Cylindrical Coordinates#
in \(\mathbb{R}^3\)
\( \boxed{ \begin{aligned} &z\text{-axis of rotational symmetry} \\ T&:(r,\theta,z)\mapsto(x,y,z) \\ T^{-1}&:(x,y,z)\mapsto(r,\theta,z) \\ T(r,\theta,z) &=(r\cos\theta,r\sin\theta,z) \\ &=(x(r,\theta,z),y(r,\theta,z),z(r,\theta,z)) &&\text{transformation} \\ \hline \\ x(r,\theta,z)&=r\cos\theta \\ y(r,\theta,z)&=r\sin\theta \\ z(r,\theta,z)&=z &&\text{transformation equations} \\ \hline \\ x^2+y^2&=r^2 &&\text{Pythagorean identity} \\ dV&=r\,dz\,dr\,d\theta &&\text{Jacobian} \end{aligned} } \boxed{ \begin{aligned} &y\text{-axis} \\ T&:(r,\theta,y)\mapsto(x,y,z) \\ T^{-1}&:(x,y,z)\mapsto(r,\theta,y) \\ T(r,\theta,y) &=(r\sin\theta,y,r\cos\theta,) \\ &=(x(r,\theta,y),y(r,\theta,y),z(r,\theta,y)) \\ \hline \\ x(r,\theta,y)&=r\sin\theta \\ y(r,\theta,y)&=y \\ z(r,\theta,y)&=r\cos\theta \\ \hline \\ x^2+z^2&=r^2 \\ dV&=r\,dy\,dr\,d\theta \end{aligned} } \boxed{ \begin{aligned} &x\text{-axis} \\ T&:(r,\theta,x)\mapsto(x,y,z) \\ T^{-1}&:(x,y,z)\mapsto(r,\theta,x) \\ T(r,\theta,x) &=(x,r\cos\theta,r\sin\theta) \\ &=(x(r,\theta,x),y(r,\theta,x),z(r,\theta,x)) \\ \hline \\ x(r,\theta,x)&=x \\ y(r,\theta,x)&=r\cos\theta \\ z(r,\theta,x)&=r\sin\theta \\ \hline \\ y^2+z^2&=r^2 \\ dV&=r\,dx\,dr\,d\theta \end{aligned} } \)
Revised
02 Apr 2023
Cylindrical Coordinate System#
Cylindrical coordinates are good to use for regions with a single axis of rotational symmetry.
In other words, one of the cross sections of the region is circular and the other ones aren’t.
Circular Cylinders
Cones
Elliptic Paraboloids
Hyperboloids
We might think of the cylindrical coordinate system as a set of stacked polar coordinate systems indexed by \(z\).
Transformation Equations \((x,y,z)\leftrightarrow(r,\theta,z)\)#
Infinitesimals#
\(dr\,d\theta\,dz\)-voxels in the rectangular \(r\theta z\)-grid are transformed into 3D wedges \(dV\) in the cylindrical \(xyz\)-grid.
\( \begin{aligned} dV\ne dr\,d\theta\,dz \end{aligned} \)
How big is cylindrical \(dV\) in relation to rectangular \(dr\,d\theta\,dz\)-voxel?
The further the \(dr\,d\theta\,dz\)-voxel is in the \(r\)-direction in the rectangular \(r\theta z\)-grid the larger the 3D wedge \(dV\) is in the cylindrical \(xyz\)-grid.
\( \begin{aligned} dV=r\,dr\,d\theta\,dz &&\text{from the Jacobian} \end{aligned} \)
Jacobian#
\( \begin{aligned} \frac{\partial(x,y,z)}{\partial(r,\theta,z)} =\begin{vmatrix} \dfrac{\partial x}{\partial r} & \dfrac{\partial x}{\partial \theta} & \dfrac{\partial x}{\partial z} \\\\ \dfrac{\partial y}{\partial r} & \dfrac{\partial y}{\partial \theta} & \dfrac{\partial y}{\partial z} \\\\ \dfrac{\partial z}{\partial r} & \dfrac{\partial z}{\partial \theta} & \dfrac{\partial z}{\partial z} \end{vmatrix} =\begin{vmatrix} \cos\theta & -r\sin\theta & 0 \\\\ \sin\theta & r\cos\theta & 0 \\\\ 0 & 0 & 1 \end{vmatrix} =\begin{vmatrix} \cos\theta & -r\sin\theta \\\\ \sin\theta & r\cos\theta \end{vmatrix} =r(\cos^2\theta+\sin^2\theta) =r \end{aligned} \)
\(z\)-axis Transformation Equations#
The following transformations take \(r,\theta,z\) as input.
\( \begin{aligned} r\ge0&\,\,\,\text{distance from the origin in the xy-plane} \\ 0\le\theta\le2\pi&\,\,\,\text{angle from the positive x-axis in the xy-plane} \\ z&\,\,\,\text{distance from the xy-plane} \end{aligned} \)
\( \begin{aligned} \text{polar}\,\,\, x(r,\theta,z)&=r\cos\theta \\ \text{polar}\,\,\, y(r,\theta,z)&=r\sin\theta \\ \text{rectangular}\,\,\, z(r,\theta,z)&=z \\ x^2+y^2&=r^2 \\ dV&=r\,dz\,dr\,d\theta \end{aligned} \)
Fixed \(r=k\) values produce concentric cylinders along the \(z\)-axis.
Fixed \(\theta=k\) values produce planes along the \(z\)-axis orthogonal to the \(xy\)-plane.
Fixed \(z=k\) values produce stacked planes parallel to the \(xy\)-plane orthogonal to the \(z\)-axis.
\(y\)-axis Transformation Equations#
The following transformations take \(r,\theta,z\) as input.
\( \begin{aligned} r\ge0&\,\,\,\text{distance from the origin in the xz-plane} \\ 0\le\theta\le2\pi&\,\,\,\text{angle from the positive z-axis in the xz-plane} \\ y&\,\,\,\text{distance from the xz-plane} \end{aligned} \)
\( \begin{aligned} \text{polar}\,\,\, x(r,\theta,y)&=r\sin\theta \\ \text{rectangular}\,\,\, y(r,\theta,y)&=y \\ \text{polar}\,\,\, z(r,\theta,y)&=r\cos\theta \\ x^2+z^2&=r^2 \\ dV&=r\,dy\,dr\,d\theta \end{aligned} \)
\(x\)-axis Transformation Equations#
The following transformations take \(r,\theta,z\) as input.
\( \begin{aligned} r\ge0&\,\,\,\text{distance from the origin in the yz-plane} \\ 0\le\theta\le2\pi&\,\,\,\text{angle from the positive y-axis in the yz-plane} \\ x&\,\,\,\text{distance from the yz-plane} \end{aligned} \)
\( \begin{aligned} \text{rectangular}\,\,\, x(r,\theta,x)&=x \\ \text{polar}\,\,\, y(r,\theta,x)&=r\cos\theta \\ \text{polar}\,\,\, z(r,\theta,x)&=r\sin\theta \\ y^2+z^2&=r^2 \\ dV&=r\,dx\,dr\,d\theta \end{aligned} \)
Examples#
[EXAMPLE]
An object occupies the region
\( R=\{0\le z\le9-x^2-y^2\} \)
and has density
\( \begin{aligned} \rho(x,y,z)=\frac{x^2z}{x^2+y^2} \,\,\,\text{g cm}^3 \end{aligned} \)
Calculate its mass.
Draw the region in \(\mathbb{R}^3\).
\( \begin{aligned} z&=0 &&xy\text{-plane} \\ z&=9-x^2-y^2 &&\text{elliptic paraboloid with z-intercept of 9 and concave down} \end{aligned} \)
Find the new bounds.
\( \begin{aligned} 0\le z\le9-x^2-y^2 \implies 0\le z\le9-r^2 \end{aligned} \)
\(0\le\theta\le2\pi\)
Project the domain onto the polar \(xy\)-plane to yield a circle of intersection \(r=3\).
\( 0=9-r^2 \implies r^2=9 \implies r=3 \,\,\,\text{where r is always nonnegative} \)
\( \begin{aligned} \text{mass}\,\,\, m &=\underset{R}{\iiint}\rho(x,y,z)\,dV =\underset{R}{\iiint}\frac{x^2z}{x^2+y^2}\,dx\,dy\,dz =\underset{R}{\iiint}\frac{(r\cos\theta)^2z}{r^2}\,r\,dz\,dr\,d\theta \\ &=\int_{\theta=0}^{\theta=2\pi}\int_{r=0}^{r=3}\int_{z=0}^{z=9-r^2}rz\cos^2\theta\,dz\,dr\,d\theta \\ &=\int_{\theta=0}^{\theta=2\pi}\cos^2\theta\,d\theta\int_{r=0}^{r=3}\int_{z=0}^{z=9-r^2}rz\,dz\,dr &&\text{by factoring} \\ &=\pi\int_{r=0}^{r=3}\int_{z=0}^{z=9-r^2}rz\,dz\,dr &&\pi=\int_{\theta=0}^{\theta=2\pi}\cos^2\theta\,d\theta \\ \end{aligned} \)
[EXAMPLE]
\( \begin{aligned} \underset{R}{\iiint}x^2y\,dV \,\,\,\text{where}\,\,\, R=\{x^2+z^2\le25,z\ge0,0\le y\le3\} \end{aligned} \)
Draw the region in \(\mathbb{R}^3\).
Half cylinder three units along the \(y\)-axis.
Find the new bounds.
\( 0\le y\le3 \)
\( \begin{aligned} z\le0 \implies r\cos\theta\le0 \implies \cos\theta\le0 \implies -\frac{\pi}{2}\le\theta\le\frac{\pi}{2} \end{aligned} \)
Project the domain onto the polar \(xz\)-plane to yield a semicircle of intersection \(r=5\).
\( x^2+z^2\le25 \implies r^2\le25 \implies r\le5 \,\,\,\text{where r is always nonnegative} \)
\( \begin{aligned} &\underset{R}{\iiint}x^2y\,dV =\underset{R}{\iiint}(r\sin\theta)^2y\,r\,dy\,dr\,d\theta =\underset{R}{\iiint}r^3y\sin^2\theta\,dy\,dr\,d\theta \\ &=\int_{\theta=-\frac{\pi}{2}}^{\theta=\frac{\pi}{2}}\int_{r=0}^{r=5}\int_{y=0}^{y=3}r^3y\sin^2\theta\,dy\,dr\,d\theta \end{aligned} \)