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Questions tagged [differential-forms]

For questions about differential forms, a class of objects in differential geometry and multivariable calculus that can be integrated.

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Differential 1-form & 2-form with 4 variables and are identified with 3 function-coefficient

I am working on a problem that requires me to compute curl(V) div(V) where V: R4 -> R3, (coordinates are (t,x,y,z) - where x,y,z are time dependent variable) and basis of V =[Vx Vy Vz]. I know I can ...
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1answer
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Confused with a spherical coordinate system surface element

I can not understand how a particular surface element is derived in spherical coordinates. The equation expressing the surface element vector is given as $$r_s = (\sqrt{R^2-z_s^2} \cos \phi,\sqrt{R^...
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Hodge theory: $\Delta \alpha = 0$ iff $d\alpha = d^* \alpha = 0$ on a noncompact manifold?

Let $M$ be a Riemannian manifold (connected, oriented). One can define the co-differential $d^* : \Omega^k(M, \mathbb{R}) \to \Omega^{k-1}(M, \mathbb{R})$ even if $M$ is not compact (for example use ...
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Integrate 2-Form Along Loop to get 1-Form?

I would like generalize the following theorem from Calc III: "If $U$ is an open, connected subset of $\mathbb{R}^2$, $\omega$ is a 1-form on $U$, and $d\omega = 0$ on $U$, then TFAE 1) $\omega = df$ ...
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Directly calculating integral of l-form over an sphere

If you have $\omega=x_3 dx_1\wedge dx_2$, $\sigma(\theta,\phi) = (\sin\phi\cos\theta,\sin\phi\sin\theta,\cos\phi)$ defined on $[0,2\pi]\times[-\pi/2,\pi/2]$, calculate $\int_\sigma \omega$ directly. ...
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Formula of 1 form in R3

I want to solve the following problem: Given $d ◦ d = 0$, d acting on $0$-form and $1$-form, I want to show that both $1$-form and $0$-form yield $0$. I finished in the case of $0$-form using that ...
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Cartan homotopy formula and curl

In Topological Methods in Hydrodynamics, V. I. Arnol'd writes that the following expression $$curl(\mathbf a \times \mathbf b)=[\mathbf a, \mathbf b]+ \mathbf a \ div \ \mathbf b - \mathbf b \ div \ \...
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Curvature forms as exterior covariant derivative?

I have read on several forums like this one, that given a connection form $\omega$ on a principal bundle and its curvature form $\Omega$, I can state that $\Omega=d_\omega\omega$ alike I do in the ...
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Existence of a countable locally finite cover with nonempty intersection of two adjacent elements

Let $\Omega$ be an open connected set in $\mathbb{C}$, not necessarily bounded. Does there exist a countable locally finite cover of $\Omega$ consisting of only open discs $\{ B(z_i, r_i): i\geq 1\}$ ...
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Weitzenböck identity for $TM$-valued differential forms

Let $M$ be a Riemannian manifold, and let $\nabla$ denote its Levi-Civita connection. We have two second order differential operators $\Gamma(T^*M \otimes TM) \to \Gamma(T^*M \otimes TM)$: The ...
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Exterior derivative of a 1-form on a surface for non-regular mappings

I would appreciate some help for this problem. I have no idea how to start. Let $M\subset \mathbb{R}^3$ be a smooth surface. Let $\phi$ be a $1$-form on $M$. By definition, the exterior derivative of ...
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Question related to orientation on a arbitrary oriented manifold

This is a section from Loring Tu's book Introduction to Manifolds page 244 Second Edition. My question is as follows: Towards the end of the text in the image he says that an oriented manifold can be ...
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1answer
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How to calculate this Differential Form

Let $d\omega=(\frac{-3}{2}u^2D - Su + Tu - u^2 - Qu)(u^3D + Su^2 - Tu^2)^{-1}du \wedge \omega$ Where $D$, $S$, $T$ and $Q$ are constants. I don't know how to calculate $\omega$
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Forward kolmogrov differential difference equation

While reading research papers related to Queueing models, without solving the equation, authors have plotted graph for kolmogrov forward differential difference equation using matlab.How it's ...
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How to Find a Vector Field(/Differential Form) with Div(F) = 0 but F \ne Curl(A) on R^3\(S^1x{0})

In this post, it is outlined how to find a differential $n$-form on $U_0 = \mathbb{R}^n\backslash\{\text{pt}\}$ whose exterior derivative is zero but which is not the exterior derivative of an $(n-1)$-...
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Understanding of Differential one Form [Spivak]

I am using Spivak Calculus on Manifolds to understand differential form. I understand that a k-form is defined to be the function sending a point to this point paired with an alternating k-tensor. I ...
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Topological invariance of compactly supported de Rham cohomology

It is well-known that if we are given two smooth manifolds (without) boundary, whose underlying topological spaces are homotopic, then the de Rham cohomologies $H^k_{dR}$ of $M$ and $N$ are isomorphic ...
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39 views

Integration problem with full derivative

I am trying to find out what type of integration is used to solve this problem and what are the rules behind it. I am sorry, if this question is quite basic, but I do not study Mathematics. The ...
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find flow of vector field $\vec{F}(x,y,z)=\sin(y^{2}z^{2})\vec{i}+(2-xy)\vec{j}+z^{2}\vec{k}$

Quarter of a Sphere S given by the equation : $ x^{2}+y^{2}+z^{2} = 16 \; \; $ with $, \; \; y\geq 0 \; \; ,\; \; z\geq 0$ oriented to the point $(0,2\sqrt{2},2\sqrt{2})$ with a normal vector $\...
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On the proof of localization in symplectic geometry

I was working on the proof of Duistermaat-Heckman theorem in Introduction to Symplectic Topology by Dusa McDuff. He used a lemma called localization. It can be found on page 194. You can find the ...
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1answer
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When can a differential form descend to the quotient manifold

Let $G$ be a finite group acting freely on a manifold $M$, then we know that $\pi: M \to M/G$ is a covering map for manifolds. Now if I have a differential $k$-form $\omega$ on $M$, such that $g^{\ast}...
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If antisymmetric tensors are differential forms, what are symmetric tensors?

Let $\mathbf{T}$ be (for example) a rank-2 antisymmetric covariant tensor, with components $T_{ij}$. In the language of differential forms, we can represent $\mathbf{T}$ as $$ \mathbf{T}=\sum_{i,j}\...
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2answers
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How to evaluate $\int_{S^2} \sin(\theta) d\theta \wedge d\phi$?

I'm kind of stuck here. I try to apply Stokes Theorem, so that $\int_{S^2} \sin(\theta) d\theta \wedge d\phi = \int_{D^2} d(\sin(\theta) d\theta \wedge d\phi) = 0 $ which is not the answer?
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1answer
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Equivalent Definitions of Compactly Supported Forms in The Vertical Direction

I've come across two definitions of compactly supported forms in the vertical direction and I'm trying to show they are equivalent. For the setup, let $\pi:E \to M$ be a vector bundle of smooth ...
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1answer
25 views

Identity on the set of forms

Given a manifold $M$, let $f:M\to M$ have the property that $f \circ f = id_M$, and that $f$ has no fixed points. Suppose there exists another manifold $N$ and a surjective local diffeomorphism $\pi:M\...
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1answer
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Alternating functions and differential forms

Can someone help me? Any sugestion will be helpful. According to Courant and John in Introduction to Calculus and Analysis, Vol. 2 chapter 5 section 5.1, if we take: \begin{equation} L = f(x,y)dy - g(...
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Derivative and Optimization as Quotient of Differential 1-Forms

Although it makes some people a bit anxious, apparently it is possible to take a derivative of one expression with respect to another, as demonstrated here. In fact, it is suggested here that an ...
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1answer
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Write Formula using differential forms

let $\alpha, \beta \in C^2(\Omega)$ be zero forms Where $\Omega$ is a regular surface with boundary $\partial{\Omega}$. I have to write the following formula using differential forms \begin{...
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line element as a 1 form

I'm studying differential forms and I know how to manipulate all the equations. On trying to find a pictorial understanding, I am a bit stuck on the following. A one-form is suppose to assign a ...
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closed differential 2-forms

Let $\psi$ be a fixed differential 1-form on $\mathbb{R}^n$. I'm looking for an emplicit differential 1-form on $\mathbb{R}^n$, $\phi$, such that $\psi\wedge \phi$ is closed. If $\psi=\sum^n_{i=1}A_i\,...
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Time derivative of a differential form

Maybe a very silly question, but I've found the following equation: $$\frac{d}{dt} \omega(p+tV|_p) = \sum_i\frac{d}{dt} (p_i+tV_i(p)) \partial_i \omega (p)$$ I suppose it's a kind of chain rule, and ...
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Exterior derivative of a vector valued function

I'm trying to understand the exterior derivative in the simplest context that I can. I feel like I understand how an exterior derivative should behave for a function. For example, let's take a simple ...
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Checking Holomorphicity/Meromorphicity of a Differential 1-Form On A Curve

Let $f\left(x,y\right)$ be an elementary function of $x$ and $y$. Examples: $$4x^{3}-ax-b-y^{2}$$ $$a^{x}-y^{2}+x^{3}y-1$$ $$x^{3}-3xy+y^{3}$$ $$\frac{\sin^{2}x}{x^{2}+\sin^{2}y}-\frac{1}{3}$$ Let ...
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Calculus of variation and isoperimetric problem with differential forms and moving frames

This question follows this one. I want to apply the calculus of variation with differential forms to three classical problem: 1. arc-length minimizing curve (geodesics) 2. area-minimizing surfaces (...
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1answer
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Explanation of defination of Manifold

While reading the book on Forms and connection ,I am stuck with following defination of manifold.I am stuck at the part after defining function $f$ for submersion. Can anyone explain me this ...
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Codifferential of k-th exterior power endmorphism applied to differential form $\delta(\wedge^p A\alpha)$?

Let $(M,g)$ be an $n$-dimensional smooth Riemannian manifold without boundary. Given an endomorphism $A(x) : \sf T_x^*M \to \sf T_x^*M$ on the cotangent bundle, we can extend $A$ to an endomorphism on ...
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Computing a Basis of Holomorphic Differential Forms for a Given Curve

I find myself repeatedly having to ask this question, because no one seems to have answered it. I put this up on Math Overflow with a bounty, got a guy who said he would answer it, but—though he ...
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Integrating Factor for Closed but not Exact Differential Form

(I kind-of skipped ODEs in undergrad, so my knowledge is a little sketchy; please excuse me if this question is elementary.) (In addition, this problem is a bit of a sketch; I'll flesh it out with ...
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1answer
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Pullback of Fubini-Study form on $\mathbb {CP}^1$

Question: Let $\varphi_S: S^2\setminus \{N\} \to \mathbb C$ be given by $\varphi_S (x_1, x_2, x_3) = \left(\frac{x_1}{1 - x_3}, \frac{x_2}{1 - x_3}\right)$, i.e., the stereographic projection. Also ...
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Prove Green Theorem using Gauss Theorem

Problem. Consider the General Stokes Theorem and $M$ a submanifold of $\mathbb{R}^{n}$, with boundary orientable. (a) Prove the Green Theorem $$\int_{M}(g_{x} - f_{y})dxdy = \int_{\partial M} ...
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Can we extend the idea of contour independence for complex contour integrals to several complex variables?

That is, given some function $f:\mathbb{C}^n \to \mathbb{C}$ entire/ sufficiently holomorphic, if we have two domains $D,D'$ in $\mathbb{C}^n$ with the same boundary i.e $\delta D = \delta D' $, will ...
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A coordinate-independent formula for a potential of an exact 2-form

If I somehow know that a given 1-form $\omega$ on a contractible set $U$ on a manifold $M$ is exact (there exist a 0-form $f$ such that $\omega=\rm{d}f$), then I can find $f$ from the following ...
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1answer
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The cohomological self-dual and anti-self-dual decomposition

The following statement is from Lübke's The Kobayashi-Hitchin Correspondence pp.222: If $a\in A^2(X)$ is harmonic, and $a=a^++a^-$ with $a^{\pm}\in A^2_{\pm}(X)$, then $a^+$ and $a^-$ are also ...
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63 views

Why isn’t Brownian motion differentiable?

Intuitively, if increments become infinitesimally small, why doesn’t Brownian motion become a differentiable function?
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78 views

Calculus of variations with differential forms

I want to generalize calculus of variations with differential forms. Or better, I saw it somewhere some time ago, but now I cannot re-build it. Here is what I remember. Let be $(M, I, \Lambda)$ a ...
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1answer
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Exterior derivative of a two-form as a divergence times the volume form

In a textbook, I have found the following relation. $$ d \stackrel{2}{\omega_V} = \frac{1}{3!} (\mathrm{div} V) \epsilon_{ijk} dx^i \wedge dx^j \wedge dx^k . $$ It is cool, but I don't know how to ...
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Show a particular differential 1-form on $S^1$ is closed but not exact

Consider the differential 1-form $\omega$ on $S^1$ given by $$\omega_{a}(\lambda (-a_2,a_1))=\lambda$$ for all $a=(a_1,a_2)$ in $S^1$. Show that $\omega$ is closed, but not exact. I know I have to ...
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Prove that $\omega$ is closed if $\int_{c}\omega \in \mathbb{Q}$.

Let $\omega$ be a differentiable 1-from defined on an open subset $U \subset \mathbb{R}^{n}$. Suppose that for each closed differential curve $c$ in $U$, $\int_{c}\omega \in \mathbb{Q}$. Prove that $\...
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1answer
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Calculate a Line Integral of differentiable form of degree $1$.

Show that the form $\omega = 2xy^3dx + 3x^2y^2dy$ is closed and calculate $\int_{c}\omega$, where $c$ is given by $y=x^2$, $(0,0)$ to $(x,y)$. My attempt. $\begin{eqnarray*} d\omega &=& d(...
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1answer
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Section and pull-back bundle

I find in general that the pull-back of a section of a vector bundle is a section of the pull-back bundle, but this seems to be false for the cotangent bundle. Let $\phi:M\to N$ be a smooth map, $\...