Stokes Definition and 295 Threads

Homework Statement Verify Stokes' theorem ∫c F • t ds = ∫∫s n ∇ × F dS in each of the following cases: (a) F=i z2 + j y2 C, the square of side 1 lying in the x,z-plane and directed as shown S, the five squares S1, S2, S3, S4, S5 as shown in the figure. (b) F = iy + jz + kx C, the three...

Hello, does anyone have reference to(or care to write out) fully rigorous proof of Stokes theorem which does not reference Differential Forms? I'm reviewing some physics stuff and I want to relearn it. I honestly will never use the higher dimensional version but I still want to see a full proof...

I am pretty confused about how to write Navier-Stokes Equation into conservation form, it seems that from my notes, first, the density term with the pressure gradient dropped out. and second, du^2/dx seems to be equal to udu/dx. Why is it so? I attached my notes here for your reference.

Hello, I have a question... With the Stokes Number you can derive if particles follow a streamline or not, right? Let's say I am in a windtunnel, the wind is coming from the east. And I have a tube in the wind tunnel (horizontally located) which is 180° opposite to the flow (the opening of the...

Can anyone recommend a good (peer reviewed) reference that discusses low Reynolds number flow around a cylinder? I'm specifically looking for derivations of the drag coefficient. The usual 'gold standard' references...

Let's say there is a 5 sided cube that is missing the bottom face. Obviously, there is a boundary curve at the middle of this cube that goes around the 4 sides, front, right, back, and left. This boundary curve forms the boundary of the top half of the cube with the 5 faces and the bottom...

Today I heard the claim that its wrong to use Stokes(more specificly divergence/Gauss) theorem when trying to get the Einstein equations from the Einstein-Hilbert action and the correct way is using the non-Abelian stokes theorem. I can't give any reference because it was in a talk. It was the...

Hello :) I am an engineer and I am trying to analyse a system which basically contains a cylindrical body free-falling through a body of static water beginning with zero velocity. I am ultimately trying to find what the velocity of the object would be at a depth of 20m. In order to do this I...

Can you derive the heat conduction equation from the navier stokes equations (particularly the energy eqn)?

Homework Statement Homework Equations The Attempt at a Solution I only know that they gave the parameterization of the circle: r(t) = <cost, sint, 2>. My problem is, did they already give the curl of F in the line integral? I don't understand why dx, dy, and dz are separated like that.

Hi all I am conducting a fluid analysis on water flowing through a subsea pipe. Having used navier stokes equation, i derived the equation for velocity in the r-direction (using cylindrical coordinates. But when initially solving the energy equation to determine temperature distribution I...

Hi there, I have recently been experimenting with solid metal spheres being let to fall through fluids of different viscosities and have recently been introduced to the 'Ladenburg correction'. This correction multiplies the measured velocity of the sphere to obtain the 'correct' velocity used...

This is a question on fluorescence spectroscopy so physics/chemistry. What causes a large Stokes shift in the spectra? I know what causes the shift in wavelength, i.e., a relaxation of vibrational states before de-excitation to the ground state, but what actually causes a (very) large Stokes...

So you can solve the Stokes equation for flow around a sphere to obtain the pressure in the fluid: p=p0-3nuacosθ/2r2 where n is the viscosity, u is the speed of the fluid (along the z axis) far away from the sphere, a is the radius of the sphere and r,θ are the usual spherical polar coordinates...

Divergence can be defined as the net outward flux per unit volume and can be explained using Gauss' theorem. (I read this in Feynman lectures Vol. 2) In the next page, He derives Stokes' theorem using small squares. The left side of equation represents the total circulation of a vector...

Hi, A potential magnetic field has no curl. According to the "curl theorem" or stokes theorem, a vector field with no curl does not close. Yet, Maxwell's equation tell us we shall not have magnetic monopoles, so the loops have to close... ? What am I missing to remove this apparent paradox of a...

Homework Statement F[/B]=(y + yz- z, 5x+zx, 2y+xy ) use stokes on the line C that intersects: x^2 + y^2 + z^2 = 1 and y=1-x C is in the direction so that the positive direction in the point (1,0,0) is given by a vector (0,0,1) 2. The attempt at a solution I was thinking that I could decide...

Homework Statement Homework Equations The path integral equation, Stokes Theorem, the curl The Attempt at a Solution [/B] sorry to put it in like this but it seemed easier than typing it all out. I have a couple of questions regarding this problem that I hope can be answered. First...

Homework Statement Evaluate the line integral of F dot dr by evaluating the surface integral in Stokes' Theorem with an appropriate choice of S. Assume that C has a counterclockwise orientation. F = (y2, - z2, x); C is the circle r(t) = < 3cos(t), 4cos(t), 5sin(t) >Homework Equations F[/B]...

Hello, In most of the papers like http://eprints.qut.edu.au/4763/1/4763.pdf one sees the raman spectra of crystals or molecules and the peaks are not further described (if they are stokes or antistokes). Do all the Peaks belong to the Anti-stokes-lines? And if it is so, why don't we see any...

I was watching a lecture in which the professor derived the Navier Stokes Equations for const density and viscosity. He however skipped a step and directly went from one equation to another without giving any explanation. I have attached an image file in which the 2nd equation is derived from...

Why do peaks of Stokes higher than peaks of anti-Stokes in graph of absorbance unit against wave number?

Homework Statement Hello i am trying to write the Stokes parameters as the amplitude of the averaged EM-wave We know the stokes parameters are {I,M,C,S} or {S0,S1,S2,S3} thus a column vector. And S0 is defined as S0=<E2x0>T+<E2y0>T S0 i understand how it was derived since Filter 0 is an...

Homework Statement Let a spherical object move through a fluid in R3. For slow velocities, assume Stokes’ equations apply. Take the point of view that the object is stationary and the fluid streams by. The setup for the boundary value problem is as follows: given U = (U, 0, 0), U constant, find...

I have problems understanding the proof of this lemma: $$\lambda \in \Lambda (m, n), \ \ \text{this means that it is an increasing function} \ \ \lambda: \{1,2,...,m\} \rightarrow \{1,...,n\}, \ \ \text{so} \ \ \lambda(1) < ... < \lambda(m)$$ $$p_{\lambda} : \mathbb{R}^m \ni (x_1, ..., x_m)...

Hi, I'm studying multivariable analysis using Spivak's book "calculus on manifolds" When I see this book, one strange problem arouse. Thank you for seeing this. Here is the problem. c0 , c1 : [0,1] → ℝ2 - {0} c : [0,1]2 → ℝ2 - {0} given by c0(s) = (cos2πs,sin2πs) : a circle of radius 1 c1(s) =...

Homework Statement Verify Stokes' theorem for the following: F=[y^2, x^2, -x+z] Around the triangle with vertices (0,0,1),(1,0,1),(1,1,1) Homework Equations \int\int_S(curlF)\cdot ndA=\int_C F\cdot r' ds The Attempt at a Solution [/B] For the LHS: curlF\cdot n=2x-2y \int\int_S(curlF)\cdot...

Homework Statement [/B] (a) Show that for an incompressible flow the velocity potential satisfies ##\nabla^2 \phi = 0##. Show further the relation for the potential to be ## \frac{\partial \phi}{\partial t} + \frac{\nabla \phi \dot \nabla \phi}{2} + \frac{p}{\rho} + gz = const.## (b)Write out...

Homework Statement Verify Stokes' theorem for the given surface S and boundary dS and vector fields F S = x2+y2+z2, z≥0 dS= x2+y2=1 F = <y,z,x> Homework Equations Stokes' theorem: ∫∫(∇×F)dS = ∫F⋅ds The Attempt at a Solution 1. Curl of F: ∇×F = <-1,-1,-1> 2. After getting the curl, I just...

Hi there, I have a question about incompressible Stokes flow in a channel between solid walls (with no-slip boundary conditions at ##y = 0, L_y##). It is my intuition that, if the flow direction is ##x## (periodic), and the direction normal to the walls is ##y##, then there cannot be a net...

I was doing some research into my coursework (don't worry, this isn't a real 'help!' thread) on Stokes law, and I found this equation on this forum (posted in 2010), regarding the effect of the walls of a pipe on the calculated value for viscosity when a sphere is dropped through that pipe...

Homework Statement Hi there. For my A2 physics coursework I have been doing an experiment into stokes law, in which I dropped ball bearings of various diameters into a tube filled with a liquid, worked out their terminal velocities and then used Stokes Law to calculate the viscosity of the...

Homework Statement . In the diagram below, the line AB is at x = 1 and the line BD is at y = 1. Use Stokes’ theorem to find the value of the integral ∫s2(∇xV)⋅dS where S2 where S2 is the curved surface ABEF, given that AF and BE are straight lines, and the curve EF is in the y-z plane (i.e...

Homework Statement Hello I was given the vector field: \vec A (\vec r) =(−y(x^2+y^2),x(x^2+y^2),xyz) and had to calculate the line integral \oint \vec A \cdot d \vec r over a circle centered at the origin in the xy-plane, with radius R , by using the theorem of Stokes. Another thing, when...

this is a rather stupid question regarding preliminaries for the definition of boundaries the question is whether every closed n-1 dim. closed submanifold C of an arbitrary n-dim. manifold defines a volume V; i.e. whether \partial V = C can be turned around such that V is defined as the...

Hi! I am looking for a very rigorous book on some of the topics covered in Calculus of Multiple Variables. My University uses the last part of Adams "Calculus: a complete course" and I found the presentation therein more fit for people needing to know enough to perform the calculations than for...

It is a nonsense use the generalized Stokes' theorem in right side of Faraday's Law? we know this is true...\displaystyle \oint_{\partial \Sigma}\vec{E}\cdot dl=-\int_{\Sigma} \frac{d\vec{B}}{dt}\cdot d\vec{A}\Rightarrow \int_{\Sigma}\vec{\nabla}\times\vec{E}\cdot d\vec{A}=-\int_{\Sigma}...

Homework Statement Find ∫CF⃗ ⋅dr⃗ where C is a circle of radius 2 in the plane x+y+z=3, centered at (2,4,−3) and oriented clockwise when viewed from the origin, if F⃗ =5yi⃗ −5xj⃗ +4(y−x)k⃗ Homework Equations Stokes theorem. ∫curl F ⋅dS The Attempt at a Solution For the curl I...

Would one expect the emission and absorption spectral lines for Hydrogen to be at slightly different frequencies ? So if electrons are excited to higher energy levels on absorption of photons and then fall back down again emitting photons, would there be any difference in frequency. Assuming...

Definition/Summary Stokes' Theorem (sometimes called the "Generalized Stokes' Theorem") is a theorem pertaining to integration of differential forms in differential geometry that vastly generalizes several theorems in analysis and calculus. Simply stated, it says that the integral of the...

Homework Statement I have been given an experiment to perform, whereby I drop steel ball bearings into a glass measuring cylinder, and time how long it takes for the ball bearing to fall a set distance. From this I should be able to work out the viscosity coeffienct of the glycerol. I have...

OK, so I know stokes theorem states that I can turn ##\int \int_{S}F\cdot dS## into ##\int \int (-P\frac{\partial g(x,y)}{\partial x} -Q\frac{\partial g(x,y)}{\partial y} + R)dxdy## but I have a problem that I'm working on that's screaming change of variables (to spherical), but I'm not sure...

Homework Statement F= <y,z,x> S is the hemisphere x^2 + y^2 + z^2 = 1, y ≥ 0, oriented in the direction of the positive y-axis. Verify Stokes' theorem. Homework Equations The Attempt at a Solution So I completed the surface integral part. I'm trying to do the line integral part of Stokes'...

Hello. My first time posting here. So... My question is kinda hard to explain but I will try to. So we all know about the Kelvin-Stokes theorem (not talking about manifolds here) : And we also know about Ostrogradsky/Gauss Theorem ...

hey pf! i had a question. namely, in the continuity equation we see that \frac{\partial}{\partial t}\iiint_V \rho dV = -\iint_{S} \rho \vec{v} \cdot d\vec{S} and we may use the divergence theorem to have: \frac{\partial}{\partial t}\iiint_V \rho dV = -\iiint_{V} \nabla \cdot \big( \rho...

Homework Statement Compare the COMSOL results to the analytical solution for laminar flow between flat plates. Assume no effect of gravity on the flow (g = 0). The comparison will involve obtaining the velocity at a point in the flow field and the ΔP/L term. For example, you can compare the...

Homework Statement Consider steady, incompressible, parallel, laminar flow of a film of oil falling down an infinite vertical wall (Figure P-1). The oil film thickness is “h” and gravity acts in the negative Z-direction (downward on the figure). There is no applied pressure driving the flow –...

Hey! :o $\overrightarrow{F}=M \hat{i}+ N \hat{j}+ P \hat{k}$ To prove the Stokes' Theorem we apply Green's Theroem at $ABE$, $BCE$, $CDE$. $(\oint_{ABE}+\oint_{BCE}+\oint_{CDE}){ \overrightarrow{F}}d \overrightarrow{R}=\iint_{ABCDE}{ \nabla \times \overrightarrow{F} \cdot \hat{n}}d \sigma$...

How would one prove Stokes' Theorem? I'm 15. I learned about Stokes' Theorem recently and I have a decent understand of it, but I thought that it would be useful to know it's derivation. Thanks for your help, PF.

hey pf! so when deriving navier stokes we have, from Newton's second law, \sum \vec{F} = m\frac{d \vec{V}}{dt} when deriving the full navier stokes (constant density) the acceleration term can be thought of as two pieces: a body change of velocity within the control volume and a mass flow...