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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How to prove that solution to ODE in spherical coordinate is equivalent to the ODE in cartesian coordinates if it is a thin shell

Solving a diffusion-type ODE across a spherical shell, the equation is: $$\frac{d}{dr}\left(r^2\frac{df}{dr}\right)=0\tag{1}$$ with boundary conditions $f(r_1)=f_1$ and $f(r_2)=f_2$. The solution is: ...
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1answer
49 views

Minimum of a potential function

I'm looking for extremes (minimum) of $$V = \frac{\alpha}{|\vec{r}_1-\vec{r}_2|} + \beta (\vec{r}_1 + \vec{r}_2)\cdot \vec{e}_z$$ where $\vec{r}_i = ...
2
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1answer
48 views

Del on Riemannian manifolds

I was supposed to figure out what $grad(div(e_r))$ is, where $e_r$ is the unit vector in spherical coordinates. In the following I assume $g:=diag(1,r^2,r^2sin^2(\theta))$ be the metric tensor. My ...
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2answers
2k views

Volume of a sphere (r=2a) with hole(r=a) drilled through centre, using spherical polar coordinates.

Need help solving 11.bi), A cylindrical hole of radius a is bored through the center of a sphere of radius 2a. Find the volume of the remaining material, using spherical polar coordinates. (You ...
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2answers
122 views

can a great circle route be predicted from initial condition?

Assuming the world is a sphere with no wind, can the great circle route of a vessel be predicted from the current position $\{\phi_i,\lambda_i\}$ and the current true course $\theta_i$? Presently, ...
2
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2answers
398 views

Vector Picking on the Unit Sphere

Imagine a vector from the center of a unit sphere to its surface: Now imagine a second vector generated in indentical fashion. Given the first vector, how can I generate vectors to uniformally ...
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1answer
3k views

Find volume of a cone $z=k\sqrt{x^2+y^2}$ bounded by $z=h$ using spherical coordinates

We were given this exercise in class to take home but I am a bit confused with it. If anyone could help I would appreciate it. Let $C$ be a conical solid bounded above by $z=h$ and below by the cone ...
2
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1answer
212 views

Triple Integral in Spherical Co-ordinates

Find the volume bounded by the surface $(x^2 + y^2 + z^2)^2 = 2z(x^2 + y^2)$ I have $x = \rho \sin\phi \cos\theta$, $y = \rho \sin\phi \sin\theta$, $z = \rho \cos\phi$. Therefore, $(x^2 + y^2 + ...
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1answer
3k views

Angle between GPS coordinates

I realize GPS Coordinates are spherical coordinates. However I know the earth is more of an ellipsoid. I need to compute with a fairly high degree of accuracy the pitch and yaw between two objects ...
2
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1answer
489 views

Equation for the sensitivity pattern of a bi-directional microphone?

Can anyone give me an equation that expresses the sensitivity pattern of a bi-directional microphone, as a function of azimuth and elevation angle? A bi-directional microphone pattern looks something ...
2
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1answer
422 views

Can you formulate a $ \phi, \theta $ restriction in spherical coordinates for a great circle?

Further to this question Quaternion rotation has a nice property that you can trace any great circle you like. You specify the axis of rotation, and you will automatically follow the great circle ...
2
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1answer
643 views

integral of a spherically symmetric 3-dimensional function over all space

I'm very sorry because it may be a very basic question but I'm not able whether to solve it for sure, nor to find an answer in stackexchange or elsewhere. I have to calculate $ \int \int ...
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2answers
30 views

Equation used to represent a disc galaxy

I'm trying to create a solid which looks something like a disc galaxy: Key features are: Bulge in the middle Tapered "width" as it extends to a disc shape The end goal would be to use Python to ...
2
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1answer
32 views

Inclusion, pullback of differential form

Let $\omega=x\,dy\wedge dz +y\,dz\wedge dx+z\,dx\wedge dy$ or in spherical coordinates (unless I had made some mistake) $\omega=r^3\cos \theta\, d\phi\wedge d\theta$. Now I want to find $i^*\omega$ ...
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2answers
49 views

cylindrical coordinates - base vector integral

Hello I have a problem where $\hat{p}$ is a basevector in a cylindrical system $$\int_{0}^{\pi/4}\hat{p} d\theta$$ I know that; $\hat{p} = \hat{x}\cos\theta + \hat{y}\sin \theta$ $$ ...
2
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1answer
31 views

Conversion of spherical coordinates to cartesian

For the flow $A = \frac{c}{r}$ with $r=\sqrt{x^2+y^2+z^2}$ I wanted to calculate the velocity field with $\nabla A$ As a result I get $(-\frac{c}{r^2},0,0)$. So far so good. When I tried converting it ...
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1answer
75 views

Calculus 3 Spherical coordinates: I'm not sure how to set this up.

find the volume of the region enclosed by the sphere $x^2+y^2+z^2=324$ and the cylinder $(x-9)^2+y^2=81$ by using spherical coordinates. I'm just not seeing how to convert this into a form where ...
2
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1answer
38 views

Triple Integrals: Conversion

I'm currently in second year calculus and have come across a problem that I'm struggling badly to try and understand. The question is as follows: Sketch the region of integration of the following ...
2
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1answer
162 views

Volume of sphere with triple integral

Using the same notations as in this picture : The element of volume is: $r^2 \sin(\theta) \, dr \, d\theta \, d\phi$ If I try to create the volume visually, I begin with integrating $r$ between ...
2
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3answers
56 views

triple integral and limits

$$\iiint\limits_H (x^2+y^2) \, dx \, dy \, dz \\ H=\{(x,y,z) \in R^3: 1 \le x^2+y^2+z^2 \le9, z \le 0 \}$$ I'm using Spherical coordinate system: $$x=r\cos \theta \cos\phi $$ $$y=r\cos \theta ...
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3answers
1k views

Equation of a straight line in spherical coordinates

I'm trying to prove the angle sum formula for a triangle on the surface of a sphere. In order to do this I wanted to create a general triangle on the sphere, with one vertex at $\theta = 0$ and one ...
2
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1answer
68 views

Spherical-Coordinate Reference Frame

my problem looks apparently easy but I can figure out a solution. I'm going through a paper about the Boltzmann equation and I got stuck with this change of coordinates. The original formula for Q ...
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2answers
3k views

Different ways for calculating distance between two geodetic points give me different results

I'm trying to calculate the distance between two geodetic points in two different ways. The points are: A:(41.466138, 15.547839) B:(41.467216, 15.547025) The ...
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2answers
623 views

How to find all 3 orthogonal vectors to a 4D vector

For a program I'm writing, I need to find the vectors orthogonal to a given vector rotated at an arbitrary angle, and in 4D. It is a unit vector. For 3D, I found the two orthogonal vectors like ...
2
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3answers
151 views

Writing triple integrals in spherical coordinates over nonspherical/nonconical regions

Defining upper and lower limits of integration for $\rho$, $\theta$, and $\phi$ is relatively easy when writing a triple integral in spherical coordinates if the region of integration is defined by ...
2
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2answers
311 views

Proving that $\nabla \times (U(r) \hat{r} = 0 $

I was just checking to see if I wsa doing this right, as it isn't a formal proof. Just showing the identity. Let $U(r) \hat{r}$ b a vector in spherical coordinates. Given that the vector is only ...
2
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1answer
639 views

Triple Integral over a shifted sphere

I am interested in the following: Let $f(x,y,z)$ be a given (known) function in Cartesian coordinates, and let $B_d(p_0) = \{ y: ||y-p_0|| < d \}$ (i.e., a sphere centered at p_0). I want to ...
2
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2answers
181 views

Are there derivations of equations of non-degenerate real quadric surfaces

Take the ellipsoid for example $$(x^2/a^2)+(y^2/b^2)+(z^2/c^2)=1$$ in the x-y plane you have an ellipse described by $$(x^2/a^2)+(y^2/b^2)=1$$ (suppose z=constant) in the y-z plane you have an ellipse ...
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1answer
473 views

Change of coordinate system on a sphere

This might take a while to explain, so bear with me: I've got a perfect sphere. I've set up an arbitrary longitude/latitude ("angle") coordinate system on it (imagine an equator around the middle, ...
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2answers
584 views

Evaluating an integral in spherical coordinates over on odd shaped region.

I have to evaluate this integral: $$ \int_{0}^{\sqrt{2}}\int_{y}^{\sqrt{4-y^2}}\int_{0}^{\sqrt{4-x^2-y^2}} \sqrt{x^2+y^2+z^2}dzdxdy $$ in spherical coordinates. I see that the region in the xy ...
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1answer
4k views

Transforming from one spherical coordinate system to another

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 ...
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2answers
517 views

How to use a Rhumb Line?

I am new to working with coordinate data and figured out the equation I am looking for is the Rhumb Line. I went to go research it and found a lot of equations and I still have no idea where to start. ...
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0answers
25 views

Pattern of collision of bouncy balls in a sphere?

Suppose that you have two infinitely bouncy golf balls that exist inside a perfect sphere in weightless suspension, and both golf balls start bouncing at a random angle and are 10 or 100 times ...
2
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1answer
51 views

Spherical distance between two points in terms of latitude and longitude

I have seen the answer to this question - Great arc distance between two points on a unit sphere However in a fortran program that I have this is the code to calculate spherical distance between two ...
2
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1answer
54 views

How to integrate a part of a sphere limited by a rectangle?

I have a function which depends on the solid angle $\frac{df}{d \Omega} = \frac{df}{d\phi d\theta} $. I want to integrate it over a part of the sphere limited by a rectangle. How should I set the ...
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0answers
48 views

Integration over n-sphere

I am trying to integrate squares of sum of coordinates in n-dimensional sphere with radius bounded by r. $f(x,y)=x^2+y^2$. In spherical coordinates when $N=2,\;y=a\cos\theta$ and $x=a\sin\theta$ So ...
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0answers
38 views

Smart coordinates for six-dimensional integral

I have a (hopefully) simple question: I am dealing with a definite (on all of $\mathbb{R}^6$) six-dimensional integral $$\int_{\mathbb{R}^6} F(\vec{x}_1,\vec{x}_2)d^3x_1d^3x_2$$ where the function ...
2
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1answer
57 views

Why must power series count up by integers 0,1,2.. in 3D harmonic oscillator in spherical coordinates?

http://www.physicspages.com/2013/01/17/harmonic-oscillator-in-3-d-spherical-coordinates/ http://quantummechanics.ucsd.edu/ph130a/130_notes/node244.html These are two links that have roughly the same ...
2
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1answer
26 views

Determine the volume of a solid given specific bounds

Determine the volume of the solid enclosed by the paraboloid $z = x^2 + y^2$ and the plane with equation $4x − 2y + z = 0$. Could someone explain to me whether I use double integral polar coordinates ...
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0answers
52 views

Intersection of two spherical caps in $(n+1)$-dimensional Euclidian space

I want to compute the intersection area of two spherical caps whose tops are placed on two orthogonal axes in $\mathbb{R}^{n+1}$. In spherical coordinates, the surface element is given by $$ \,dS = ...
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2answers
32 views

Evaluating integrals in R^m

Let $|\cdot|_m$ denote the Euclidean norm in $\mathbb{R}^m$. Then I wish to prove that $\displaystyle\int\limits_{\mathbb{R}^m}|x|_me^{-|x|_m}dx<\infty$ It's kinda embarrassing to say this, but ...
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0answers
61 views

Can anyone check if this correct?

Convert to spherical coordinates and evaluate:$$\iiint_{E}z(x^2+y^2+z^2)^{-3/2}dV$$ where E is the region satisfying the following inequalities:$$x^2+y^2+z^2\le16,z\ge 2$$ This is what i have done so ...
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2answers
53 views

How to tell if two spherical coordinates lie on the same plane

I have the rho, theta, and phi values of two points, how can one tell that two vectors are normal to the same plane by looking at their spherical coordinates?
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0answers
116 views

Fourier transform of an exponentially singular radial function

I am trying to compute the 3D Fourier transform of a spherically symmetric function of the form $$f(r) = e^{\frac{1}{r} e^{-r}} - 1\, ,$$ which entails the integral $$\begin{aligned}F(k) =& \int ...
2
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1answer
83 views

Coordinates of tilted circle.

The original question is as follows: Imagine a wire located at the intersection of $x^2+y^2+z^2=1$ and $x+y+z=0$, whose density depends on position according to $\rho({\bf x})=x^2$ per unit length. ...
2
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2answers
76 views

Laplacian $\frac{1}{r^2}\frac{\partial}{\partial r}(r^2 \frac{\partial \phi }{\partial r})= \frac{1}{r} \frac{\partial ^2 }{\partial r^2}(r \phi )$

Does anyone have any intuition on remembering or very quickly deriving that $$\frac{1}{r^2}\frac{\partial}{\partial r}(r^2 \frac{\partial \phi }{\partial r}) = \frac{1}{r} \frac{\partial ^2 ...
2
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1answer
359 views

Find the volume of the solid bounded by the surface given in spherical coordinates by $R = 4-3\cos(\phi)$.

It is worth noting that $R$ in this case denotes the distance from origin to a point $P$ in space. You may be more familiar with $\rho$ instead of $R$. Here is my attempted solution: I am assuming ...
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0answers
193 views

Given 3 Vertices of a Tetrahedron, Find the 4th

A regular tetrahedron is circumscribed by the Earth (assume spherical). You are given 3 of the 4 vertices (as latitude and longitude in decimal format), and asked to find the 4th. Any help is most ...
2
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1answer
187 views

Divergence in spherical coordinates

On the one hand there is an explicit formula for divergence in spherical coordinates, namely: $$ \nabla \cdot \vec{F} = \frac{1}{r^2} \partial_r (r^2 F^r) + \frac{1}{r \sin \theta} \partial_\theta ...
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0answers
117 views

How do you set up the integral in spherical coordinates in the following problem?

Find the volume bounded by the surface $z = x^2 + y^2$ and $x^2+y^2 = 1$ in the first quadrant. The answer is $\pi/8$ using rectangular and cylindrical coordinates and that is the correct answer, but ...