1. Coordinate Systems

d. Cylindrical and Spherical Coordinates - 3D and nD

3. Cylindrical and Spherical Coordinates in Higher Dimensions (Optional)

Cylindrical and spherical coordinates can be generalized from \(3\) dimensions to higher dimensions, but this must be done recursively from one dimension to the next. To facilitate the formulas, we will use subscripted variables. Thus the rectangular coordinates are: \[ x_1=x \qquad x_2=y \qquad x_3=z \qquad x_4 \qquad x_5 \qquad \cdots \qquad x_n \] Then the radial coordinates are: \[ r_2=r=\sqrt{\left.x_1\right.^2+\left.x_2\right.^2} \qquad r_3=\rho=\sqrt{\left.x_1\right.^2+\left.x_2\right.^2+\left.x_3\right.^2} \qquad \cdots \qquad r_n=\sqrt{\left.x_1\right.^2+\left.x_2\right.^2+\cdots+\left.x_n\right.^2} \] and the polar angles are \[ \phi_3=\phi \qquad \phi_4 \qquad \phi_5 \qquad \cdots \qquad \phi_n \] Note: The azimuthal angle \(\theta\) is not renamed because it is exactly the same in all dimensions.

Using these variables, the 3D cylindrical coordinates are given by: \[\begin{array}{rlrl} x_1&=r_2\cos\theta \qquad &r_2 &=\sqrt{\left.x_1\right.^2+\left.x_2\right.^2} \\[5pt] x_2&=r_2\sin\theta&\tan\theta &=\dfrac{x_2}{x_1} \\[5pt] x_3&=x_3&x_3 &=x_3 \end{array}\] and the 3D spherical coordinates are given by: \[\begin{array}{rlrl} x_1&=r_3\sin\phi_3\cos\theta \qquad &r_3&=\sqrt{\left.x_1\right.^2+\left.x_2\right.^2+\left.x_3\right.^2} \\[5pt] x_2&=r_3\sin\phi_3\sin\theta &\phi_3&=\arccos\dfrac{x_3}{\sqrt{\left.x_1\right.^2 +\left.x_2\right.^2+\left.x_3\right.^2}} \quad \\[5pt] x_3&=r_3\cos\phi_3 &\tan\theta&=\dfrac{x_2}{x_1} \end{array}\]

We are now ready to discuss cylindrical and spherical coordinates in \(4\) dimensions:

4D Cylindrical and Spherical Coordinates

4D Cylindrical Coordinates are constructed by starting with 3D spherical coordinates and adding one new rectangular coordinate \(x_4\). Thus the rectangular - cylindrical transformations are \[\begin{array}{rlrl} x_1&=r_3\sin\phi_3\cos\theta \qquad &r_3&=\sqrt{\left.x_1\right.^2+\left.x_2\right.^2+\left.x_3\right.^2} \\[5pt] x_2&=r_3\sin\phi_3\sin\theta &\phi_3&=\arccos\dfrac{x_3}{\sqrt{\left.x_1\right.^2 +\left.x_2\right.^2+\left.x_3\right.^2}} \quad \\[5pt] x_3&=r_3\cos\phi_3 &\tan\theta&=\dfrac{x_2}{x_1} \\[5pt] x_4&=x_4&x_4&=x_4 \end{array}\] Nothing much new!

4D Spherical Coordinates are constructed by starting with 4D cylindrical coordinates and replacing \(x_4\) and \(r_3\) by a new spherical radius \(r_4\) and a new polar angle \(\phi_4\) defined by the relations \[ r_4=\sqrt{\left.r_3\right.^2+\left.x_4\right.^2} \qquad \cos\phi_4=\dfrac{x_4}{r_4} \qquad \sin\phi_4=\dfrac{r_3}{r_4} \] Thus the rectangular - spherical transformations are \[\begin{array}{rlrl} x_1&=r_4\sin\phi_4\sin\phi_3\cos\theta \qquad &r_4&=\sqrt{\left.x_1\right.^2+\left.x_2\right.^2 +\left.x_3\right.^2+\left.x_4\right.^2} \\[5pt] x_2&=r_4\sin\phi_4\sin\phi_3\sin\theta &\phi_4&=\arccos \dfrac{x_4}{\sqrt{\left. x_1\right.^2 +\left.x_2\right.^2+\left. x_3\right.^2+\left. x_4\right.^2}} \quad \\[5pt] x_3&=r_4\sin\phi_4\cos\phi_3&\phi_3 &=\arccos\dfrac{x_3}{\sqrt{\left.x_1\right.^2+\left.x_2\right.^2 +\left.x_3\right.^2}} \\[5pt] x_4&=r_4\cos\phi_4&\tan\theta&=\dfrac{x_2}{x_1} \end{array}\]

Finally, we can recursively define cylindrical and spherical coordinates in \(n\) dimensions:

\(n\)D Cylindrical and Spherical Coordinates

In each new dimension, the \(n\)D Cylindrical Coordinates are constructed by taking the \(n-1\) spherical coordinates and adding one new rectangular coordinate \(x_n\). (We won't write these down.)

Then the \(n\)D Spherical Coordinates are constructed by starting with \(n\)D cylindrical coordinates and replacing \(x_n\) and \(r_{n-1}\) by a new spherical radius \(r_n\) and a new polar angle \(\phi_n\) defined by the relations: \[ r_n=\sqrt{\left. r_{n-1}\right.^2+\left. x_n\right.^2} \qquad \cos\phi_n=\dfrac{x_n}{r_n} \qquad \sin\phi_n=\dfrac{r_{n-1}}{r_n} \] Thus the rectangular - spherical transformations are: \[\begin{array}{rlrl} x_1&=r_n\sin\phi_n\cdots\sin\phi_4\sin\phi_3\cos\theta \qquad &r_n &=\sqrt{\left.x_1\right.^2+\left.x_2\right.^2+\cdots+\left.x_n\right.^2} \\ x_2&=r_n\sin\phi_n\cdots\sin\phi_4\sin\phi_3\sin\theta&\phi_n &=\arccos\dfrac{x_n}{\sqrt{\left.x_1\right.^2 +\left.x_2\right.^2+\cdots+\left.x_n\right.^2}} \\ x_3&=r_n\sin\phi_n\cdots\sin\phi_4\cos\phi_3&&\vdots \\ &\vdots&\phi_4 &=\arccos\dfrac{x_4}{\sqrt{\left.x_1\right.^2+\left.x_2\right.^2 +\left.x_3\right.^2+\left.x_4\right.^2}} \quad \\ x_{n-1}&=r_n\sin\phi_n\cos\phi_{n-1}&\phi_3 &=\arccos\dfrac{x_3}{\sqrt{\left.x_1\right.^2+\left.x_2\right.^2 +\left.x_3\right.^2}} \\ x_n&=r_n\cos\phi_n&\tan\theta&=\dfrac{x_2}{x_1} \end{array}\]