Editing Polar coordinate system (section)

== Differential geometry ==
In the modern terminology of [[differential geometry]], polar coordinates provide [[coordinate charts]] for the [[differentiable manifold]] {{math|'''R'''<sup>2</sup> \ {(0,0)}<nowiki/>}}, the plane minus the origin. In these coordinates, the Euclidean [[metric tensor]] is given by<math display="block">ds^2 = dr^2 + r^2 d\theta^2.</math>This can be seen via the change of variables formula for the metric tensor, or by computing the [[differential form]]s ''dx'', ''dy'' via the [[exterior derivative]] of the 0-forms {{math|1=''x'' = ''r'' cos(''θ'')}}, {{math|1=''y'' = ''r'' sin(''θ'')}} and substituting them in the Euclidean metric tensor {{math|1=''ds''<sup>2</sup> = ''dx''<sup>2</sup> + ''dy''<sup>2</sup>}}.

{{Collapse top|title=An elementary proof of the formula}}
Let <math>p_1=(x_1,y_1)=(r_1,\theta_1)</math>, and <math>p_2=(x_2,y_2)=(r_2,\theta_2)</math> be two points in the plane given by their cartesian and polar coordinates. Then
:<math>ds^2=dx^2+dy^2=(x_2-x_1)^2+(y_2-y_1)^2.</math>
Since <math>dx^2=(r_2\cos\theta_2-r_1\cos\theta_1)^2</math>, and <math>dy^2=(r_2\sin\theta_2-r_1\sin\theta_1)^2</math>, we get that
:<math>ds^2=r_2^2\cos^2\theta_2-2r_1r_2\cos\theta_1\cos\theta_2+r_1^2\cos^2\theta_1+r_2^2\sin^2\theta_2-2r_1r_2\sin\theta_1\sin\theta_2+r_1^2\sin^2\theta_1=</math>
:<math>r_2^2(\cos^2\theta_2+\sin^2\theta_2)+r_1^2(\cos^2\theta_1+\sin^2\theta_1)-2r_1r_2(\cos\theta_1\cos\theta_2+\sin\theta_1\sin\theta_2)=</math>
:<math>r_1^2+r_2^2-2r_1r_2(1-1+\cos\theta_1\cos\theta_2+\sin\theta_1\sin\theta_2)=</math>
:<math>(r_2-r_1)^2+2r_1r_2(1-\cos\theta_1\cos\theta_2-\sin\theta_1\sin\theta_2).</math>
Now we use the trigonometric identity <math>\cos(\theta_2-\theta_1)=\cos\theta_1\cos\theta_2+\sin\theta_1\sin\theta_2</math> to proceed:
:<math>ds^2=dr^2+2r_1r_2(1-\cos d\theta).</math>
If the radial and angular quantities are near to each other, and therefore near to a common quantity <math>r</math> and <math>\theta</math>, we have that <math>r_1r_2\approx r^2</math>. Moreover, the cosine of <math>d\theta</math> can be approximated with the Taylor series of the cosine up to linear terms:
:<math>\cos d\theta\approx1-\frac{d\theta^2}{2},</math>
so that <math>1-\cos d\theta\approx\frac{d\theta^2}{2}</math>, and <math>2r_1r_2(1-\cos d\theta)\approx2r^2\frac{d\theta^2}{2}=r^2d\theta^2</math>. Therefore, around an infinitesimally small domain of any point,
:<math>ds^2=dr^2+r^2d\theta^2,</math>
as stated.

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An [[Orthonormality|orthonormal]] [[Moving frame|frame]] with respect to this metric is given by<math display="block">e_r = \frac{\partial}{\partial r}, \quad e_\theta = \frac1r \frac{\partial}{\partial \theta},</math>with [[Moving frame#Coframes|dual coframe]]<math display="block">e^r = dr, \quad e^\theta = r d\theta.</math>The [[connection form]] relative to this frame and the [[Levi-Civita connection]] is given by the skew-symmetric matrix of 1-forms<math display="block">{\omega^i}_j = \begin{pmatrix} 0 & -d\theta \\ d\theta & 0\end{pmatrix}</math>and hence the [[curvature form]] {{math|1=Ω = ''dω'' + ''ω''∧''ω''}} vanishes. Therefore, as expected, the punctured plane is a [[flat manifold]].<!-- Rather advanced -->