What is Cylindrical: Definition and 821 Discussions
A cylinder (from Greek κύλινδρος – kulindros, "roller", "tumbler") has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. It is the idealized version of a solid physical tin can having lids on top and bottom.
This traditional view is still used in elementary treatments of geometry, but the advanced mathematical viewpoint has shifted to the infinite curvilinear surface and this is how a cylinder is now defined in various modern branches of geometry and topology.
The shift in the basic meaning (solid versus surface) has created some ambiguity with terminology. It is generally hoped that context makes the meaning clear. Both points of view are typically presented and distinguished by referring to solid cylinders and cylindrical surfaces, but in the literature the unadorned term cylinder could refer to either of these or to an even more specialized object, the right circular cylinder.
Suppose I have a cylindrical shell of radius r, height h. I can easily express the surface as
$$(r cos(\theta)) i + (r sin(\theta)) j + t k$$
$$0<\theta<2π , 0<t<h$$
For a conical surface of base rad ρ and height h,
$$z=kr -> z=k, r=ρ$$
$$k=\frac{h}{ρ}$$
Then the surface is
$$ \frac...
As you can see in this picture: This explanation "relation between the normal and the slope of a curve" is formulated here:
$$\frac{1}{\rho} \frac{d\rho }{d\psi }=\tan\left(\frac{\theta+\psi}{2}\right)$$
I got confused because I don't have the curve equation(regarding the slope of the curve...
We were taught that in cylindrical coodrinates, the position vector can be expressed as
And then we can write the line element by differentiating to get
.
We can then use this to do a line integral with a vector field along any path. And this seems to be what is done on all questions I've...
we know that flux is equal to the area integral of electric field dotted with dA and we can set this equal to charge enclosed divided by epsilon naught. Thus, in this case, the integral simplifies to E * A = (q_enclosed)/(ε_naught) when we choose a cylindrical gaussian surface with radius of r...
Hi everyone,
I am using MCNP6.2 and trying to set up a cylindrical coordinate in a reactor channel. The origin as the midplane of the channel.
In my attempt of setting up a cylindrical FMESH with the origin on the z-axis at the bottom of the channel (so z<0) I got this fatal error message...
Here is figure 2.16.6
Here is the picture I drew to set up the problem
My first question is if the reasoning and integrals are correct. I used Maple to compute the three integrals. The first two result in 0, which makes sense by symmetry.
Maple can't seem to solve the last integral.
Would method of separation of variables lead to a solution to the following PDE?
$$ \frac{1}{r} \frac{ \partial}{\partial r} \left( kr \frac{ \partial T}{ \partial r}\right) = \rho c_p \frac{\partial T }{ \partial t }$$
This would be for the transient conduction of a hollow cylinder, of wall...
so I was wondering. there is this normal force on the can from the path. And there's this formula to find the angle between the radial line and the tangent or also between the normal force and either the radial or theta axis. the formula is ##\psi = r/dr/d\theta##. The thing is that here they...
Hello !
I would like to consult you about this cylindrical nuclear reactor model that I have been thinking of with the idea of reducing the friction of the plasmas with the walls of the Toroidal nuclear fusion reactors that causes the plasma temperature to drop and the nuclear reactions to stop...
All the inductor components I’ve see are made with a circular core instead of a cylindrical core. Are there any advantages to this design in terms of field strength relative to input current (assuming the same number of turns of wire)?
In the field strength equation, is “coil length” always...
I know that Ienl for the inner cylinder is just I and the current density for the outer tube is J1= -I/(pi(Ra^2-Rb^2). I assume that the current through the enclosed portion of the conducting tube (I1) is equal to J1(A1) where A1 is the area of the enclosed portion of the conducting tube. I...
Using the equation above, I plugged in 5.5 inches for the radiu and 0.5 inches for the value of dr and then solved for the estimate of the change in volume, dV. However, the solution instead uses a value of 6 inches for the radius receiving a different estimate for the problem than I did. Is my...
Hi PF!
I have a function ##f(s,\theta) = r(s,\theta)\hat r + t(s,\theta)\hat \theta + z(s,\theta)\hat z##. How can I plot such a thing in Mathematica? Surely there's an easier way than decomposing ##\hat r, \hat \theta## into their ##\hat x,\hat y## components and then using ParametricPlot3D?
I calculate that \mbox{curl}(\vec{e}_{\varphi})=\frac{1}{\rho}\vec{e}_z, where ##\vec{e}_{\rho}##, ##\vec{e}_{\varphi}##, ##\vec{e}_z## are unit vectors of cylindrical coordinate system. Is there any method to spot immediately that ##\mbox{curl}(\vec{e}_{\varphi}) \neq 0 ## without employing...
What I have done:
(a) If we start at ##R_5## then we have ##\Delta V=-\int_{R_5}^{R_1}\vec{E}\cdot d\vec{l}=-(\int_{R_5}^{R_4}\vec{0}\cdot d\vec{l}+\int_{R_4}^{R_3}\frac{\lambda}{\varepsilon_0}dl+\int_{R_3}^{R_2}\vec{0}\cdot d\vec{l}+\int_{R_2}^{R_1}\frac{\lambda}{\varepsilon_0}dl=-\lambda(...
When I try to derive Gauss's law with a straight line of charge with density ##\lambda## through a cylindrical surface of length L and radius R,
$$\vec E = \frac{\lambda*L}{4\pi\epsilon*r^2}$$
$$A = 2\pi*r*L$$
$$\vec E*A = \frac{\lambda *L^2}{2\epsilon*r} \neq \frac{q_{enc}}{\epsilon}$$
What am...
Good day!
I am currently struggling with a very trivial question. During my studies, I operated with a parameter called "geometrical buckling" for neutrons and determined it in cylindrical coordinates. But thing is that we usually do not consider buckling's dependence on angle so its angular...
I was asked to give an estimate for a three dimensional permeate problem and I need an assist in how to setup a model equation.
Picture a 6" underground cylindrical bore. The depth of the cylindrical bore is 150 feet. The bore is sealed at the bottom and the permeability below the bore bottom...
Hey all,
I was citing a result from a review paper in my paper, and I think it's wrong. I would really appreciate an outside perspective if anyone has the time!
The result was for the electric field outside a metal rod (cylindrical waveguide, if you prefer) in vacuum. Here's the picture (you...
Hi, I´m quite lost and would appreciate guidance
I have solved for 2 tubes using Bernoulli´s equation before, but now how does it change?
Is it really going to rise water level inside? Why?
Let's say we have a cylindrical fuel pin with fissile material in the middle, followed by a gas gap and cladding material. It is being cooled by water on the outside. The temperature drop through the fissile material should be parabolic due to heat generation, and the temperature drops through...
Here is the initial problem and my attempt at getting Laplace solution. I get lost near the end and after some research, ended up with the Bessel equation and function. I don't completely understand what this is or even if this i the direction I go in.
This is a supplemental thing that I want to...
I pretend to use the ecuation twice, once for the interior and another for the vaccum, so if I use the cilindrical coordinates for \nabla_t^2 it results in two Bessel equations, one for the interior and another fot the vaccum.
In the vaccum, the fields should experiment a exponential decay, in...
Hi
If i calculate the vector product of a and b in cartesian coordinates i write it as a determinant with i , j , k in the top row. The 2nd row is the 3 components of a and the 3rd row is the components of b.
Does this work for sphericals or cylindricals eg . can i put er , eθ , eφ in the top...
Hi all
I am trying to calculate Source Level 'SL' of a underwater acoustic Cylindrical Array with multiple Transducers. The array has 03 Rings with 32 transducers in each Ring. The spacing between each ring is around 0.18m whereas Dia of each Ring is about 1m. Each transducer is being driven...
For question, in the solution when setting up the moment equation, why does the solution use Fa and Fb? Friction is coefficient times normal force and that can be found with the info given so why does the solution take it to be an variable?
I have a vector in cylindrical Coordinates:
$$\vec{V} = \left < 0 ,V_{\theta},0 \right> $$
where ##V_\theta = V(r,t)##.
The Del operator in ##\{r,\theta,z\}$ is: $\vec{\nabla} = \left< \frac{\partial}{\partial r}, \frac{1}{r}\frac{\partial}{\partial \theta}, \frac{\partial}{\partial z}...
In physics is usually defined that in cylindrical coordinates ##\varphi \in [0,2 \pi)##. In relation with Deckart coordinates it is usually written that
\varphi=\text{arctg}(\frac{y}{x}).
Problem is of course because arctg takes values from ##-\frac{\pi}{2}## to ##\frac{\pi}{2}##. What is the...
I'm trying to evaluate the following integral in cylindrical coordinates.
$$\int_0^6 \int_0^{\frac{\sqrt{2}}{2}}\int_x^{\sqrt{1-x^2}}e^{-x^2-y^2} \, dy \, dx \, dz$$
After attempting to set the bounds in cylindrical coordinates, I got
$$\int_0^6 \int_0^{\frac{\sqrt{2}}{2}}\int_{\rho \cos\varphi...
I got the answer for velocity and acceleration. But I don't know how to draw the shape of the particle's motion over time. How to draw it? should we change a,b,c,e into a numbers or not? or we may not to change a,b,c,e?
Please help me how to draw the shape of particle's motion over time?
Hi,
I was just working on a homework problem where the first part is about proving some formula related to Stokes' Theorem. If we have a vector \vec a = U \vec b , where \vec b is a constant vector, then we can get from Stokes' theorem to the following:
\iint_S U \vec{dS} = \iiint_V \nabla...
##\vec F= 2x^2y \hat i - y^2 \hat j + 4xz^2 \hat k ##
## \Rightarrow \vec \nabla \cdot \vec F= 4xy-2y+8xz##
Let's shift to a rotated cylindrical system with axis on x axis:
##x \to h, y \to \rho cos \phi, z \to \rho sin \phi ##
Then our flux, as given by the Divergence theorem is the volume...
I saw the solution of the light propagates in cylinder.. so in every solution there is the first order Gaussain function (the slandered one) times another function which gives I think the separation, both of them gives the intensity separation.. So what does that mean?! is it as I draw on the...
Let's say I have three modes in a fiber that is elliptical cylinder shaped (cylinder with elliptical facet), as in the image below (the source:Optical Engineering, 46(4), 045003 (2007)) so what is the equations that describe these fields..
1) Conservation of energy
## mg(R-r)(1-cos \theta_0) = \frac{1}{2}mv^2 + \frac{1}{2} I \omega^2 ##
because of pure rolling ## \omega = \frac{v}{r} ##
So i got:
## v = \sqrt{\frac{4}{3} g (R-r) (1-cos(\theta_0))} ##
this is how i got normal force:
2) ## N - mg = m \frac{v^2}{R-r} ##
where v is...
Similar to what is shown here, except the south side would be the weak side of the array.
A link to purchase one of these or at least the magnetic field arrangement would be very helpful. Thanks in advance.
What I've done so far:
From the problem we know that the curve c is a half-circle with radius 1 with its center at (x,y) = (0, 1).
We can rewrite x = r cos t and y = 1 + r sin t, where r = 1 and 0<t<pi. z stays the same, so z=z.
We can then write l(t) = [x(t), y(t), z ] and solve for dl/dt...
Let r = position of the electron = 6mm - 36.8μm; λ = mean free path traversed.
Integrate E(r) = Q/(2πϵLr) between the two shells gives:
V = [Q/(2πϵL)]*log(r/(r-λ))
I know that the question is asking for the voltage at which the electron energy will get to 23eV, but i am unsure how to get rid...