <math>\mathbf{r}</math>:
<math display="block">\hat{\mathbf{r}} = (\cos(\varphi), \sin(\varphi))</math>
Radial and velocity directions span the ''plane of the motion'', whose normal direction is denoted <math>\hat{\mathbf{k}}</math>:
<math display="block">\hat{\mathbf{k}} = \hat{\mathbf{v}} \times \hat{\mathbf{r}}.</math>
The ''transverse direction'' is perpendicular to both radial and normal directions:
<math display="block">\hat \boldsymbol \varphi = (-\sin(\varphi), \cos(\varphi)) = \hat{\mathbf{k}} \times \hat{\mathbf{r}} \ ,</math>

Then 
<math display="block">\begin{align}
        \mathbf{r}  &= (x,\ y) = r(\cos\varphi,\ \sin\varphi) = r \hat{\mathbf{r}}\ , \\[1.5ex]
   \dot{\mathbf{r}} &= \left(\dot{x},\ \dot{y}\right) = \dot{r}(\cos\varphi,\ \sin\varphi) + r\dot{\varphi}(-\sin\varphi,\ \cos\varphi) = \dot{r}\hat{\mathbf{r}} + r\dot{\varphi}\hat{\boldsymbol{\varphi}}\ ,\\[1.5ex]
  \ddot{\mathbf{r}} &= \left(\ddot{x},\ \ddot{y}\right) \\[1ex]
                    &= \ddot{r}(\cos\varphi,\ \sin\varphi) + 2\dot{r}\dot{\varphi}(-\sin\varphi,\ \cos\varphi) +  r\ddot{\varphi}(-\sin\varphi,\ \cos\varphi) - r\dot{\varphi}^2(\cos\varphi,\ \sin\varphi) \\[1ex]
                    &= \left(\ddot{r} - r\dot{\varphi}^2\right) \hat{\mathbf{r}} + \left(r\ddot{\varphi} + 2\dot{r}\dot{\varphi}\right) \hat{\boldsymbol{\varphi}} \\[1ex]
                    &= \left(\ddot{r} - r\dot{\varphi}^2\right) \hat{\mathbf{r}} + \frac{1}{r}\; \frac{d}{dt} \left(r^2\dot{\varphi}\right) \hat{\boldsymbol{\varphi}}.
\end{align}</math>

This equation can be obtain by taking derivative of the function and derivatives of the unit basis vectors.
 
For a curve in 2D where the parameter is <math>\theta</math> the previous equations simplify to: 
<math display="block">\begin{aligned}
\mathbf{r} &= r(\theta) \hat \mathbf{e}_r \\[1ex]
\frac {d\mathbf{r}}{d\theta} &= \frac {dr} {d\theta} \hat \mathbf{e}_r + r \hat \mathbf{e}_\theta\\[1ex]
\frac {d^2\mathbf{r}}{d\theta^2} &= \left(\frac {d^2 r} {d\theta^2}-r\right) \hat \mathbf{e}_r + \frac {dr} {d\theta} \hat \mathbf{e}_\theta
\end{aligned}</math>

====Centrifugal and Coriolis terms====
{{See also|Centrifugal force}}
{{multiple image
|align = vertical
|width1 = 100
|image1 = Position vector plane polar coords.svg
|caption1 = Position vector '''r''', always points radially from the origin.
|width2 = 150
|image2 = Velocity vector plane polar coords.svg
|caption2 = Velocity vector '''v''', always tangent to the path of motion.
|width3 = 200
|image3 = Acceleration vector plane polar coords.svg
|caption3 = Acceleration vector '''a''', not parallel to the radial motion but offset by the angular and Coriolis accelerations, nor tangent to the path but offset by the centripetal and radial accelerations.
|footer = Kinematic vectors in plane polar coordinates. Notice the setup is not restricted to 2d space, but a plane in any higher dimension.
}}

The term <math>r\dot\varphi^2</math> is sometimes referred to as the ''centripetal acceleration'', and the term <math>2\dot r \dot\varphi</math> as the ''Coriolis acceleration''. For example, see Shankar.<ref name="Shankar">{{Cite book |last=Ramamurti Shankar |url=https://books.google.com/books?id=2zypV5EbKuIC&q=Coriolis+%22polar+coordinates%22&pg=PA81 |title=Principles of Quantum Mechanics |publisher=Springer |year=1994 |isbn=0-306-44790-8 |edition=2nd |page=81}}</ref>

Note: these terms, that appear when acceleration is expressed in polar coordinates, are a mathematical consequence of differentiation; they appear whenever polar coordinates are used. In planar particle dynamics these accelerations appear when setting up Newton's [[Newton's second law|second law of motion]] in a rotating frame of reference. Here these extra terms are often called [[fictitious force]]s; fictitious because they are simply a result of a change in coordinate frame. That does not mean they do not exist, rather they exist only in the rotating frame.

[[Image:Co-rotating frame vector.svg|thumb|Inertial frame of reference ''S'' and instantaneous non-inertial co-rotating frame of reference ''S′''. The co-rotating frame rotates at angular rate Ω equal to the rate of rotation of the particle about the origin of ''S′'' at the particular moment ''t''. Particle is located at vector position ''r''(''t'') and unit vectors are shown in the radial direction to the particle from the origin, and also in the direction of increasing angle ''ϕ'' normal to the radial direction. These unit vectors need not be related to the tangent and normal to the path. Also, the radial distance ''r'' need not be related to the radius of curvature of the path.]]

=====Co-rotating frame=====
<!-- This whole section should go to hell -->For a particle in planar motion, one approach to attaching physical significance to these terms is based on the concept of an instantaneous ''co-rotating frame of reference''.<ref name="Taylor">For the following discussion, see {{Cite book |last=John R Taylor |title=Classical Mechanics |publisher=University Science Books |year=2005 |isbn=1-891389-22-X |page=§9.10, pp. 358–359}}</ref> To define a co-rotating frame, first an origin is selected from which the distance ''r''(''t'') to the particle is defined. An axis of rotation is set up that is perpendicular to the plane of motion of the particle, and passing through this origin. Then, at the selected moment ''t'', the rate of rotation of the co-rotating frame Ω is made to match the rate of rotation of the particle about this axis, ''dφ''/''dt''. Next, the terms in the acceleration in the inertial frame are related to those in the co-rotating frame. Let the location of the particle in the inertial frame be (''r''(''t''), ''φ''(''t'')), and in the co-rotating frame be (''r''&prime;(t), ''φ''&prime;(t)). Because the co-rotating frame rotates at the same rate as the particle, ''dφ''′/''dt'' = 0. The fictitious centrifugal force in the co-rotating frame is ''mr''Ω<sup>2</sup>, radially outward. The velocity of the particle in the co-rotating frame also is radially outward, because ''dφ''′/''dt'' = 0. The ''fictitious Coriolis force'' therefore has a value −2''m''(''dr''/''dt'')Ω, pointed in the direction of increasing ''φ'' only. Thus, using these forces in Newton's second law we find:
<math display="block">\mathbf{F} + \mathbf{F}_\text{cf} + \mathbf{F}_\text{Cor} = m\ddot{\mathbf{r}} \, , </math>
where over dots represent derivatives with respect to time, and '''F''' is the net real force (as opposed to the fictitious forces). In terms of components, this vector equation becomes:
<math display="block">\begin{align}
             F_r + mr\Omega^2 &= m\ddot{r} \\
  F_\varphi - 2m\dot{r}\Omega &= mr\ddot{\varphi} \ ,
\end{align}</math>
which can be compared to the equations for the inertial frame:
<math display="block">\begin{align}
        F_r &= m\ddot{r} - mr\dot{\varphi}^2 \\
  F_\varphi &= mr\ddot{\varphi} + 2m\dot{r}\dot{\varphi} \ .
\end{align}</math>

This comparison, plus the recognition that by the definition of the co-rotating frame at time ''t'' it has a rate of rotation Ω = ''dφ''/''dt'', shows that we can interpret the terms in the acceleration (multiplied by the mass of the particle) as found in the inertial frame as the negative of the centrifugal and Coriolis forces that would be seen in the instantaneous, non-inertial co-rotating frame.

For general motion of a particle (as opposed to simple circular motion), the centrifugal and Coriolis forces in a particle's frame of reference commonly are referred to the instantaneous [[osculating circle]] of its motion, not to a fixed center of polar coordinates. For more detail, see [[Centripetal force#Local coordinates|centripetal force]].