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 […]

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: $f(r, \theta, \phi)=R(r)A(\theta, \phi)$ ? Thanks in advance for any answers!

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 […]

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 & […]

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 […]

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 […]

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? 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 […]

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 $2 \pi$, […]

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