Articles of spherical coordinates

The gradient on sphere

In $n$-dimensional spherical coordinates, the gradient of a real valued function $f : \mathbb{R}^n \rightarrow \mathbb{R}$ can be represented by $\mathrm{grad} f = \left( \dfrac{\partial f}{\partial r}, \dfrac{1}{r} \nabla_\theta f \right) $, where $$ \nabla_\theta f = \dfrac{\partial f}{\partial \theta_1} \vec{\theta_1} + \dfrac{1}{\sin \theta_1} \dfrac{\partial f}{\partial \theta_2} \vec{\theta_2} + \dfrac{1}{\sin \theta_1 \sin \theta_2} \dfrac{\partial f}{\partial […]

building transformation matrix from spherical to cartesian coordinate system

How to arrive at the following from given $ x = r\sin \theta \cos \phi, y = r\sin \theta \sin \phi, z=r\cos\theta $ $$ \begin{bmatrix} A_x\\ A_y\\ A_z \end{bmatrix} = \begin{bmatrix} \sin \theta \cos \phi & \cos \theta \cos \phi & -\sin\phi\\ \sin \theta \sin \phi & \cos \theta \sin \phi & \cos\phi\\ \cos\theta & […]

How do I find the Fourier transform of a function that is separable into a radial and an angular part?

how do I find the Fourier transform of a function that is separable into a radial and an angular part: $f(r, \theta, \phi)=R(r)A(\theta, \phi)$ ? Thanks in advance for any answers!

Find volume between two spheres using cylindrical & spherical coordinates

I’ve got two spheres, one of which is the other sphere just shifted, and I’m trying to find the volume of the shared region. The spheres are $x^2 + y^2 +z^2 = 1$ and $x^2 + y^2 +(z-1)^2 = 1$ I know how to transform the variables into cylindrical and spherical coordinates but I’m having […]

What is a method for determining the radius of a sphere from distances between points of right triangles on it?

Suppose there are two right triangles formed by points {U, V, W1} and {U, V, W2} on the surface of a sphere. The distances between these points form the sides of the triangles, a, b1, c1 and a, b2, c2, where U-V is a, V-W# is b#, and U-W# is c#. The angles C1 & […]

Finding the boundaries on $\rho$ in spherical coordinates

For spherical coordinates, how would we find the boundary on $\rho$ for the volume enclosed by the surfaces $z = \sqrt{x^2 + y^2}$ and $x^2 + y^2 + (z-a)^2 = a^2$? I’m not sure how to algebraic show this, as I can’t really do it visually either. Visually, I know it should depend on the […]

conversion of laplacian from cartesian to spherical coordinates

In cartesian coordinates, the Laplacian is $$\nabla^2 = \frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}\qquad(1)$$ If it’s converted to spherical coordinates, we get $$\nabla^2=\frac{1}{r^2}\frac{\partial}{\partial r}\left( r^2 \frac{\partial}{\partial r}\right)+\frac{1}{r^2 sin\theta}\frac{\partial}{\partial \theta}\left(sin \theta \frac{\partial}{\partial \theta}\right)+\frac{1}{r^2 sin^2 \theta}\frac{\partial^2}{\partial \phi^2}\qquad(2)$$ I am following the derivation (i.e. the method of conversion from cartesian to spherical) in “Quantum physics of atoms, molecules, solids, nuclei […]

Deriving equations of motion in spherical coordinates

OK, we’ve been asked to derive the equations of motion in spherical coordinates. According to the assignment, we should end up with this: $$ \bf \vec{v} \rm = \frac{d \bf \vec{r} \rm}{dt} = \dot{r} \bf \hat{r} \rm + r \dot{\theta}\hat{\boldsymbol \theta} \rm + r \dot{\phi}\sin \theta \bf \hat{\boldsymbol\phi}\rm $$ In this case θ is the […]

How to integrate a vector function in spherical coordinates?

How to integrate a vector function in spherical coordinates? In my specific case, it’s an electric field on the axis of charged ring (see image below), the integral is pretty easy, but I don’t understand how handle the vector $(r,\theta,\phi)$ while integrating over $\phi$. I tried the following: $$\vec{E}=\int\limits_{Q}{\frac{kdq}{\|\vec{r}\|^3}\vec{r}}$$ $$ =\int_0^{2\pi}{\frac{k\lambda \vec{r}sin\theta d\phi}{\|\vec{r}\|^3}\vec{r}}$$ $$ =\int_0^{2\pi}{\frac{k\lambda […]

Find volume of the cap of a sphere of radius R with thickness h

I have to determine the volume and the formula for the volume for this spherical cap of height $h$, and the radius of the sphere is $R$: Two methods: *I just need help setting up the triple integrals 1) Cylindrical For for this method I am thinking that $\theta$ goes from $0$ to $2 \pi$, […]