Section 2.7 Cylindrical and Spherical Coordinates (Postponed)
This section will be covered later, just before this material is used in Section 5.5, “Triple Integrals in Cylindrical and Spherical Coordinates”.References.
- OpenStax Calculus Volume 3, Section 2.7.
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openstax.org/books/calculus-volume-3/pages/2-7-cylindrical-and-spherical-coordinates - Calculus, Early Transcendentals by Stewart, Sections 15.7 and 15.8.
Introduction.
Coordinate systems for three dimensional space that are convenient for working with domains and functions that have either cylindrical symmetry (depending only on distance from a certain line), or spherical symmetry (depending only on distance from a certain point).
Topics.
Subsection 2.7.1 Cylindrical Coordinates
One way to describe the location of a point in space is to gives its height \(z\) above [or below] the \(x\)-\(y\) plane as with cartesian coordinates, but then describe the location of the point below [or above] it in that plane with plane polar coordinates, \(r\) and \(\theta\text{.}\)
This gives cylindrical coordinates \((r,\theta,z)\text{,}\) (sometimes denoted \((r,\theta,z)_c\) to distinguish from cartesian coordinates), related to cartesian coordinates \((x,y,z)\) by
\begin{equation}
x = r \cos \theta, \quad y = r \sin \theta, \quad [z=z]\tag{2.7.1}
\end{equation}
To convert from rectangular to cylindrical, one uses
\begin{equation*}
r^2 = x^2+y^2 \mbox{ (so }r = \sqrt{x^2+y^2}),
\quad \tan\theta = y/x \text{ (or } \cos\theta = x/r \text{ or } \sin\theta = y/r)
\end{equation*}
A unique choice of \(\theta\) is then determined by using the smallest non-negative value, so that \(0 \leq \theta < 2\pi\) (Any such value of \(\theta\) is allowed on the \(z\)-axis \(x=y=0\text{.}\))
These coordinates are "cylindrical" because a circular cylinder of radius \(a\) around the \(z\)-axis has the simple equation \(r = a\text{.}\) Essentially, this is describing the \((x,y)\) part of the location with plane polar coordinates.
Subsection 2.7.2 Spherical Coordinates
Just as plane polar and cylindrical coordinates are based around horizontal distance \(r\) from \(x=y=0\text{,}\) spherical coordinates start with the distance \(\rho\) from the origin \(O(0,0,0)\) to a point \(P(x,y,z)\text{:}\)
\begin{equation}
\rho^2 = x^2+y^2+z^2 \mbox{ so } \rho = \sqrt{x^2+y^2+z^2}.\tag{2.7.2}
\end{equation}
Thus a sphere of radius \(a\) center the original has the simple equation \(\rho = a\text{.}\)
"Longitude" \(\theta\) and "Latitude" \(\phi\).
To complete the coordinates, we need to describe the location of the point \(P\) on the sphere of radius \(\rho\) with two numbers.
The familiar case of this is using angles of latitude and longitude to describe position on the earth’s surface. Both are done a bit differently here, to avoid negative values.
First, we consider the circle of latitude containing the point: the circle of center \(Q(0,0,z)\) on the \(z\) axis at the same "height" as \(P\) and passing through \(P\text{,}\) so that its radius is \(r=\sqrt{x^2+y^2}\text{.}\)
Position on this circle can be described with polar coordinates, using the angle \(\theta\) measured from the eastern point \(E(r,0,z)\) of this circle in the direction of positive \(y\text{,}\) and as above, using values \(\theta \in [0, 2\pi)\text{.}\)
The \(r\) and \(\theta\) here are the same as with cylindrical coordinates, so we still have
\begin{equation}
x = r \cos \theta,\quad y = r \sin \theta.\tag{2.7.3}
\end{equation}
The position of this circle of latitude on the sphere is specified with a "latitude angle", measured from the North pole \(N(0,0,\rho)\)
That is, \(\phi\) is the angle \(\angle NOP\text{,}\) which we can take in the range \(0 \leq \phi \leq \pi\text{,}\) with \(\phi=0\) the North Pole, \(\phi=\pi\) the South Pole \(S(0,0,-\rho)\text{.}\)
Note that the range only has to cover a half circle, not returning to the North pole at \(\phi=2\pi\text{.}\)
Another way to think of this is that one looks at the half plane of points with a given longitude \(\theta\text{,}\) with cartesian coordinates \((z, r)\) in the place of \((x, y)\) and describe this with a second set of polar coordinates, so now measuring angles from the positive \(z\)-axis. Thus the radius is \(\rho\text{,}\) the polar angle is \(\phi\text{,}\) and
\begin{equation}
z = \rho \cos\phi, \quad r = \rho \sin\phi\tag{2.7.4}
\end{equation}
Equations Connecting these Coordinate Systems.
\begin{equation}
x = \rho \sin \phi \cos \theta, \quad y = \rho \sin \phi \sin \theta, \quad z = \rho \cos \phi\tag{2.7.5}
\end{equation}
relating spherical coordinates \((\rho,\theta,\phi)\) to cartesian coordinates \((x,y,z)\text{.}\) (Spherical coordinates are sometimes denoted \((\rho,\theta,\phi)_s\) to distinguish from cylindrical or cartesian coordinates.)
Study Guide.
Study Section 2.7 of Calculus Volume 3; in particular
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openstax.org/books/calculus-volume-3/pages/2-7-cylindrical-and-spherical-coordinates- All the Definitions and Theorems.
- Examples 60–65 and 67, and the Checkpoints following each.
- One or several exercises from each of the following ranges: 363–366, 367–370, 371–378 (graphing not necessary), 379–384, 385–388, 389–392, 393–398, 399–402, 407–410.
