how to compute length in polar coordinates? The line element $\Delta s^2$ is suppose to be an invariant of Euclidean space. In standard coordinates $\Delta s^2=\Delta x^2+\Delta y^2$ while in polar coordinates $\Delta s^2=\Delta r^2+r^2\Delta \theta^2$. Equivalently, in standard coordinates the metric is $$\begin{bmatrix}1&0\\0&1\end{bmatrix}$$ while in polar coordinates it is   $$\begin{bmatrix}1&0\\0&r^2\end{bmatrix}.$$
I don't understand how to compute the length of a point. Suppose the point is $(a,b)$ in standard coordinates. It's squared length in standard coordinates is given by $$(a,b)\begin{bmatrix}1&0\\0&1\end{bmatrix}(a,b)^T=a^2+b^2$$But how do you compute the length in polar coordinates? Wouldn't it be $$(\sqrt{a^2+b^2},\arctan(b/a))\begin{bmatrix}1&0\\0&a^2+b^2\end{bmatrix}(\sqrt{a^2+b^2},\arctan(b/a))^T$$which is definitely not the same as $a^2+b^2$. 
How do you get the same line element in polar coordinates versus standard coordinates?
 A: $\newcommand{\dd}{\partial}$To say the Euclidean metric in polar coordinates is
$$
\left[\begin{array}{cc}
1 & 0 \\
0 & r^{2}
\end{array}\right]
$$
really means that $ds^{2} = dr^{2} + (r\, d\theta)^{2}$. In other words, if
$$
v = a\, \dd_{r} + b\, \dd_{\theta}
\tag{1}
$$
is a tangent vector at some point $(r, \theta)$ in polar coordinates, then the Euclidean length of $v$ is
$$
\sqrt{a^{2} + (rb)^{2}}.
\tag{2}
$$
To calculate the arc length of a curve, use (2) to calculate the curve's speed, then integrate.
For example, to calculate the length of the parametrized line segment
$$
\gamma(t) = (r_{0}t, \theta_{0}),\quad 0 \leq t \leq 1
$$
joining the origin to the point with polar coordinates $(r_{0}, \theta_{0})$, note that


*

*The velocity of gamma is $v = r_{0}\, \dd_{r}$.

*The speed of $\gamma$ is $\sqrt{r_{0}^{2}} = |r_{0}|$ by (2).

*The length of $\gamma$ is the integral of the speed,
$$
\int_{0}^{1} \|\gamma'(t)\|\, dt
  = \int_{0}^{1} |r_{0}|\, dt
  = |r_{0}|,
$$
the Euclidean distance.
A: It seems that you are talking about two dimensional coordinate. No matter 2D or 3D polar coordinate, the length, $r$, is given.
The relation for a 2D standard coordinate and polar coordinate is: 
$x = r \cos\theta, y = r \sin\theta $. The squared length is $x^2 + y^2 = r^2$ since $\sin^2\theta+\cos^2\theta=$1
  More common is the relation of 3D polar coordinates, $r, \theta, \phi$ to Cartesian axes $x, y, z$.
$x=r\sin\theta \cos\phi,y=r\sin\theta \sin\phi, z=r\cos\theta$
Hope you can picture the coordinates in mind with the equations.
