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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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covariantly constant one form

Let $v$ be a vector field. We say that $v$ is covariantly constant iff \begin{equation} \nabla v=0, \end{equation} where $\nabla v= (\partial_j v^i+\Gamma^i_{jk} v^k) \ \partial_i \otimes dx^j$, i.e....
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Two related(?) definitions on the wedge product

I read from a textbook that one defines the wedge product of basis elements as, for example, $$e^{i_1}\wedge...\wedge e^{i_k}\equiv k!\text{Alt}(e^{i_1}\otimes...\otimes e^{i_k})$$ and of two forms $$...
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1answer
17 views

Wrong sign in the Hodge Laplacian of a product of functions

If $f,g : \mathbb R \to \mathbb R$ are smooth, then $(fg)^{\prime \prime} = f^{\prime \prime} g + 2 f^\prime g^\prime + f g^{\prime \prime}$ and, if we define a scalar product on $1$-forms by $\langle ...
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23 views

Why intermediate map in Gysin sequence is multiplication by Euler class

I am reading Bott and Tu, Differential Form in Algebraic Topology. At page 178, they constructed Gysin sequence of Sphere Bundle. I am having trouble understanding the argument, $d_{k+1}$ is ...
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1answer
43 views

Looking at the connection 1-form on a principal G-bundle in coordinates

I'm reading this paper, and I'm confused about something. Let $A$ be a connection on a principal $G$-bundle $P$ over $\mathbb{R}^4$, and $F(A)$ its curvature. Let $\mathrm{ad}(P)=P\times_G\mathfrak{g}...
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Exercise on differential forms.

Let $\theta: S^1 \rightarrow \mathbb{R}:(x,y) \mapsto \arctan(\dfrac{y}x)$. Prove that $d\theta$ is a closed 1-form which is not exact. I managed to prove that it was closed but how can it possibly ...
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1answer
50 views

On an abelian variety, every element of $\Omega_0 \otimes k$ extends to a holomorphisc differential form

Let $X$ be an abelian variety over an algebraically closed field, $\alpha \in \Omega_{X,0} \otimes_{\mathscr{O}_{X,0} } k(0) $ (= the dual of $T_{X,0}$) Then does this $\alpha$ extend to an element of ...
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Laplacian in different coordinates

Lets consider the following coordinates on $\Bbb R^3$: $$x_1=r\cos(\xi),x_2=r\sin(\xi) w_2,x_3=r\sin(\xi) w_3$$ with $w_2^2+w_3^2=1,r=||x||$ and $0\leq \xi\leq \pi$. Further let $f$ be a function on $\...
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38 views

Why do we need the multiple $\frac{1}{k! l!}$ in the definition of wedge product?

In the book of Analysis On Manifolds by Munkres, at page 238, it is claimed that in the definition of wedge product For alternating k-tensor $f$ and alternating l-tensor $g$ on $V$, $$f \wedge g =...
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Doubt about differentia operatorl in polar coordinates

The differential operator $d$ can be written as $$d=\partial_i dx^i$$ where $\partial_i\equiv\frac{\partial}{\partial x_i}$. If I have $d$ expressed in cartesian coordinate and I want to obtain the ...
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1answer
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Is a 3-form left invariant on SL(2,$\mathbb{R}$)?

This is from Spivak "Intro to Differential Geometry" Chapter 10 Exercise 26. We are given G as SL(2,$\mathbb{R}$). P is the inclusion map from G to $\mathbb{R}^4$. $x,y,u,v$ are the coordinate ...
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Do Maxwell's equations (generalized) apply to _every_ $k$-form on a pseudo-Riemannian manifold?

Given a pseudo-Riemannian $n$-manifold and a $k$-form $F$ on the manifold, I will call its exterior derivative $J=dF$ the source of $F$ and the differential $K=dG$ the dual source of $F$, where $G=​{\...
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Clock form and derivative

Consider $X,Y$ two smooth vector fields of $\mathbb{R}^2$ and the set $C=\{z\in \mathbb{R}^2;\, det(X,Y)(z)\neq 0\}$. On the set $C$, we define the form $\alpha$ by : $ \alpha(X)=1,\quad \alpha(Y)=0....
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Calculating The First Fundamental Form of a Generalized Cone

$\newcommand{\bs}{\boldsymbol{\sigma}} \newcommand{\bg}{\boldsymbol{\gamma}}\newcommand{\bp}{\textbf{p}}$ Let $\bs(u,v)=(1-v)\bp+v\bg(u)$ be a generalized cone, where $\bg$ is unit speed curve and $\...
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Solving differential forms?

I am trying to solve this equation: $dz = \frac{x^2+z-y^2}{x} dx + \frac{y^2+xy-z}{x} dy$ With the following form: $dq_3 = B_1 dq_1 + B_2 dq_2$ condition (taking dq1.dq2=dq2.dq1)is: $\frac{\...
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Calculate the flow of vector field $\operatorname{curl} F$ over surface

$P = \{(x,y,z)\in \mathbb{R}^3 : x^2+y^2 = 1, |z| \leq 1\}$ In each point the normal vector which defines the positive side of $P$ is $[x,y,0]^T$. Calculate the flow of vector field $\...
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The integral over the oriented curve which is given as the edge of the area

Let's suppose $ D = \{(x,y)\in \mathbb{R}^2:x^2-1\leq y\leq 1 \}$ has the orientation defines as following: in each point $(x,y) \in \partial D$ normal vector of positive side points to the outside of ...
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1answer
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Integral of 1-form along any path between two points

Let $K$ be an oriented curve starting at $a = (\pi, -1)$ and ending at $b = (3\pi, 1)$. $$ I = \int_K (e^y+e^{-y})\cos xdx + (e^y-e^{-y})\sin xdy $$ Does the value of the integral depend on the ...
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1answer
48 views

Derivative with regard to time wedge product

Could anyone help me understand the second equality of the expression below? I have already tried to interpret as the derivative in relation to the time of the composition of two functions, ie the ...
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Evaluate integral (Chern article)

My question is evaluate some integral of the article "A Simple Intrinsic Proof of the Gauss-Bonnet Formula for Closed Riemannian Manifolds" write by Chern. Let's go: If $(M^n,g)$ is a closed even ...
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Question with differential form : What mean “for $n=3$ the exterior differentiation gives again the curling operation”?

Q1) Let $n=3$ and let $$\omega =F_1dx_1+F_2dx_2+f_3dx_3.$$ Then $$d \omega =K_1 dx_2\wedge dx_3+K_2 dx_3\wedge dx_1+K_3 dx_1\wedge dx_2,$$ where $(K_1,K_2,K_3)=\operatorname{Curl}(F)$. Therefore $d\...
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1answer
39 views

Commutator of Lie derivative and Hodge star operator

I want to derive and expression for the commutator $[\mathcal{L}_Z,\star]\omega$. I found this post of mathoverflw that answers this question, but I have a few questions about Willie Wong's proof. How ...
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2answers
71 views

Evaluate $\int_{S^2} \frac{dx \wedge dy}{z} e^{1 - (x^2 + y^2)}$

Any ideas how to evaluate this integral over the sphere? $$ \int_{S^2} \frac{dx \wedge dy}{z} e^{1 - (x^2 + y^2)} = \underbrace{\cdots }_{ (0,0,1)} + \underbrace{\cdots }_{ (0,0,-1)}$$ I considered ...
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1answer
28 views

Orientation forms on compact smooth manifolds are equivalenf if they have the same integral

I have a problem solving the following exercise (Ex. 22-15) from John M. Lee‘s „Introduction to smooth manifolds“. The problem is the following: Use the same technique as in the proof of the Darboux ...
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1answer
26 views

Purely Inseparable Field Extension Differentials

Let $K$ a field of characteristic $p$ and oneconsider a purely inseparable field extension $L = K(t)$. Therefore there exist a minimal $n \in \mathbb{N}$ with $t^{p^n} := k \in K$ (equivalently: $P(X)=...
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group acts on the contract operator $\iota_{X_M}$

I am working on equvariant cohomology and get some problem. Give $M$ as a manifold and the action of a Lie group $G$ acts on $M$, denote the Lie algebra of $G$ by $\mathfrak{g}$. Let $\alpha$ be a ...
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1answer
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Differentials of non-smooth functions, wedge products of currents?

In a paper of McMullen he considers foliations on a manifold determined by a closed 1-form $\rho$. He says an $L^\infty$ function $f$ is constant on the leaves of the foliation if "$df \wedge \rho = ...
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1answer
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Why does Green's theorem fail for this exact differential form $\int Mdx + Ndy$

Why does Green's theorem fail for this exact differential form $\int {Mdx + Ndy} = 0$ since it's an exact differential and M and N both are not functions of z ( the case for which i want to ask the ...
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1answer
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A Hodge dual computation on a $4$-dimensional Riemannian manifold

Let $(M,g)$ be a $4$-dimensional smooth Riemannian manifold. I am trying to understand the following exterior algebra computation: Let $x^1,x^2,x^3,x^4$ be local coordinates on $M$ such that the ...
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1answer
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Definition of $dz_i\otimes d\bar{z_j}(\frac{\partial}{\partial z_m},c\frac{\partial}{\partial z_n})$

Is $dz_i\otimes d\bar{z_j}(\frac{\partial}{\partial z_m},c\frac{\partial}{\partial z_n}):=dz_i(\frac{\partial}{ \partial z_m})d\bar{z_j}(\bar{c}\frac{\partial}{\partial \bar{z_n}})$? It seems that by ...
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34 views

Notation of the Pullback of a $1$-form

Let $\omega = \mathrm{d}y-A(t,y)\,\mathrm{d}t$ and $\gamma(t)=(t,y(t))^t$ be given. I want to compute $\gamma^*\omega(t)=\omega_{\gamma(t)}(\dot\gamma(t))$. Now the actual problem I have, is to ...
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1answer
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How to determine an integrating factor which is both function of x and y but not an exponential function e.g.($x^my^n$)

I was watching https://m.youtube.com/watch?v=efBdnK1q504&list=PLj7p5OoL6vGykv4JM5MpY3I5j6mkLyKQU&index=8&t=0s video. Now I would wonder what if special integrating factor had been a ...
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1answer
49 views

Differential forms and partial derivatives

If I have a vector valued function $f(\vec R)$ and I imagine that $R$ is also a function of time, so that $R(t)$ Then I'm struggling to prove that: $$ df \left({\partial R \over \partial t}\right) = {...
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Pullback of a 1-form

Let $f\colon \mathbb{R}^n\rightarrow \mathbb{R}^N$ be a parametrisation of a manifold $M$. Given the form $\displaystyle \omega = \sum_{j=1}^N w^j\mathrm{d}x_j$ where $f_j\in C^\infty (M)$. The ...
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Find conditions for which 1-form is closed

Let $\omega = \alpha \,\mathrm{d}r+r\beta\,\mathrm{d}\theta$ a 1-form on $\mathbb{R}^2$, where $\alpha,\beta$ are pulled back by polar coordinates. I want to find equations for $\alpha,\beta$ where $\...
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1answer
26 views

Exterior Derivatives

I am self-studying differentialgeometry and tried to compute the exterior derivatives of the following 2-forms: $\omega_1 = x\,\mathrm{d}y\wedge \mathrm{d}z-y\,\mathrm{d}x\wedge \mathrm{d}z+z\,\...
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1answer
35 views

Differential form residue

It is possible to write $$ \sqrt{ 2 + \cos(x) + \cos(y) } -2 = \frac{-a(x,y)\sin(x)}{\sqrt{ 2 + \cos(x) + \cos(y) }} + \frac {b(x,y)sin(y)}{\sqrt{ 2 + \cos(x) + \cos(y) }}$ $$ for some functions $a,b ...
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1answer
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Differential of a function on $\mathcal{R}^n$ covector field

At the moment I have some trouble with the differential of a function $df$ as covector field. Most of the explanations are on manifolds but I am at the moment interested in $\mathcal{R}^n$. The ...
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1answer
54 views

Specific formula of Hodge star operator $\bar{*}$

Hodge star operator is an operator: $\bar{*}:\epsilon^{p,q}=\Gamma(X,\bigwedge^p\Omega^1_X\otimes\overline{\bigwedge^p\Omega^1_X})\to\epsilon^{n-q,n-p}$ with the relation $$\alpha\wedge\bar{*}({\beta})...
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1answer
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Hodge star operator $\bar{*}$ and volume form

Hodge star operator is an operator: $\bar{*}:\epsilon^{p,q}=\Gamma(X,\bigwedge^p\Omega^1_X\otimes\overline{\bigwedge^p\Omega^1_X})\to\epsilon^{n-q,n-p}$ with the relation $$\alpha\wedge\bar{*}({\beta})...
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2answers
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Integral along different loops on torus gives a full lattice in $\mathbb C$

Let $S$ be a compact Riemann surface we $g=1$, $\alpha,\beta$ be the generator of $\pi_1(S)$, $\omega\neq 0$ be a fixed holomorphic $1$-form. How can we know $\int_\alpha \omega$ and $\int_\beta \...
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1answer
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Why is $\epsilon^{p,q}(X):=\Gamma(X,\bigwedge^p\Omega^1_X\otimes\overline{\bigwedge^p\Omega^1_X})$?

Here $\Omega^1_X=(T^*X)^{1,0}$, from : this notes we have $\epsilon^{p,q}(X):=\Gamma(X,\bigwedge^p\Omega^1_X\otimes\overline{\bigwedge^p\Omega^1_X})$ But I wonder why it's tensor not wedge? Shouldn'...
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1answer
56 views

Compact surface with constant strictly positive curvature is a sphere

I'm following Cartan's Differential forms. I'm trying to do exercise 8 on page 161. The chapter is about moving frames and differential forms in surface theory. Consider the frame of Ex. 2 (...
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2answers
85 views

Integral of 1-form $\omega=\dfrac{-y \,dx + x \,dy}{x^2 +y^2}$ over a triangle. [duplicate]

I'm trying to evaluate the integral of the $1$-form $$\omega=\dfrac{-y \,dx +x\,dy}{x^2 +y^2}$$ through the corners of a triangle with the vertices $A= (-5,-2)$, $B=(5,-2)$, $C=(0,3)$. I've ...
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0answers
77 views

Lie algebra-valued differential forms, exactness, closedness, and the Moyal product

This is my attempt to prove something (I'm not even sure if it's true to begin with) using somewhat loose arguments. I present here all the steps and ideas and I would be extremely grateful if someone ...
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0answers
61 views

Finding an Antiderivative for an Exact Form

Let $\omega$ be an exact $1$-form on a path-connected manifold $M$. For a fixed $x \in M$ we can define for all $y \in M$. $$ g(y) := \int_\gamma \gamma^*(\omega), $$ where $\gamma$ is a path from $x$ ...
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1answer
58 views

Integral of differential forms $\int (-y+\sin x^2)dx + xdy$.

I should calculate $$\int (-y+\sin x^2)dx + xdy$$ on the curve $c=c_{1}+c_{2}-c_{3}-c_{4}$ where it doesn't give me any parametrisation mappings; only the normal $$ \begin{cases} c_{1}:& [0,...
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0answers
44 views

Hessian matrix vs differential 2-form

Could someone clarify the convention that the second derivative of a scalar function $f: \Bbb R^n \rightarrow \Bbb R$ is sometimes defined as a linear operator $D^2f : \Bbb R^n \rightarrow L(\Bbb R^n, ...
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0answers
46 views

Exercise about differential form

(1) Determine $g:(0, +\infty)\to \mathbb{R}$ such that $g(1)=1$ and the differential form $\omega=2xg(x^2+y^2)dx + \frac{2y}{(x^2+y^2)^2}dy$ is closed. (2) Before determining the potentials, ...
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1answer
52 views

Prove: $\omega$ Is Exact

Let $\omega=\omega_1dx_1+...+\omega_ndx_n$ be a continuous form on an open and connected set $\Omega \subset \mathbb{R}^n$. Let assume that for all $x,y\in \Omega$ there is $c\in \mathbb{R}$ such ...