What is Stokes theorem: Definition and 80 Discussions
In vector calculus and differential geometry, the generalized Stokes theorem (sometimes with apostrophe as Stokes' theorem or Stokes's theorem), also called the Stokes–Cartan theorem, is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. It is a generalization of Isaac Newton's fundamental theorem of calculus that relates two-dimensional line integrals to three-dimensional surface integrals.Stokes' theorem says that the integral of a differential form ω over the boundary of some orientable manifold Ω is equal to the integral of its exterior derivative dω over the whole of Ω, i.e.,
∫
∂
Ω
ω
=
∫
Ω
d
ω
.
{\displaystyle \int _{\partial \Omega }\omega =\int _{\Omega }d\omega \,.}
Stokes' theorem was formulated in its modern form by Élie Cartan in 1945, following earlier work on the generalization of the theorems of vector calculus by Vito Volterra, Édouard Goursat, and Henri Poincaré.This modern form of Stokes' theorem is a vast generalization of a classical result that Lord Kelvin communicated to George Stokes in a letter dated July 2, 1850. Stokes set the theorem as a question on the 1854 Smith's Prize exam, which led to the result bearing his name. It was first published by Hermann Hankel in 1861. This classical Kelvin–Stokes theorem relates the surface integral of the curl of a vector field F over a surface (that is, the flux of curl F) in Euclidean three-space to the line integral of the vector field over its boundary (also known as the loop integral).
Simple classical vector analysis example
Let γ: [a, b] → R2 be a piecewise smooth Jordan plane curve. The Jordan curve theorem implies that γ divides R2 into two components, a compact one and another that is non-compact. Let D denote the compact part that is bounded by γ and suppose ψ: D → R3 is smooth, with S := ψ(D). If Γ is the space curve defined by Γ(t) = ψ(γ(t)) and F is a smooth vector field on R3, then:
∮
Γ
F
⋅
d
Γ
=
∬
S
∇
×
F
⋅
d
S
{\displaystyle \oint _{\Gamma }\mathbf {F} \,\cdot \,d{\mathbf {\Gamma } }=\iint _{S}\nabla \times \mathbf {F} \,\cdot \,d\mathbf {S} }
This classical statement, is a special case of the general formulation stated above after making an identification of vector field with a 1-form and its curl with a two form through
{\displaystyle d(F_{x}\,dx+F_{y}\,dy+F_{z}\,dz)=(\partial _{y}F_{z}-\partial _{z}F_{y})\,dy\wedge dz+(\partial _{z}F_{x}-\partial _{x}F_{z})\,dz\wedge dx+(\partial _{x}F_{y}-\partial _{y}F_{x})\,dx\wedge dy}
.Other classical generalisations of the fundamental theorem of calculus like the divergence theorem, and Green's theorem are special cases of the general formulation stated above after making a standard identification of vector fields with differential forms (different for each of the classical theorems).
##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...
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
\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 don't know...
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
given a vector field v[/B=]Kθ/s θ (which is a two dimensional vector field in the direction of the angle, θ with a distance s from the origin) find the curl of the field and verify stokes theorem applies to this field, using a circle of radius R around the origin
Homework...
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...
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
Let F(x, y, z) = \left ( e^{-y^2} + y^{1+x^2} +cos(z), -z, y \right)
Let s Be the portion of the paraboloid y^2+z^2=4(x+1) for 0 \leq x \leq 3
and the portion of the sphere x^2 + y^2 +z^2 = 4 for x \leq 0
Find \iint\limits_s curl(\vec{F}) d \vec{s}
Homework...
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 =...
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...
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