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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2answers
511 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
18 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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1answer
42 views

Confusion about the $x=\cos(\theta)$ substitution in Legendre polynomials derivation [closed]

I am confused about the argument they make in http://glennrowe.net/physicspages/2011/03/08/legendre-equation-legendre-polynomials/ They break up the proposed solution to the Legendre equation into ...
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0answers
47 views

Cyclindrical coordinates and Spherical coordinates [closed]

Consider the solid body which lies above the upper half of the cone $x^2 +y^2= 3z^2$ and below the sphere $x^2 + y^2 + z^2 = 4z$. Assume this body is of constant density. (a) Use cylindrical ...
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1answer
11 views

How do I get vectors orthogonal to the one generated by the spherical coordinate formula?

Given a formula: F : ℝ → ℝ → ℝ3 F(θ,φ) = (cos(φ)*sin(θ), sin(φ)*sin(θ), cos(θ)) what are the formulas: ...
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0answers
11 views

Proving that the Jacobian for spherical coordinates and the Jacobian for integrating over a sphere coincide

I want to use the formula for switching an integration from spherical to cartesian coordinates : $\int_{RS^{n-1}} \! f(x) \, \mathrm{d}\sigma (x)$=$\int\int...\int ...
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2answers
379 views

Center of mass of one octant of a non-homogenous sphere

Find the center of mass of that part of the sphere $x^2+y^2+z^2 \le a^2$ having $x,y,z \ge 0$ (that is, the part in the first octant) With density given by $\rho(x,y,z)=(x^2+y^2+z^2)^{3/2}$ It ...
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2answers
300 views

converting kph and heading to xyz velocity vector

I am writing software (in C++) that is required to send out messages from our simulation system to another simulation system. Problem is we track the simulation object's current speed (kph) and ...
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1answer
106 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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1answer
51 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 ...
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1answer
18 views

volume using spherical coordinates

Let $$V = \{(x, y, z): x^2 + y^2 ≤ 4 , 0 ≤ z ≤ 4\}$$ be a cylinder and let $P$ be the plane through $(4, 0, 2), (0, 4, 2)$ and $(−4, −4, 4)$. Compute the volume of $C$ below the plane $P$. I'm having ...
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0answers
27 views

Solving LES containg spherical coordinates

i have a three-dimensional parametric equation of a line, where the directional vector is normalized and converted to spherical coordinates to calculate a angle offset. It looks like this: ...
5
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5answers
14k views

What is the general formula for calculating dot and cross products in spherical coordinates?

I was writing a C++ class for working with 3D vectors. I have written operations in the Cartesian coordinates easily, but I'm stuck and very confused at spherical coordinates. I googled my question ...
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3answers
75 views

Triple Integral in Spherical Coordinates.

$\newcommand{\de}{\operatorname{d}}$A little stuck on this one. $$\iiint_V ye^{-(x^2+y^2+z^2)^2}\,{\rm d} V$$ Use Spherical Coordinates to evaluate where V is the solid that lies between y=0 and the ...
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0answers
20 views

Normal form of plane

I propose a simple equation of plane generalized from 2D Normal form of straight line as follows, starting from Straight line Normal form: $$ x\, \cos \alpha + y \sin \alpha = p. \tag{1}$$ ...
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2answers
210 views

How many coordinates are necessary to determine a sphere?

Do determine a circle, you would need at least three coordinates. How many are necessary to determine a sphere?
3
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1answer
56 views

Circle from three points on surface of sphere

I need to compute the circle on the surface of a sphere given three points on that very surface. It is very easy to do that in Euclidean geometry, but the sphere has no x and y, but just two angles ...
2
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1answer
1k views

Rotating a point in spherical coordinates around Cartesian axis

If I have a point in spherical coordinates, and I rotate it around one of the Cartesian axes, what will be the new spherical coordinates for the point? Both spherical and Cartesian coordinate systems ...
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0answers
10 views

Comparing paired objects in 3d space using direction angles.

I have data about paired objects in 3d space, where each object is defined by three components. What I would like to do is compare the orientations of the paired objects and see if one group of those ...
0
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0answers
6 views

Midpoint of two n-vectors

Currently I am using n-vectors (more information can be found here) for an accurate and singularity-free representation of coordinates on the earth's surface. For various reasons, I need to compute ...
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0answers
34 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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1answer
650 views

Rotation matrix in spherical coordinates

When given arbitrary point on a unit sphere $a = (\theta, \phi)$ and an arbitrary axis $\vec{A}=(\Theta, \Phi)$, can we have an algebraic expression for $a_1=(\theta_1, \phi_1)$ which is a rotation of ...
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1answer
3k views

Three Dimensional Fourier Transform of Radial Function without Bessel and Neumann

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: ...
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0answers
25 views

Heat Equation: Remains finite?

I am doing a PDE question. It's about heat equation, spherical coordinates (the usual stuff). The boundary condition is $\frac {\partial T}{\partial r} (1,t) = 0 $ and it also said for $T$ to remain ...
3
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1answer
24 views

Integrating a sphere by discs vs shells (spherical coordinates)

I am getting very confused about the following. Let's say I want to find the volume of a sphere. I can start with a circle having circumference $2\pi R\cos\theta$. I can multiply by $R d\theta$ and ...
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0answers
15 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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0answers
23 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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2answers
50 views

Volume of solid by Spherical

Trouble setting up the integrals for this problem. Find the volume of the solid bounded by $x^2 + y^2 = 1, z = 0$, $z = 6$, $y\geq 1/2$. Use integration with Spherical coordinates. (Hint: Use two ...
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0answers
45 views

Is there a spherical coordinates system for vectors of complex numbers?

Suppose I have a scalar field $f(\vec{x})$, where $\vec{x}\in\mathbb{R}_3$, and I wish to average $f$ over a sphere $|\vec{x}|=R$: $\displaystyle\langle f\rangle_{R} = \frac{\int_{S} f(\vec{x})\, ...
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1answer
59 views

Radius of the Earth at N32.704220, W90.000000?

I want to express a point on a map in radian spherical coordinates. By Google maps, this location is north of Canton, MS, USA just a few hundred feet from US 51. In radian spherical coordinates, ...
2
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1answer
305 views

Prove that hyperspherical coordinates are a diffeomorphism, derive Jacobian

The explicit form for the transformation into hyperspherical coordinates is $$x_1 = r\sin\theta_1 \sin\theta_2 \dotsb \sin \theta_{n-1} \\ x_2 = r\sin\theta_1 \sin\theta_2 \dotsb \cos \theta_{n-1} \\ ...
2
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1answer
1k views

Reverse use of Haversine formula

Alright the title is not the best. What I want to do is to change the given parameters in Haversine's formula. If we know the lat,lng of two points we can calculate their distance. I found the ...
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1answer
28 views

Short question about spherical coordinates

If I have a vector orthogonal to the $x$-$y$ plane of an $xyz$ axis system, I mean, a vector with just $z$ component: How can I express it in spherical coordinates?
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2answers
237 views

Integrating a jacobian to find the volume.

I want to solve the following: Prove that $$\displaystyle \int_R \sin^{n-2}\phi_1 \sin^{n-3}\phi_2\cdots\sin \phi_{n-2} d\theta d\phi_1\cdots d\phi_{n-2} = \frac{2\pi^{n/2}}{\Gamma(n/2)}$$ where $ ...
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0answers
9 views

Setting up triple integral in spherical coordinates

Integrate $f(x, y, z) = z^2$ over $A = \{(x, y, z) \in \mathbb{R^3} | x^2+y^2+z^2 \leq R^2, x^2 + y^2 + z^2 \leq 2RZ\}$ I know $A$ is the intersection between two spheres but I am unable to figure ...
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0answers
20 views

How to compute the following Jacobian

I need to show that the Jacobian of the n-dimensional spherical coordinates is $$\displaystyle r^{n-1}\sin^{n-2}\phi_1\sin^{n-3}\phi_2\cdots\sin\phi_{n-2}$$ then I have computed the Jacobian matrix, ...
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0answers
18 views

Translation matrix in spherical coordinates system

I'm using WorldWind software to draw segments (polyline) on the globe to materialize an aircraft flightplan. Each point in a flightplan is named waypoint. Waypoints are expressed in geographical ...
0
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1answer
42 views

sphere-sphere intersection

Let $ S_1 : (x-1)^2 +y^2+z^2=1 $ $S_2 : x^2 +y^2 +z^2 =1$ $S_3 : (x+1)^2 +y^2 +z^2 =1 $ Find the volume of the solid inside $S_2$ and outside $S_1$ and $S_3$, using triple integrals. I have try ...
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1answer
20 views

x1, y1 and radius are given - can anything be assumed about x2, y2?

I have a list of lat/lng coordinates. Given the coordinates x1, y1, and a radius r -- is there anything I can assume about the coordinates that fall within the radius of x1, y1? For example, can I ...
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1answer
50 views

Find geo coordinate by a coordinate and an angle

I need some help with this problem. I have a GPS coordinate and an angle in degrees. I need a new GPS coordinate x km away from the point I already have. Degree is counted clockwise and y-axis is ...
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0answers
25 views

Does anybody know how to actually derive spherical harmonics in a way that is historically accurate and intuitive?

And by "historically accurate", I mean without resorting to techniques of derivation which were developed after the fact or explanations which use the very concept they're trying to explain. The few ...
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1answer
24 views

Spherical Coordinates ( plane y = -x)

I am attempting to express the plane y = -x in spherical coordinates. Is there any clean way to do this? I have expressions for rho, theta, and phi in my text book but I don't think anyone of those ...
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0answers
23 views

On the equidistant distribution of $n$ points on a sphere $S^2$ by algorithm and their “validity” measures by statistical methods

I have found an algorithm for distributing $n$ points $P_0, P_1, ..., P_n$ (approximately) equidstantly on a sphere where $$\varphi_i = \pi(\phi - 1)i \qquad \theta_i= \mathrm {asin} (2i/n - 1), ...
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2answers
31 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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1answer
88 views

Manifolds, coordinate systems, books

Which books, say Lee's Introduction to Smooth Manifolds or Munkres' Analysis on Manifolds explains how the theory of a differentiable manifolds can be used to solve a problem that is expressed in a ...
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0answers
21 views

Parametrization of sphere including constant inclination $(\theta, i)$ geodesics

Find parametrization of sphere with respect to $\theta$ = constant meridians and i = constant inclination geodesic circles passing through N-S axis and E-W axis respectively. The Earth does not rotate ...
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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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1answer
33 views

Cartesian to Spherical Coordinate Conversion for Triple Integral

I have a question regarding what happens to the boundaries when converting a triple integral from Cartesian to Spherical Coordinates. Example ...
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1answer
84 views

Finding a 3D co-ordinate using triangluation

I'm trying to find a real world coordinate where 3 spheres collide and interact. At the moment I have been able to set up my triangulation equations so that I can work out the 2D position of where my ...
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0answers
35 views

Solutions of the Laplace Equation in spherical coordinates

I would like some help with the following problem. Thanks for any help in advance. Use spherical coordinates to find all solutions of the Laplace equation ∆u(x)=0, u∈Ω⊂R3 that depend only on the ...