Questions on spherical coordinates, a three-dimensional coordinate system where a point is represented in terms of its distance from the origin, and its latitude and longitude angles (or complements thereof).

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Numerical method to solve a trigonometric (cotangent) function - transient heat transfer problem

I was trying to develop a mathematical model for transient one-dimensional heat conduction of spheres using approximate analytical solution as mentioned in Cengel{refer page number 229 in that ...
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1answer
57 views

How to resolve this equation to another value?

Sorry guys, I don't know how to be more specific in the question without writing a way too long question... Anyway my problem: I have this formula to calculate the distance between two points on the ...
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1answer
189 views

Changing from rectangular coordinates to spherical coordinates (integration)

I am taking calculus 3 and I have problems understanding how to change from rectangular coordinates to spherial ones (integration). For example, I have this problem: Find the volume of the solid $T$ ...
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1answer
188 views

Law sines in Spherical Triangle $\rightarrow$ Law sines in plane triangle

Could any one tell me how to estimate or get law of sines in Spherical Triangle to The Law of Sines in Plane Triangle? i.e $\frac{\sin a}{\sin A}=\frac{sin b}{\sin B}=\frac{\sin c}{\sin C}$ to $\frac{...
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2answers
276 views

Calculating spherical distance between two geo-locations

I wanted to show my nephew(16) a simple approach to calculate the distance between two geo-locations. The mathematical knowledge of a 16-year old boy is limited to simple geometrical shapes like ...
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1answer
334 views

Distance measurement between latitude/longiture pairs.

I need to calculate the distance between two lat/lng coordinate pairs. In addition, If given an initial lat/lng coordinate, angle of travel, and distance, I need to calculate the resulting lat/lng ...
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1answer
423 views

Integration on the unit sphere

I have an integral on the unit sphere as follows. $$I(\mathbf{s}_1, \mathbf{s}_2) = \int_{\mathbb{S}^2} f(\mathbf{x} \cdot \mathbf{s}_1)f(\mathbf{x}\cdot\mathbf{s}_2)d\mathbf{x} $$ where the ...
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1answer
460 views

Converting between spherical coordinate systems

Say I have the spherical coordinates of some locations, specifically their longitude ($0$ to 360) and latitude (latitude = $0$ at equator, $90$ at north pole, $-90$ at south pole) on a sphere with a ...
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1answer
330 views

Analytically derive n-spherical coordinates conversions from cartesian coordinates

I'm finding it difficult to find any non-geometrical derivation of coordinate conversions from cartisan to spherical. I can understand the derivations geometrically, because I can visualize the ...
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1answer
1k views

Divergence Theorem to evaluate integral where S is a sphere

Use the Divergence Theorem to evaluate $$\iint_S x^2y^2+y^2z^2+z^2x^2 dS$$ where $S$ is the surface of the sphere $x^2+y^2+z^2=1$. So the divergence of F is $2y^2x+2z^2y+2x^2z$. Then I tried to ...
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Monte Carlo simulation on sphere: unbiased random steps

Im doing a Metropolis Monte Carlo simulation with particles on a sphere and have a question concerning the random movement in a given time step. I understand that to obtain a uniform distribution of ...
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1answer
922 views

Coordinates of interception point Y with XY being the shortest distance of X to AB on sphere

How would one calculate the interception point $Y$ with $\overleftrightarrow{XY}$ being the shortest distance of $X$ to $\overleftrightarrow{AB}$? This answer to the question How to find the ...
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1answer
227 views

Parametric representation of rectangular form in terms of parameters $\rho$ & $\theta$

I need to represent the cone $z=\sqrt{3x^2+3y^2}$ parametrically in terms of $\rho$ and $\theta$ where $(\rho,\theta,\phi)$ are spherical coordinates. Attempt. I tried using: $$x=\rho\sin\phi\cos\...
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1answer
78 views

Integral from Sphere to Disc

Suppose one has an integral of the form $\int_{S_1^{d-1}} f(\phi(v)) d \text{vol}_{S_1^{d-1}}(v)$. Here $S_1^{d-1}\subset \mathbb{R}^d$ is the unit sphere. Let $B_1^{d-1}\subset\mathbb{R}^{d-1}$ be ...
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1answer
160 views

Find third point in mapping system

I have two points defined: $A$ and $B$ For both I know $x,y$, longitude, and latitude (gps coordinates). How do I calculate $x,y$ of a third point $C$ when I know its longitude and latitude? I know ...
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Looking for a particular parameterization of $S^n$

Say we have take vectors $(x_1,..,x_d) \in S^{d-1}$ and we look at vectors $(a_1,..,a_d) \in (\mathbb{Z^+ \cup \{0\}})^d$ such that $\sum_{i=1}^da_i =k$ for some positive integer $k$. Is there any ...
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How to sketch functions in polar and spherical coordinates by hand on paper?

I've been practicing drawing surfaces in different coordinates. I can do the easier ones but no I am completely stuck on the following two: Say we define spherical coordinates as follows: so that $...
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1answer
25 views

Converting coordinates

I am having huge trouble converting between cartesian to polar and to spherical. I need these methods for my limits of integration in most cases but using the formulas never seem to work, I just ...
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0answers
19 views

shperical coordinates limits

how to write the limits for the integration in spherical coordinates to get the volume of 1- the solid inside the cylinder $$r= \sin(\theta)$$ bounded from above by $$z=\sqrt{1-x^2-y^2}$$ and ...
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24 views

Getting topological objects from the “cube” of $T^3$

One can imagine $T^3$ much like he can imagine $T^2$: as a flat box with opposite faces identified. One may put coordinates on $T^3$, each of which would logically range from $0$ to $2\pi$. To get $S^...
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34 views

Laplacian in spherical coordinates - brownian motion

Consider the Laplacian equation on the unit sphere, for a vector $f$. $\theta$ is polar angle, and $\phi$ is azimuthal angle. The Laplacian in spherical coordinate is : $$ \Delta f = {1 \over r^2} {\...
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1answer
41 views

Is there a solution to the equation $tan({\phi})=\frac{0}{0}$

I've been reading about conversion from Cartesian ($x,y,z$) to Spherical (r, $\theta$, $\phi$) coordinates. The formula to find the value of ${\phi}$ is given as: $\tan({\phi})=\frac{y}{x}$ My ...
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50 views

Estimating the distance between two coordinates but without using Euclidean distance

Bill opens up "Café Finder" on his phone, and it tells him that it will take him 10 minutes to get to his nearest Starbucks to grab a triple-shot frapa-crapa-flat-white, so he decides to walk. 20 ...
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Integration in spherical coordinates

I needed to solve the following integral on one of my exercise sheets, which seemed not too difficult: $ \phi(\vec{r}) = \dfrac{1}{4\pi\epsilon_0} \int\limits_0^{\infty} dr' \int\limits_0^{\pi} d\...
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How to setup/evaluate a triple integral to show an interesting result in physics?

I know this isn't the physics forum, but the task i'm struggling with is purely mathematical. My task is as follows; Let $A$ be a sphere centered at origin with radius $R$ and assume $a \geq R$. ...
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33 views

Marginals/conditionals of a normalized Gaussian vector

It is well - known that if $x=(x_1,...,x_n)^T\sim{N(0, \sigma^2I)}$, then its normalized version is uniformly distributed on the unit $n-1$ - sphere: $$ y:=\frac{x}{||x||_2}\sim{\text{Uniform}}(S_{n-...
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1answer
228 views

Triple integral to find the mass of the intersection between two spheres

I've got two unit spheres, one is centered at $(0,0,0)$ and the other at $(0,0,1)$, the intersection of these two spheres is my region $R$. I would like to find the integral: $$\iiint\limits_R z\;dV$...
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20 views

Vectors and dot product in spherical coordinate system [duplicate]

Let $\vec{v_1}$ and $\vec{v_2}$ given: $\overrightarrow{V_1} = r_1\hat{u_r} + \theta_1\hat{u_\theta} + \phi_1\hat{u_\phi} \\ \overrightarrow{V_2} = r_2\hat{u_r} + \theta_2\hat{u_\theta} + \phi_2\hat{...
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angle between hrizontal and a line connecting the center of an oblate ellipse to a point in space

I would like to know how I can calculate the angle $\alpha$ in an oblate ellipse similarly to the sphere.
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38 views

rotation of spherical surface in spherical coordinates

I need to plot a spherical surface in computer (like the surface of a lens). I know the normal vector (as an example, say $\ n=(1,2,3) $) of this surface and it originates from the centre of the ...
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1answer
19 views

rotate a scalar valued spherical function

I want to rotate a function $f(\theta,\phi)$ around an arbitrary angle in 3D space. (Assuming $\phi$ is in the $xy$ plane and goes from $0$ to $2\pi$, and $\theta$ starts from $+z$ and goes from $0$ ...
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25 views

Spherical Triangle — Angle from one Point without using North

I'm designing a tool for students and right now im working with coordinates. I have the plan to shift the north pole in my code to Munich and I would like to do this by working with a spherical ...
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1answer
27 views

Find critical points using spherical coordinate

again! Let $E:=\left\{ (x,y,z) \in \mathbb{R}^3 \mid x^2+y^2+z^2 \le 4 \right\}$ and $f:E \to \mathbb{R}$ the function define by $$f(x,y,z):= \frac{z}{1+x^2+y^2+z^2}.$$ How can I determine the ...
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39 views

Locate a point on sphere with equal distance

Given 3 points A (lat1, lon1), B(lat2, lon2), O(lat3,lon3) on earth with geometric location longitude and latitude and a distance d, where O is middle point of A and B. Let GCD denote the great circle ...
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1answer
25 views

Volume of Region using Spherical Polar Coordinates

Find the volume of the region D bounded by the hemisphere $y=\sqrt{4-x^2-z^2}$ and the planes $y=x\ $,$\ y= \sqrt3x$ by using polar coordinates. Working: I have calculated $x^2+y^2+z^2=4$ so am I ...
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49 views

How to get parallels of tilted Equator?

I have a Great Circle on Earth, which is not an Equator nor Meridian, and it's not parallel to these. I have four geographical coordinate pairs for it, separated by 90 degrees, so I can use these in ...
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59 views

Coordinates of an n-sphere [duplicate]

I'm a little embarrassed to ask this question because it should be easy but it's stumped me for over a week now. The answer will determine how I write some code, so it matters. According to Wikipedia ...
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32 views

Numerical evaluation of the Laplace operator near the singularity points in spherical coordinates

If one had to evaluate $\Delta Y_l^m$ numerically everywhere on the unit sphere, including the singularity points $\theta = 0,\pi$, how would they do it? Let's say $Y_l^m$ is a spherical harmonic. I'm ...
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3answers
61 views

Polar Co-ordinate proofs

The expression for acceleration in spherical polars is $$ \ddot{\mathbf r} =( \ddot r -r\dot\theta^2-r\dot\phi^2\sin^2\theta) \mathbf e_r + (r\ddot\theta+2\dot r \dot\theta-r\dot\phi^2\sin\theta\cos\...
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31 views

Integrating over particular grids to obtain Spherical Harmonic coefficients

Theoretically the spherical harmonic expansion coefficients of a function $f$ should be calculated via a continuous integration: $$F_{lm} = \int_{0}^{2\pi}\int_{0}^{\pi} f(\theta,\phi)Y_{lm}^*(\theta,\...
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1answer
65 views

Spherical Parametrization of a Cholesky Decomposition

[Background: Trying to build up my math base knowledge (self-taught) to follow an explanation on page 291 of this document, dealing with spherical parametrization to estimate unknown variance-...
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1answer
161 views

PDE Heat Equation with Variable Coefficient {Second ODE Variable Coefficient}

Another PDE question: If I have a non constant coefficients in my heat equation (PDE), how do I solve it? For example we have: $\frac {\partial T}{\partial t} =\frac {\partial ^2 T}{\partial r^2} + \...
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21 views

Cover the entire sphere with a moving cone

Assume you have a cone with half-angle theta in a 3D cartesian space, with the vertex of the cone in the origin, and that you want to rotate the cone along a curve so that it covers the entire sphere. ...
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83 views

Problem with center of mass in polar coordinates

When we calculate center of mass using rectangular coordinates, we find the average values in each coordinate. Obviously we can't do this very same thing in polar coordinates: if we integrated a ...
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149 views

distance in n-dimensional space

According to answer of this question : Distance between 2 points in 3D space (in spherical polar coordinates) The distance between 2 points in 3 dimensional space is : $$ \sqrt{r_1^2+r_2^2-2r_1r_2\...
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1answer
261 views

Center of mass of a trick sphere-cone intersection

B is the solid region occupying the space situated inside the sphere of radius R centered at the origin and above the cone of equation $z = \sqrt{x^2 + y^2}$. The B density is proportional to the ...
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How do you represent a vector that points to an arbitrary point on a sphere of radius R?

Is it just $$v = R \hat{r} + \theta\hat{\theta} + \psi \hat{\psi}$$ That doesn't seem right as the unit is not correct.
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412 views

Set up integral in spherical coordinates outside cylinder but inside sphere

I have the equation of a cylinder and the equation of a sphere given: Cylinder: $x^2+y^2=4$ Sphere: $x^2+y^2+z^2=25$ I'm asked to set this up in cylindrical and spherical coordinates. Cylindrical ...
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105 views

3D rotational matrix between two spherical co-ordinate systems.

So I have a classical mechanics problem where I have worked out the azimuthal and altitude angle for a vector, I then want to apply rotational matrices so that the vector is realigned with the z axis ...
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58 views

Convert from Spherical to Cylindrical Coordinates

The following integral is given in Spherical Coordinates, which procedure should I follow to express it in Cylindrical Coordinates? $$\int_{0}^\pi \int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \int_{\frac{2}{\...