Stokes theorem Definition and 81 Threads

I identified this as a Stokes theorem problem. I first took the curl of the vector field and got ##\langle4,4,-6\rangle##. The surface integral becomes $$\int_S\langle4,4,-6\rangle\cdot\text{d}^{2}\textbf{r}$$ Here, I define ##\text{d}^{2}\textbf{r}## to be the differential area for an...

Does this mean we can write the following? $$\mathcal{E}=\oint_C \vec{E}\cdot d\vec{r}+\oint_C \vec{v}\times\vec{B}\cdot d\vec{r}\tag{3}$$ I haven't seen an equation like the above in my books and notes yet. What I have seen are two cases. In one case, we have a uniform magnetic field and we...

To my understanding, an orientation can be expressed by choosing a no-where vanishing top form, say ##\eta := f(x^1,...,x^n) dx^1 \wedge ... \wedge dx^n## with ##f \neq 0## everywhere on some manifold ##M##, which is ##\mathbb{H}^n := \{ x \in \mathbb{R}^n : x^n \geq 0 \}## here specifically. To...

Hello everyone, as part of my bachelor studies, I need to attend a seminar with the aim to prepare a presentation of about an hour on a certain topic. I have chosen the presentation about the generalized Stokes theorem, i.e. $$ \int_M d\omega = \int_{\partial M} \omega. $$ After hours of...

##curl([x^2z, 3x , -y^3],[x,y,z]) =[-3y^2 ,x^2,3]## The unit normal vector to the surface ##z(x,y)=x^2+y^2## is ##n= \frac{-2xi -2yj +k}{\sqrt{1+4x^2 +4y^2}}## ##[-3y^2,x^2,3]\cdot n= \frac{-6x^2y +6xy^2}{\sqrt{1+4x^2 + 4y^2}}## Since ##\Sigma## can be parametrized as ##r(x,y) = xi + yj +(x^2...

My answer:

I do not understand how can I parameterize the surface and area and line differentials.

Hi, My question pertains to the question in the image attached. My current method: Part (a) of the question was to state what Stokes' theorem was, so I am assuming that this part is using Stokes' Theorem in some way, but I fail to see all the steps. I noted that \nabla \times \vec F = \nabla...

The question provides the vector field (xy, 2yz, 3zx) and asks me to confirm Stokes' theorem (the vector calc version) but I am trying to use the generalized differential forms version. So, I am trying to integrate \omega = xy\,dx + 2yz\,dy + 3zx\,dz along the following triangular boundary...

Homework Statement Homework Equations Stokes theorem $$\int_C \textbf{F} . \textbf{dr} = \int_S \nabla \times \textbf{F} . \textbf{ds}$$ The Attempt at a Solution I have the answer to the problem but mine is missing a factor of$$\sqrt 2 $$ I can't seem to find my error

Homework Statement [/B] Homework Equations Navier strokes theorem The Attempt at a Solution May I ask why would there suddenly a "h" in the highlighted part? "h" wasnt existed in the previous steps, which C2=0 shouldn't add height of the liquid as a constant in the formula... thanks

Homework Statement Homework Equations Stokes Theorem The Attempt at a Solution I'm having a tough time "cancelling" out integrals from both sides of an equation (if possible). On the right hand of the equation, we know since it is a closed curve, that Stoke's Theorem applies and we can...

Hey! :o I want to calculate $\int_{\sigma}\left (-y^3dx+x^3dy-^3dz\right )$ using the fomula of Stokes, when $\sigma$ is the curve that is defined by the relations $x^2+y^2=1$ and $x+y+z=1$. Is the curve not closed? Because we have an integral of the form $\int_{\sigma}$ and not of the form...

Hello! I am reading this paper and on page 9 it defines the De Rham's period as ##\int_C \omega = <C,\omega>##, where C is a cycle and ##\omega## is a closed one form i.e. ##d\omega = 0##. The author says that ##<C,\omega>:\Omega^p(M) \times C_p(M) \to R##. I am a bit confused by this, as...

So the main thing I'm wondering is given a question how do we determine whether to use one of the fundamentals theorems of vector calculus or just directly evaluate the integral, and if usage of one of the theorems is required how do we determine which one to use in the situation? Examples are...

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...

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...

I have this question: Is it possible to define an orientation for a null submanifold with boundary? In that case, is possible to use Stokes' theorem? In particular, there is a way to define a volume form on that submanifold?

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...

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 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...

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 Calculate the line integral: F = <xz, (xy2 + 2z), (xy + z)> along the curve given by: 1) x = 0, y2 + z2 = 1, z > 0, y: -1 → 1 2) z = 0, x + y = 1, y: 1→0 3) z = 0, x-y = 1, y: 0 → -1 Homework Equations The Attempt at a Solution I don't think the...

Homework Statement I have to control stokes theorem( I have to calculate line-and surface integral. I have a vectorfield a=(3y,xz,yz^2). And surface S is a paraboloid 2z=x^2+y^2. And it is limited by plane z=2. For line integral the line is a circle C: x^2+y^2=4 on the plane z=2. Vector n is...

Homework Statement Use the Stokes' Theorem to show that [SIZE="4"]\intf(∇ X A) dS = \int(A X ∇f) dS + \ointf A dl Homework Equations Use vector calculus identities. Hint given : Start with the last integral in the above relation. The Attempt at a Solution To be honest, I really...

I am wondering why z = -sin(t) and not sin(t)

Homework Statement Hey guys, I'm having trouble finding the n in stokes theorem. For example, F(x,y,z)= z2i+2xj-y3; C is the circle x2 + y2=1 in the xy-plane with counterclockwise orientation looking down the positive z-axis. ∫∫CurlF*n I know the curl is -3y2i+2zj+2k The...

Homework Statement Verify stokes theorem where F(xyz) = -yi+xj-2k and s is the cone z^2 = x^2 + y^2 , 0≤ Z ≤ 4 oriented downwards Homework Equations \oint_{c} F.dr = \int\int_{s} (curlF).dS The Attempt at a Solution Firstly the image of the widest part of the cone on the xy plane is the...

Homework Statement Use Stokes Theorem to evaluate the integral\oint_{C} F.dr where F(x,y,z) = e^{-x} i + e^x j + e^z k and C is the boundary of that part of the plane 2x+y+2z=2 in the first octant Homework Equations \oint_{C} F.dr = \int\int curlF . dS The Attempt at a Solution So first...

Homework Statement Use stokes theorem to elaluate to integral \int\int_{s} curlF.dS where F(x,y,z)= x^2 z^2 i + y^2 z^2 j + xyz k and s is the part of the paraboliod z=x^2+ y^2 that lies inside the cylinder x^2 +y^2 =4 and is orientated upwards Homework Equations The Attempt at a...

I am a little confused about how to generally go about applying Stokes's Theorem to cylinders, in order to calculate a line integral. If, for example you have a cylinder whose height is about the z axis, I get perfectly well how to parameterize the x and y components, using polar coordinates...

Homework Statement Prove that ## \oint_{\partial S} ||\vec{F}||^2 d\vec{F} = -\int\int_S 2 \vec{F}\times d\vec{A} ## Homework Equations Identities: ##\nabla \times (||\vec{F}||^2 \vec{k}) = 2\vec{F} \times \vec{k} ## For ##\vec{k} ## constant i.e. ## \nabla \times \vec{k} = 0 ## Stokes...

We're given x^2+2*y^2=1. so x^2=1-2y^2 now using distance formula d^2=x^2+y^2 since x^2=1-2y^2, substituting it in the distance formula we get: d^2=1-2y^2+y^2=1-y^2; differentiating and then setting the eq to 0 we get; 0=-4y or y=0. now x^2=1-2y^2=1 so x=+-1 so point having...

Homework Statement The vector field F is defined in 3-D Cartesian space as F = y(z^2−a^2)i + x(a^2− z^2)j, where i and j are unit vectors in the x and y directions respectively, and a is a real constant. Evaluate the integral  Integral:(∇ ×F)·dS, where S is the open surface of the...

Homework Statement F= xi + x3y2j + zk C is the boundary of the semi-ellipsoid z=√(4-4x2-y2) in the plane z=0 Homework Equations Stokes theorem states: ∫∫(curlF ° n)dS The Attempt at a Solution I found the curl of the F to be 3x2y2k I found that the dot product of CurlF and n =...

Homework Statement This is not actually a homework question, just a question I ran into while studying for my math final. When I am using stokes theorem: ∫∫(curlF ° n)dS I have listed in my notes from lecture that there are time when it is applicable to replace dS with an easier...

Acording to the non-Abelian stokes thoerem http://arxiv.org/abs/math-ph/0012035 I can transform a path ordered exponential to a surface ordered one. P e\oint\tilde{A}= P e∫F where F is some twisted curvature;F=U-1FU, and U is a path dependet operator.So, I have a system where every element...

Homework Statement V.Field F(x,y,z)=<x^2 z, xy^2, z^2> where S is part of the plane x+y+z=1 inside cylinder x2 + y2 =9 Homework Equations Line integrals, Stokes Theorem, Parametrizing intersections... The Attempt at a Solution I found the answer to be 81pi/2 using stoke's theorem...

Homework Statement verify Stokes theorem for the given Surface and VECTOR FIELD x2 + y2+z2=4, z≤0 oriented by a downward normal. F=(2y-z)i+(x+y2-z)j+(4y-3x)k Homework Equations ∫∫S Δ χ F dS=∫ ∂SF.ds the triangle is supposed to be upside down. The Attempt at a Solution myΔχF =...

Is it? If so, can you show how? Thanks :smile:

So I'm self teaching myself Multivariable Calculus from UCBerkeley's Youtube series and an online textbook. I'm up to Stokes Theorem and I'm getting conflicting definitions. UCBerkeley Youtube series says that Stokes Theorem is defined by: \int {(Curl f)} {ds} And then the textbook says that...

Homework Statement for the vector field E=x(xy)-y(x^2 +2y^2) find E.dl along the contour find (nabla)xE along the surface x=0 and x=1 y=0 and y=1 Homework Equations The Attempt at a Solution i tried the second question (nabla)xE over the surface by finding the...

Homework Statement Use stokes theorem to find double integral curlF.dS where S is the part of the sphere x2+y2+z2=5 that lies above plane z=1. F(x,y,z)=x2yzi+yz2j+z3exyk Homework Equations stokes theorem says double integral of curlF.dS = \intC F.dr The Attempt at a Solution...

Homework Statement See figure attached for problem statement Homework Equations The Attempt at a Solution See figure attached for my attempt. I found this problem to be a little long and drawn out, which leads me to believe I made a mistake somewhere. Is this the case? Or...

Homework Statement Use Stoke's theorem to evaluate the line integral \oint y^{3}zdx - x^{3}zdy + 4dz where C is the curve of intersection of the paraboloid z = 2 + x^{2} + y^{2} and the plane z=5, directed clockwise as viewed from the point (0,0,7). Homework Equations The...

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...