Stokes Definition and 268 Threads

Hello, I read some fluidmechanics and there was something I didn't understand. The shear stress in a Newtonian fluid is tau=viscosity*dV/dy, (no need to be dy, but dx and dz also can do.) A shear component called tau(xx) came up, I have two questions about this component: 1. Shear is...

Homework Statement Use the surface integral in stokes theorem to find circulation of field F around the curve C. F=x^2i+2xj+z^2k C: the ellipse 4x^2+y^2=4 in the xy plane, counterclockwise when viewed from above Homework Equations stokes theroem: cirlulation=double integral of nabla...

Homework Statement I have to use stokes' theorem and calculate the surface integral, where the function F = <xy,2yz,3zx> and the surface is the cube bounded by the points (2,0,0), (0,2,0),(0,0,2),(0,2,2),(2,0,2),(2,2,0),(2,2,2). The back side of the cube is open. [/B] Homework...

Hi I'm practicing for my exam but I totally suck at the vector fields stuff. I have three questions: 1. Compute the surface integral \int_{}^{} F \cdot dS F vector is=(x,y,z) dS is the area differential Calculate the integral over a hemispherical cap defined by x ^{2}+y ^{2}+z...

use the stokes theorem to evaluate the surface integral [curl F dot dS] where F=(x^2+y^2; x; 2xyz) and S is an open surface x^2+y^2+z^2=a^2 for z>=0. So i guess its a hemisphere of radius a lying on x-y plane. I don't see however how to take F dot dr. What is this closed curve dr bounding...

Homework Statement Use stokes theorem F = xyi + yzj + zxk on triangle 1,0,0,,,,,,,0,1,0,,,,,,0,0,1 Homework Equations The Attempt at a Solution First i found the curl F curl F = -yi - zj - xk Then i found the equation of the plane for the triangle z = g(xy) = 1 -...

Homework Statement Given F = xyz i + (y^2 + 1) j + z^3 k Let S be the surface of the unit cube 0 ≤ x, y, z ≤ 1. Evaluate the surface integral ∫∫(∇xF).n dS using a) the divergence theorem b) using Stokes' theorem Homework Equations Divergence theorem: ∫∫∫∇.FdV = ∫∫∇.ndS Stokes...

Homework Statement F = xi + x3y2j + zk; C the boundary of the semi-ellispoid z = (4 - 4x2 - y2)1/2 in the plane z = 0Homework Equations (don't know how to write integrals on here, sorry) double integral (curl F) . n dsThe Attempt at a Solution curl F = 3y2x2k n = k curl F . n = 3y2x2 So I...

Homework Statement Calculate \int \int _{S}\nabla \times \overline{F} \cdot \hat{N}dS where \overline{F} = 3y\hat{i} - 2xz\hat{j} + (x^{2}-y^{2})\hat{k} and S is a hemispherical surface x2 + y2 + z2 = a2, z ≥ 0 and \hat{N} is a normal of the surface outwards. Can you use Stokes' theorem...

Homework Statement Please help me to check whether I did the right working for this problem. Thanks. The numerical answer is correct but I'm not very sure if the working is correct also. Find \int y dx + z dy + x dz over the closed curve C which is the intersection of the surfaces whose...

Hello all. I'm investigating a little bit about stokes law in order to understand the settling velocity of falling particles and on the net i encountered with 2 different formulas and i simply can't find the reason why they are different. every formula gives me a different answer. The 2...

Homework Statement This is a question about stokes theorem in general, not about a specific problem. Directly from lecture: "If S has no boundry (eg. if S is the boundry of a solid region) then \int\int_{S}Curl(\stackrel{\rightarrow}{F})\bullet ds = 0 " because apparently "no boundry C...

Homework Statement Faraday’s Law can be written as: \oint_P \vec{E} \cdot \vec{dl} = -\frac{d}{dt}\Phi Where \Phi is the magnetic flux. Use Stokes’ theorem to obtain the equvilant Maxwell equation (i.e. Faraday’s Law in differential form). Homework Equations Stokes' Law...

in my GR book, it claims that integral of a covariant divergence reduces to a surface term. I'm not sure if I see this... So, is it true that: \int_{\Sigma}\sqrt{-g}\nabla_{\mu} V^{\mu} d^nx= \int_{\partial\Sigma}\sqrt{-g} V^{\mu} d^{n-1}x if so, how do I make sense of the d^{n-1}x term? would...

Homework Statement let F be vector field: \[\vec F = \cos (xyz)\hat j + (\cos (xyz) - 2x)\hat k\] let L be the the curve that intersects between the cylinder \[(x - 1)^2 + (y - 2)^2 = 4 \] and the plane y+z=3/2 calculate: \[\left| {\int {\vec Fd\vec r} } \right|\] Homework Equations...

Homework Statement let F be vector field: \[\vec F = \cos (xyz)\hat j + (\cos (xyz) - 2x)\hat k\] let L be the the curve that intersects between the cylinder (x - 1)^2 + (y - 2)^2 = 4 and the plane y+z=3/2 calculate: \[\left| {\int {\vec Fd\vec r} } \right|\] Homework Equations in...

Homework Statement Use Stokes Theorem to compute \int_{L}^{} y dx + z dy + x dx where L is the circle x2 + y2 + z2 = a2, x + y + z = 0 The Attempt at a Solution I feel like this problem shouldn't be that hard but I can't get the right answer: (pi)a2/3. I calculated the curl of F as...

Homework Statement Hey. I need to find the circulation of F through out the line L. I know I need to use stokes theorem, the problem is, how do I find the area of L? I mean, I know the intersection line of the sphere and the plot looks like an ellipse on the XY surface, but how do I find...

Homework Statement I got stuck using the Stokes' theorem, the problem is at the bottom of the pic. I found the Curl of F, and also the normal of the Triangle. As you can see, I ended up with an area integer with 3 variables, how do I solve this? did I do it right? Homework Equations...

"Stokes theorem" equivalent for cross product line integral Homework Statement I am aware that the vector path integral of a closed curve under certain conditions is equivalent to the flux of of the curl of the vector field through any surface bound by the closed path. In other words, Stokes...

Homework Statement A = sin(\phi/z)* a(\phi) I'm having problem verifying Stokes Theorem. I have to verify the theorem over the upper half of the sphere with radius b and the sphere is centered at the origin. The problem also says z > = 0 Could someone help me with this.

I wasn't sure whether to put this in Aerospace, but decided on physics in the end. 1.) How do you factor a chemical reaction into the solution for the Navier Stokes equations? More precisely, how can you include the affects of a heat absorbing (endothermic), or heat releasing (exothermic)...

Hi, I've been doing some work with the NS equations. I've read a few papers by fellow undergrads that imply a relationship between the helmholtz-hodge decomposition and the pressure equation. As far as I can see, they're both separate ways of resolving the problem of keeping the flow...

Homework Statement Let \vec{F}=xyz\vec{i}+(y^{2}+1)\vec{j}+z^{3}\vec{k} And let S be the surface of the unit cube in the first octant. Evaluate the surface integral: \int\int_{S} \nabla\times \vec{F} \cdot \vec{n} dS using: a) The divergence theorem b) Stoke's theorem c)...

[SOLVED] Calculus - Stokes' theorem Homework Statement I have F in Cartesian coordinates (F is a vector): F = (y , x , x*z) and a curve C given by the quarter-circle in the z-plane z=1 (so t : (cos(t) , sin(t) , 1) for t between 0 and Pi/4). I have found the line integral, and it equals...

I'm trying to find the viscosity of some glycerol that we dropped various steel balls down using the equation: V = [2r^2 (p – σ) g] / 9η I put in these values: p = 7800 kg m-3 σ = 1200 kg m-3 g = 9.8 m s-2 And ended up with the equation. η = 129360r^2 / 9V My problem is that I...

(This is from the perspective of Geophysical Fluid Dynamics) In the Navier Stokes equations I am confused as to why there is both a pressure term and a gravity term. Is this pressure resulting from differences in densities and temperature differences alone? I would think that the gravity term...

If you start with the two dimensional green's theorem, and you want to extend this three dimensions. F=<P,Q> Closed line integral = Surface Integral of the partials (dP/dx + dQ/dy) da seems to leads the divergence theorem, When the space is extended to three dimensions. On the...

Homework Statement Compute the line integral of v = 6i + yz^2j + (3y + z)k along the path (0,0,0) -> (0,1,0) -> (0,0,2) -> (0,0,0). Check your answer using Stokes' Thm Homework Equations The Attempt at a Solution I've tried breaking into three pieces. The first with dx = dz =...

-------------------------------------------------------------------------------- I am having some issues with this problem... F=( x+y, y+z, z+x) bounded by the plane with vertices at {2,0,0},{0,2,0},{0,0,2} I need to do both sides of stokes thm and I am running into problems when I try...

Hi - Is it possible to derive Stokes law or is it an emprirical law.? http://en.wikipedia.org/wiki/Stokes'_law I was thinking of using the Navier-Stokes equations but i don't want to start out if it impossible.. Thx.

Homework Statement Use the surface integral in Stokes' theorem to calculate the circulation of field F around the curve C in the indicated direction. (3) F = (y)i + (xz)j + (x^2)k. C: Boundary of the triangle cut from the plane x+y+z=1 by the first octant, counterclockwise as seen...

Homework Statement Let C be the closed curve that goes along straight lines from (0,0,0) to (1,0,1) to (1,1,1) to (0,1,0) and back to (0,0,0). Let F be the vector field F = (x^2 + y^2 + z^2) i + (xy + xz + yz) j + (x + y + z)k. Find \int F \cdot dr By Stokes Theorem, I know that I can...

Hi, I have some questions which I encountered during my thesis-writing, I hope some-one can help me out on this :) First, I have some problems interpreting coordinate-transformations ( "active and passive") and the derivation of the Equations of Motion. We have S = \int L(\phi...

(1) using stokes theorem and cutting the surface into 2 parts how can we prove that \int curl A.n dS = 0 assume the surface "S" to be smooth and closed, and "n" is the unit outward normal as usual. (2) How can you prove \int curl A.n dS = 0 using the divergence theorem?

Homework Statement Solve the following question by using Stokes' Theorem. (Line integral on C) 2zdx + xdy + 3ydz = ? where C is the ellipse formed by z = x, x^2 + y^2 = 4. Homework Equations The Attempt at a Solution We have the vector A=(2z,x,3y) which is cont...

Hello, I want, for obscur reasons which would lead us too far to explain, to split my flow into two component, one steady and another one non-steadyv = v_0 + v' I'm looking for a simple equation governing the evolution of this non steady components. The complete momentum equation gives...

Hi, I have 2 little questions and hope to find some clarity here. It concerns some mathematics. 1) Is the variation of the metric again a tensor? I have the suspicion that it's not, because i would say that it doesn't transform like one. How can i get a sensible expression then for the...

I was wondering as to how to prove stokes theorem in its general and smexy form.Also what is the intuition behind it(more important) aside from the fact that its a more general form of the other theorems from vector calculus?

Stokes' Theorem - Limits of Integration - Urgent! Please give a hand :) Homework Statement Assume the vector function \vec{A} = \hat{a}_x \left( 3x^2 y^3 \right) + \hat{a}_y \left( -x^3 y^2 \right) Evaluate \int \left( \nabla \times \vec{A} \right) \cdot d\vec{s} over the triangular...

Here's my problem: Take u=(x^3)+(y^3)+(z^2) and v=x+y+z and evaluate the surface integral double integral of grad(u) x grad(v) ndS where x is the cross product and between the cross product and the ndS there should be a dot product sign. The region S is the hemisphere x^2+y^2+z^2=1 with z...

Hello, I haven't studied PDEs much yet, but checked out what the Navier Stokes equations are. I think I understood meaning of the terms in Navier Stokes equations, and what is their purpose in defining the time evolution of velocity of the fluid, but I couldn't see any conditions for the...

For the surface S (helicoid or spiral ramp) swept out by the line segment joining the point (2t, cost, sint) to (2t,0,0) where 0 is less than or equal to t less than or equal to pi. (a) Find a parametrisation for this surface S and of the boundary A of this surface. I can only guess that...

Please check my work for the following problem: Homework Statement By subsituting A(r) = c \phi(r) in Gauss's and Stokes theorems, where c is an arbitrary constant vector, find these two other "fundamental theorems": a) \int_{\tau} \nabla \phi d \tau = \int_{S} \phi ds b) - \int_{S} \nabla...

Just a couple quick conceptual questions about Stokes' Theorem (maybe this belongs in the non-homework math forum?). Does Stokes' theorem say anything about circulation in a field for which the curl is zero? I would think that all it says is that there is no net circulation. Also, if F is a...

Would anybody derive stokes law for me or show me how to do it?

Test Stokes' Theorem for the function \vec{v} = (xy) \hat{x} + (2yz) \hat{y} + (3yz) \hat{z} / for the triangular shaded region \int_{S} (\grad \times v) \dot da = \oint_{P} v \bullet dl for the left hand side \int_{0}^{2} \int_{0}^{2} (-2y \hat{x} - 3z \hat{y} - x \hat{z})...

Hello all, http://img244.imageshack.us/img244/218/picture8ce5.png I am completely new to this stokes theorem bussiness..what i have got so far is the nabla x F part, but i am unsure of how to find N (the unit normal field i think its called). any suggestions people? i get that nabla...

Hi, i can't seem to figure out how stokes theorem works. I've run through a lot of examples but i still am not having any luck. Anyway, some advice on a particular problem would be greatly appreciated. The problem is: F(x,y,z) = <2y,3z,-2x>. The surface is the part of the unit...

For the following vector fields F , decide whether it is conservative or not by computing curl F . Type in a potential function f. F(x,y,z)=-3xi-2yj+k f(x,y,z)=? I'm not sure what the problem is asking. calcualting curl F needs integration and a boundary. I don't know why they ask for...