2
votes
2answers
66 views

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: ...
1
vote
1answer
27 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
2answers
54 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 ...
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
1answer
29 views

Curl of a function with only angular dependence

Let a function in spherical coordinates $$\vec F(\vec r) = \int{ d^3\vec r\,' \vec j(\vec r\,') } \,e^{-ik\hat r \cdot \vec r \,'}$$ Where $\vec j$ is a vector function. So $\vec F$ only depends on ...
2
votes
1answer
43 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 ...
0
votes
0answers
45 views

How to find $\nabla$ in spherical coordinates

I want to derive(!) just a few components like the $\hat{e}_r$ component of the divergence operator in spherical coordinates and the $\hat{e}_{\phi}$ component of the curl operator in spherical ...
0
votes
0answers
49 views

Mean value of a function over the n-sphere's surface.

We know that we can use the bloch sphere to represent an unitary vectors $v$ in $\mathbb{C}^{2}$, due to the fact $su(2) \approx so(3)$. Then, if we have the function $f:\mathbb{C}^{2} \rightarrow ...
1
vote
1answer
75 views

Laplace operator in Spherical Coordinates, a formal approach

I'd like to show the well-known formula of the Laplacian operator for euclidean $\mathbb{R}^3$ in spherical coordinates: $$ \Delta U = \frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial ...
0
votes
2answers
126 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 ...
2
votes
3answers
183 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 ...
1
vote
2answers
215 views

How to calculate volume of a cylinder using triple integration in “spherical” co-ordinate system?

Lets have a cylinder given by $x^2+y^2=1$ which is cut from the top by plane $z=2$ and bottom by $z=-2$.I am having problem regarding the limits of ρ for the equation ∭ ρ sin^2ϕ dρ dϕ dθ where ϕ is ...
1
vote
1answer
52 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
1answer
125 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 ...
1
vote
1answer
416 views

Express spherical coordinates with different centers in terms of each other.

Imagine that you have two spheres with a distance $R$ from one center to the other one. Now, it is well known how one would get the cartesian position vector of each point in sphere 1 by using ...
0
votes
1answer
242 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 ...
0
votes
1answer
74 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 ...
0
votes
0answers
104 views

two points on a unit sphere

Consider the two vectors to the points on the unit sphere, $${\bf v}_i=(\sin\theta_i\cos\varphi_i,\sin\theta_i\sin\varphi_i,\cos\theta_i)$$ with $i=1,2$. Use the dot product to get the angle $\psi$ ...
0
votes
2answers
478 views

Identifying a surface $\rho^2\cos(2\phi)-1=0$

I need convert this spherical expression, to a rectangular form (specific surface). $$\rho^2\cos(2\phi)-1=0$$ Thanks for a while.
2
votes
1answer
153 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 + ...
0
votes
1answer
76 views

Spherical Coordinates

Let Q be the region above by the plane $8z=4-x-y$ and below by the cone $64z^2=x^2+y^2$. How would I setup the triple integral to find volume of Q, using spherical coordinates? I just need help with ...
2
votes
2answers
154 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 ...
1
vote
1answer
73 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 ...
3
votes
0answers
792 views

Spherical coordinates grad and div.

Struggling with the following: Prove the identity $$ \nabla = e_{r}(e_{r} \cdot \nabla) + e_{\theta}(e_{\theta} \cdot \nabla) + e_{\phi}(e_{\phi} \cdot \nabla).$$ Given the vector fields ...
0
votes
1answer
335 views

Finding the sign of $\phi$ in spherical coordinates

I know its a little silly, but I got the wrong sign several times. Just to be clear, $z=r\cos(\phi), -\frac{\pi}{2}\leq\phi\leq\frac{\pi}{2}$ when converting from cartesian to spherical. So, how do I ...
4
votes
1answer
683 views

How to integrate by parts in spherical coordinates

I'm running into some troubles with the integration of a spherically symmetric 3D function. I'm having the following expression to evaluate : $$ I=\int_0^{2\pi} d\phi \int_0^\pi \sin\theta ...