Element of Arc Length in Cylindrical Coordinates
In Cartesian coordinates the position vector of a particle is\[\mathbf{r} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k}\]
Then
\[d \mathbf{r} = dx \mathbf{i} + dy \mathbf{j} + dz \mathbf{k}\]
An element of arc length expressed in Cartesian coordinates is
\[ds = \sqrt{(dx)^2 + (dy)^2 +(dz)^2}\]
In cylindrical polar coordinates
\[x= r cos \theta \rightarrow dx = dr cos \theta - r sin \theta d \theta\]
\[y= r sin \theta \rightarrow dy = dr sin \theta + r cos \theta d \theta\]
\[z= z \rightarrow dz = dz\]
Then an element of arc length expressed in Cartesian coordinates is
\[\begin{equation} \begin{aligned} ds &= \sqrt{(dr cos \theta - r sin \theta d \theta)^2 + (dr sin \theta + r cos \theta d \theta)^2 +(dz)^2} \\ &= \sqrt{(dr)^2 cos^2 \theta -2r dr d \theta cos \theta cos \theta + r^2 sin^2 \theta (d \theta)^2 +(dr)^2 sin^2 \theta +2rdr d \theta sin \theta cos \theta +r^2 cos^2 \theta (d \theta)^2 + (dz)^2} \\m &= \sqrt{(dr)^2 (cos^2 \theta + sin^2 \theta) + d \theta)^2 (r^2 sin^2 \theta + r^2 cos^2 \theta ) + (dz)^2} \\ &= \sqrt{(dr)^2 + r^2d \theta)^2 + (dz)^2} \end{aligned} \end{equation}\]
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