0
votes
2answers
19 views

Derivation for the integrating term in line integrals and volume integrals in spherical coordinates

Can anyone refer me to, or respond with, the derivation for the integrating term in line integrals $dl=dr\hat{r}+rd\theta\hat{\theta}+r\sin\theta\ d\phi\hat{\phi}$ and volume integrals ...
0
votes
2answers
41 views

How can I solve these two tough integrals?

\begin{equation*} J_{1} = \int_{0}^{\sqrt{{\pi}/{6}}} \int_{y}^{\sqrt{{\pi}/{6}}} \cos{(x^2)}\,dx\,dy \end{equation*} \begin{equation*} J_{2} = \int\int_{E}\int z e^{(x^2+y^2)} + xe^{x^8}\,dV, ...
1
vote
1answer
24 views

How to describe the region inside a sphere and below a cone in cylindrical and spherical coordinates?

If E is the region of space located inside the sphere $x^2 + y^2 + z^2 = 4$ and below the cone $z = \sqrt{3x^2 + 3y^2}$ How may I describe E in cylindrical and spherical coordinates? And how may I ...
2
votes
0answers
28 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 ...
1
vote
1answer
41 views

Writing the squared sine as a Legendre polynomial of cosine

I'm just getting learning about how Legendre polynomials come about when considering product solutions in spherical coordinates with azimuthal symmetry. I'm trying a problem on my own, and I'm a bit ...
1
vote
2answers
37 views

Finding the Limits of the Triple Integral (Spherical Coordinates)

Let $D$ be the region in $\mathbb{R}^3$ below $z=-\sqrt{x^2 + y^2}$ and above $z=-\sqrt{4-x^2 -y^2}$. Rewrite \begin{align*}\iiint \limits_D z^2 dV\end{align*} using Spherical Coordinates. I ...
1
vote
3answers
44 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 ...
3
votes
1answer
37 views

Evaluate the integral $\iiint\limits_E x^2 \,\, \mathrm{d}V$

Where E is the region bounded by the xz-plane and the hemispheres $y=\sqrt{9-x^2-z^2}$ and $y=\sqrt{16-x^2-z^2}$. This is an exercise from my professor guide. What I tried so far: These exercise ...
0
votes
1answer
48 views

Flow of fluid through a really tricky closed surface S (divergence theorem)

Considering a fluid whose velocity field is $\vec{v}(x,y,z)= (y^{3}e^{-z^{2}} + x)\vec{i} + (ze^{x} + y^{2})\vec{j} + (cos(x^{2}+ y^{2}) +2z)\vec{k}$ Calculate the flow of fluid through the closed ...
1
vote
0answers
25 views

Parametrization of a bounded solid.

So, I have a solid bounded by $z=\sqrt{x^2+y^2}, z=\sqrt{1-x^2-y^2}, z=2$ I had to parametrize it using spherical coordinates so I used $$\begin{cases} x(\rho, \theta, ...
3
votes
0answers
21 views

Converting a polar integral to spherical

$$\int_0^{2\pi} \int_0^{\sqrt{2}}\int_r^{\sqrt{4-r^2}}\mathrm{d}z \, r \, \mathrm{d}r \, \mathrm{d}\theta$$ So in spherical this would become: $$\int_0^{2\pi} \int_0^{\pi/4}\int_0^2 \rho^2\sin\phi \, ...
0
votes
1answer
57 views

Find the volume inside

Find the volume inside the torus $\rho=\sin\phi$. First of all how can $\rho=\sin\phi$ represent a torus? I can't even visualise that. All Ideas are welcome, this looks like a 'food for thought ...
0
votes
0answers
22 views

Volume of a shape using spherical coordinates and integrals

A solid is described in spherical coordinates by the inequality ρ ≤ sin(φ). Find its volume. So, I took ρ from 0 to sin(φ), φ from 0 to pi, and theta from 0 to 2Pi and formed the integral: integral ...
2
votes
1answer
49 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 ...
1
vote
1answer
41 views

Implicit partial derivative of a spherical cap

Consider a spherical cap, for which the base radius is $a$ and the height is $h$. Then, the surface area and volume is (these equations can be found on Wolfram Mathworld) $A(a,h) = \pi(a^2 +h^2)$, ...
3
votes
1answer
44 views

Surface area of the part of the sphere $x^2+y^2+z^2=a^2$ that is inside the cylinder $x^2+y^2=ax$

I've been solving some surface area problems lately, but I don't think that the same approach that I was using will work with this one (or at least will result in a lot work). So, I believe I should ...
0
votes
1answer
37 views

How do I find the limits for $\iiint_{W} \frac{dx dy dz}{(x^2 + y^2 + z^2)^{\frac{3}{2}}}$?

Evaluate $\iiint_{W} \frac{dx dy dz}{(x^2 + y^2 + z^2)^{\frac{3}{2}}}$ where $W$ is the solid bounded by the two spheres $x^2 + y^2 + z^2 = a^2$ and $x^2 + y^2 + z^2 = b^2$ where $0 < b < a$. ...
3
votes
2answers
62 views

Integral using spherical coordinates

I am trying to compute the volume of the following set : intersection of cylinder $x^2 + y^2 \leq R$ and sphere $x^2 + y^2 + z^2 \leq 4R^2$. I am having trouble setting up the integral properly ...
1
vote
0answers
49 views

volume in spherical coordinates

I am trying to find the region bounded by the sphere $p = 2\cos\psi$ and hemisphere $p=1$, $z\geq 0$. Not quiet sure not to do this problem, so please help.
0
votes
0answers
30 views

Showing the uniqueness of the solution to some problem?

I have faced problems proving all kinds of uniqueness theorems but this one I've come across seems particularly tricky to me. Can you help me? The function g(x,y,z) is zero for r²>a² (where ...
0
votes
0answers
74 views

A basic question on surface area of spherical cap of sphere

Consider a sphere of radius 1. Now chop a spherical cap with latitude line $\phi$ at the bottom of the cap is removed from top (say $0<\phi<\frac{\pi}{2}$). I want to know the surface area of ...
0
votes
0answers
270 views

Volume enclosed by two spheres (triple integral, cylindrical coordinates)

The question: Find the volume of the solid enclosed by the sphere $x^2 + y^2 + z^2 - 6z = 0$ , and the hemisphere $x^2 + y^2 + z^2 = 49 , z ≥ 0$ I set up the triple integral ...
2
votes
3answers
588 views

Find volume of the cap of a sphere of radius R with thickness h

I have to determine the volume aka 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 ...
0
votes
1answer
29 views

3D Fourier transforms of $e^{-\beta r} $ and $re^{-\beta r} $

I am trying to find the integrals $$\large\int\limits_{\mathbb{R}^3} e^{-\beta \left|\vec{r}\,\right|}e^{i \vec{q} \cdot\,\vec{r}} \mathop{d^3r}$$ $$\large\int\limits_{\mathbb{R}^3} ...
0
votes
0answers
122 views

Conversion of covariance matrix from Cartesian to Spherical coordinates for integration

I have to perform a convolution of a function in polar coordinates $\rho(\textbf{x}) = \rho(r,\theta,\phi)$ with a function $P(\textbf{x}) = P(x,y,z)$ in cartesian coordinates. $\int ...
2
votes
1answer
175 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 ...
2
votes
3answers
258 views

Find volume between two spheres using cylindrical & spherical coordinates

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 ...
2
votes
2answers
60 views

Triple integral over a sphere with parameter $2n$?

I need to intergrate $x^{2n}+y^{2n}+z^{2n}$ over a sphere of equation $x²+y²+z²=1$. I have thought of changing the coordinates from cartesian to spherical but I don't know how to deal with the ...
1
vote
2answers
75 views

Multivariable Calculus Volume of Integration Question

I have an integral $$ \iiint\sqrt{x^{2} + y^{2} + z^{2}\,}\,{\rm e}^{-\left(x^{2} + y^{2} + z^{2}\right)}\, {\rm d}x\,{\rm d}y\,{\rm d}z $$ The integration region is bounded by the sphere ...
0
votes
1answer
49 views

Spherical Coordinates Representation

I just wanted to know what the set of all points in which spherical coordinates can be shown in more than one way is? I think it is only the origin but I am not sure
2
votes
2answers
764 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 ...
1
vote
1answer
68 views

How to derive curl in spherical coordinates

This is one of those questions where I know I am making a dumb mistake someplace and I am trying to check where it is. $$ \begin{vmatrix} \frac{\bf\hat r}{r^2\sin\theta} & ...
2
votes
3answers
106 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 ...
0
votes
0answers
71 views

A problem on vector calculus on spherical coordinate systems

I'm wondering whether I could get some idea on a problem I've been working on, here it is: h and g are two functions defined on the surface of a unit sphere, and their values depend on the ...
2
votes
1answer
210 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} \\ ...
1
vote
1answer
56 views

Consider the region shared by ρ=8cos(φ) and ρ=4. Find the volume of the region.

Consider the region shared by ρ=8cos(φ) and ρ=4. Find the volume of the region. I know it involves a triple integral, but do not understand how to set up the integral.
2
votes
2answers
218 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
votes
2answers
144 views

Deriving equations of motion in spherical coordinates

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} ...
0
votes
0answers
103 views

Transforming an integral over $R^n$ to a radius and a directional vector (aka Spherical-Radial)

In several papers by John Monahan and Alan Genz there's mention of a spherical-radial transformation: $$ \int_{\mathbb{R}^n} f(\mathbf{x}) ~\mathrm{d}\mathbf{x} = \int_0^\infty ...
0
votes
0answers
57 views

“Great Circle” distance [duplicate]

Given two points on a sphere, then the "great circle distance" between two points is the length of the smallest arc of a great circle containing both points. Assume that $\Sigma$ is a sphere of radius ...
-2
votes
1answer
3k views

Volume of a Cylinder Using Cylindrical Coordinates and Triple Integration

Calculate the Volume $V$ of a right circular cylinder of radius $a$ and height $h$, using cylindrical coordinates and triple integration.
3
votes
1answer
137 views

Laplacian on Sphere of Function Only Depending on Angle Between Points

Consider a function $f:S^2 \to \mathbb{R}$ , with $S^2$ the unit $2$-sphere in $\mathbb{R}^3$. Let's say that $f$ depends only on the polar angle $\theta$ from the north pole (e.g., $f(r,\theta,\phi) ...
2
votes
1answer
358 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 ...
3
votes
1answer
411 views

integrating a vector over a sphere

I have the following triple integral in spherical coordinates $(r,\theta,\phi)$: $$\int_0^{2\pi}\int_0^\pi\int_0^RCr^3\hat\theta\cdot r^2dr\sin{\theta}d\theta d\phi$$ How do I handle the ...
1
vote
1answer
123 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 $T$ ...
0
votes
1answer
270 views

Triple Integral Spherical Coordinates

So I have to compute the triple integral of this: $\int\int\int \frac{1}{1+x^2+y^2+z^2}$ and it says the equation of the sphere is $ x^2 + y^2 + z^2 = z$ which is just an elongated sphere running ...
3
votes
1answer
878 views

How to integrate a vector function in spherical coordinates?

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 ...
3
votes
3answers
226 views

Integration with Spherical Coordinates

Use spherical coordinates to find the volume of the solid inside both $x^2+y^2+z^2=16$ and $z=(x^2+y^2)^{1/2}$.
0
votes
1answer
76 views

How to minimize the length of a curve on $S^2$

The length of a curve $\gamma$ starting from a point $p$ and ending at another point $q$ on $S^2$ is given by the formula $$l_{\gamma}(S^2)=\int_{0}^{1}\sqrt{(d\phi/dt)^2+ \sin^2\phi ...
4
votes
1answer
344 views

Stokes' and Divergence Theorem Problems

I have 2 questions on stokes and divergence theorem each. I think I have done both and I just want to make sure that I did them correctly. Question 1 Let $C$ be the boundary of the surface ...