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 $2 \pi$, […]

I have two arcs on a sphere that are defined as pair of points: (θ₀, φ₀), (θ₁, φ₁). I need to find a point where they intersect, or some indication if they don’t. What is important is that they are arcs, not circles, so it is important to find intersection that is on the arcs […]

I am trying to compute the Fourier transform of $\frac1{|\mathbf{x}|^2+1}$ where $\mathbf{x}\in\mathbb{R}^3$. Just writing out the integral: $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac1{|\mathbf{x}|^2+1}e^{-2\pi i (\mathbf{x}\cdot\mathbf{\xi})}dx_1dx_2dx_3$. Mathematica was no help with this integral. I realized though that the function is radial, so that in spherical coordinates $f(\rho,\theta,\phi)=\frac1{r^2+1}=f(\rho)$. I thought this would simplify matters because then the limits of integration are just […]

I want to calculate a unit direction vector of a direction with given the azimuth and elevation (cf. http://en.wikipedia.org/wiki/Azimuth), respectively $$\alpha \in [0^{\circ},360^{\circ}), \qquad \beta \in (-90^{\circ},90^{\circ}).$$ I have a right-handed coordinate system with z-up and looking down +y (yes, I have a graphics background :D). I got the hint that it’s easy to calculate […]

I have a set of points on the surface of a sphere specified in one coordinate system (specifically, the equatorial coordinate system), and for each point I need to work on all its neighbouring points as if it were on the equator (i.e. as if its elevation were zero). Specifically, each point is specified by […]

I’m having an issue with accuracy when converting Lat/Long coordinates to X,Y and then finding the shortest distance from a Point to a Line with said coordinates. The distance is off by around 40-50% of actual, which is unaccceptable for use. First I convert the coordinates (which are in decimal format) to radians, and then […]

I would like to draw a discretized circle on the surface of a sphere (the Earth in that case). The input of the algorithm would be the center of the circle (expressed as longitude and latitude), the radius (expressed in meters), and the number of points to calculate. The output would be the coordinates of […]

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