Stokes theorem Definition and 83 Threads

Homework Statement From Calculus:Concepts and Contexts 4th Edition by James Stewart. Pg.965 #13 Verify that Stokes' Theorem is true for the given vector field F and surface S F(x,y,z)= -yi+xj-2k S is the cone z^2= x^2+y^2, o<=z<=4, oriented downwards Homework Equations The Attempt at a...

Do you agree that the following identity is true: \int_S (\nabla_\mu X^\mu) \Omega = \int_{\partial S} X \invneg \lrcorner \Omega where \Omega is volume form and X\invneg \lrcorner \Omega is contraction of volume form with vector X.

I find in a homework an alternative Stokes theorem tha i wasn't knew before. I wold like to know that it is really true. Any can give me a proof please? It is: Int (line) dℓ′× A = Int (surface)dS′×∇′× A

Homework Statement Consider an imaginary circular disc, of radius R, whose arbitrary orientation is described by the unit vector, \vec {n} , perpendicular to the plane of the disc. Define the component, in the direction \vec {n} , of the angular velocity, \vec {\Omega} , at a point in the...

Homework Statement Verify Stokes Theorem ∬(∇xF).N dA where surface S is the paraboloid z = 0.5(x^2 + y^2) bound by the plane z=2, Cis its boundary, and the vector field F = 3yi - xzj + yzk. The Attempt at a Solution I had found (∇xF) = (z+x)i + (-z-3)k r = [u, v, 0.5(u^2 + v^2)]...

Homework Statement magnetic field is azimuthal B(r) = B(p,z) \phi current density J(r) = Jp(p,z) p + Jz(p,z) z = p*exp[-p] p + (p-2)*z*exp[-p] z use stokes theorem to find B-filed induced by current everywhere in space Homework Equations stokes -...

Homework Statement \nabla \times f \vec{v} = f (\nabla \times \vec{v}) + ( \nabla f) \times \vec{v} Use with Stoke's theorem \oint _C \vec{A} . \vec{dr} = \int \int _S (\nabla \times \vec{A}) . \vec{dS} to show that \oint _c f \vec{dr} = \int \int _S \vec{dS} \times \nabla...

Homework Statement Prove that 2A=\oint \vec{r}\times d\vec{r} Homework Equations The Attempt at a Solution From stokes theorem we have \oint d\vec{r}\times \vec{r}=\int _{s}(d\vec{s}\times \nabla)\times \vec{r}= \int _{s}(2ds\frac{\partial f}{\partial x},-ds+ds\frac{\partial...

Homework Statement Use Stokes's Theorem to evaluate \int F · dr In this case, C (the curve) is oriented counterclockwise as viewed from above. Homework Equations F(x,y,z) = xyzi + yj + zk, x2 + y2 ≤ a2 S: the first-octant portion of z = x2 over x2 + y2 = a2 The Attempt at a...

Homework Statement A vector field A is in cylindrical coordinates is given. A circle S of radius ρ is defined. The line integral \intA∙dl and the surface integral \int∇×A.dS are different. Homework Equations Field: A = ρcos(φ/2)uρ+ρ2 sin(φ/4) uφ+(1+z)uz (1) The Attempt at...

For stokes theorem, can someone tell me why \hat{a} \bullet \vec{ds} = ds? My notes say it's because they are parallel, but I'm not sure what that means. Also to get things clear, Stokes theorem is the generalized equation of Green's theorem. The purpose of Stokes theorem is to provide a...

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 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 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...

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...

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 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...

"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...

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 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...

(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?

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?

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...

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...

I can't figure this out, help me NOW! :-p Just kidding. So anyways, here's the question: Part of stokes theorem has the following in it: \frac {\partial } {\partial x} ( Q + R \frac{\partial z}{\partial y} ) Which is written as: \frac { \partial Q }{\partial x} + \frac {...

I'm not sure if this post should go here or in the Calc setion, but I figure more knowledgeable people browse this form. This question is relating to 'directionality' of doing closed loop integrals. If you have some 2D wire structure, let's image it looks like a square wave, or a square well...

i thought stokes theorem (green's thm) was hard after reading it in spivak, who calls it trivial nonetheless. however lang showed it is indeed trivial in his analysis I. the same proof occurs in courant. I.e. the point is that the theorem is easy for a rectangle, where it follows...