For questions about differential forms which commonly arise in differential geometry, and sometimes in multivariable calculus.

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22 views

Help in the proof of Poincaré Theorem to differential forms

I'm revising the proof of Poincaré Theorem, but I don't understand a pass of proof. Let be $E$ and $F$, Banach spaces and $U\subset E$ open set. Consider $\omega\in\Omega_p^n(U;F)$ a p-differential ...
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2answers
58 views

Self-studying differential forms and tensors

I am interested in understanding the generalized Stokes' Theorem. From my understanding, this theorem involves differential forms and exterior algebra, and tensors (to some extent). I'm not ...
2
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1answer
55 views

Tangent space and differential forms of quotient groups

I have difficulty understanding the following argument because it seems that many details are swept under the rug and I am looking for a detailed rigorous exposition on this (I'll try to make clear ...
5
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1answer
51 views

divisor of a section of the sheaf of logarithmic differentials

Let $S=\{0, 1, \infty\} \subset \mathbb{P}^1$ and let $\Omega^1_{\mathbb{P}^1}(\log S)$ be the line bundle of logarithmic differentials along $S$. Consider the form $$ ...
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1answer
32 views

$\bigwedge^nT^∗M$ is trivial $\Leftrightarrow M$ is orientable

I can't figure out how to prove the following: Let $M$ be an $n$-manifold. Then $\bigwedge^nT^∗M$ is trivial $\Leftrightarrow M$ is orientable Thank you
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2answers
66 views

Prove $\omega\wedge\eta=0$ iff there exist a $\theta\in A^3(\mathbb{R}^5)$ such that $\eta=\omega\wedge\theta?$ [closed]

Suppose $\omega\in A^1(\mathbb{R}^5)$ and $\eta\in A^4(\mathbb{R}^5)$. prove $\omega\wedge\eta=0$ iff there exist a $\theta\in A^3(\mathbb{R}^5)$ such that $\eta=\omega\wedge\theta?$
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1answer
41 views

for a vector field $v$, what is the differential 1-form obtained from $v$ by the canonical isomorphism induced by the inner product.

I am studying analysis on manifolds and am trying some problems and I am conceptually stuck on one. It is from Do Carmo's book Differential Forms and Applications, chapter one problem 11 b. I'll ...
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0answers
33 views

Locally Hamiltonian vector fields

Consider the following definitions (taken from [1]) Definition. Let $E$ be a Banach space and $B: E \times E \to \mathbb R$ a continuous bilinear mapping. Then $B$ induces a map $B^\natural: E \to ...
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0answers
26 views

Given a smooth 2-form $\varphi$ on a punctured Riemann surface, is there $\nu$ of type (1,0) such that $\varphi=d\nu$?

Let $X$ be a compact Riemann surface, $p\in X$, and $\varphi$ be a smooth 2-form on a $X-\{p\}$, and hence exact. I'm wondering if it is possible to find a form of type (1,0) whose differential is ...
4
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1answer
58 views

Second Chern class of $TS^2$

The induced metric on a sphere may be given by, $$ds^2 = d\theta^2 + \sin^2\theta\, d\phi^2$$ By using Cartan's method of moving frames, one can compute the curvature 2-form in an orthonormal basis ...
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2answers
99 views

Problem on Integration: $\Bbb R-\Bbb C$ split and pull back of forms

This post is not short. However I'm sure that a guy who good handle these concepts, could read and answer in five minutes. I only want to write my attempt, in order to understand where I'm wrong. Let ...
5
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1answer
39 views

Cotangent Fields: Exactness vs. Conservation

Given a smooth manifold. Then a cotangent field is exact iff conservative: $$\alpha\in\mathcal{X}^*(M):\quad\alpha=\mathrm{d}h\iff\oint\alpha=0$$ How to prove this properly?
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0answers
36 views

Linear independence of differential 1-forms

Let be $(E,\mathbb{K}),(F,\mathbb{K})$ Banach space and $U\subset E$ open set. If $f_1,...,f_n\in\Omega_1(U;F)$ differential 1-forms are linearly independents, where $$\Omega_1(U;F)=\{f:U\rightarrow ...
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1answer
22 views

How to prove that angular velocity is not a derivative of angular displacement?

The angular velocity $\omega$ of a two-dimensional solid body is given by $$\omega = \hat{z} \cdot \frac{\vec{r} \times \vec{v}}{r^2},$$ where $\vec{r}$ and $\vec{v}$ are the position and the velocity ...
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1answer
64 views

A proof of exactness of closed 1-forms on the two-sphere

Remark. I'm aware that here the same problem is solved. I'm in trouble with the proof I'm going to quote. Consider the following Claim. Every closed 1-form $\beta$ on $S^2$ is exact. This is an ...
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0answers
56 views

What is this inner product on differential forms?

I am trying to understand the definition of $d^\ast$ of $d$ where $d$ denotes the exterior derivative as given in these lecture notes. (please see page 3) Here are my thoughts so far: Let us ...
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2answers
59 views

Notation of coordinate representation in Lee

In Lee's Introduction to Smooth Manifolds he writes $$ \omega = \omega_i dx^i$$ where $\omega$ is a differential form. See for example page 293. What does $\omega_i dx^i$ stand for? According ...
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1answer
75 views

Cauchy integral formula: can it be proved like this?

Consider the Cauchy theorem: Let $D\subset \mathbb C$ be a domain such that $\partial D$ is smooth and $\overline{D}$ is compact. Let $f$ be holomorphic on $D$ and continuous on the closure. Then ...
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1answer
55 views

Closed non-exact $2$-form on $T = S^1 \times S^1$

I would like to find a closed non-exact $2$-form on $T = S^1 \times S^1$. Here are my thoughts: Since $d\theta$ and $d\varphi$ are closed non-exact $1$-forms an obvious candidate is $d\theta \wedge ...
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1answer
34 views

Finding two inequivalent closed, non-exact $1$-forms on $T = S^1 \times S^1$: second check

This is a follow up on my previous question. I would like to test whether I understand the first part of the answer given to me there by rewriting it in my own words. Please could someone tell me ...
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2answers
63 views

Stokes or homotopy?

The problem states Show that if $X$ is a simply connected manifold, then $\oint_{\gamma}\omega=0$ for all closed 1-forms $\omega$ on X and all closed curves $\gamma$ in $X$. However I have ...
6
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1answer
80 views

Finding two inequivalent closed, non-exact $1$-forms on $T = S^1 \times S^1$

I've been studying the torus and the first cohomology group $H^1_{dR}(T)$ for a couple of weeks now. I finally had a breakthrough of understanding and would like to kindly request the community to ...
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2answers
104 views

How to evaluate this integral: $\oint dx$?

I am trying to understand differential forms. Now I tried to evaluate $$ \oint_{S^1}dx$$ I should get anything non-zero but I don't know how to do it (even though I know the result). If $S^1$ in ...
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1answer
53 views

Relationship between differential forms and coordinates

For the purpose of this question let us restrict our considerations to smooth $3$-manifolds. So the manifold $M$ we consider here is endowed with smooth coordinate charts $(x,y,z)$. What I have ...
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2answers
40 views

$H^1_{dR}(S^1 \times S^1) = H^1_{dR}(S^1) \oplus H^1_{dR}(S^1)$ without Künneth?

As in the title: I am trying to derive $H^1_{dR}(S^1 \times S^1) = H^1_{dR}(S^1) \oplus H^1_{dR}(S^1)$. First let me share my thoughts: I am trying to derive that $$ H^1_{dR}(S^1 \times S^1) = ...
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1answer
43 views

$(2n-1)$-form is closed [closed]

Consider in $\mathbb{R}^{2n}$ differential forms$$\omega = dx_1 \wedge dx_2 \wedge \dots \wedge dx_n\text{ and }\theta = \sum_{n+1}^{2n} (-1)^{j-1}x_jdx_{n+1} \wedge \dots \wedge dx_{j-1} \wedge ...
3
votes
1answer
81 views

Exterior derivative of a 2-form

I want to prove that the exterior derivative of a 2-form in $\mathbb{R}^n$: $${\alpha = \sum a_{ij} dx_i \wedge dx_j}$$ is: $${d \alpha = \sum (\frac{\partial a_{ij}}{\partial x_k} + \frac{\partial ...
3
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1answer
110 views

What does a standalone $dx$ mean?

Some literature uses $dx$, in the context of differential equations, in a confusing way without defining what it really stands for: $Mdx + Ndy = 0$ Does it mean one of the following or something ...
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0answers
23 views

Verify stokes theorem example [duplicate]

Let $c:I^2\rightarrow\mathbb{R}^3$ be the singular $2$-cube given by $$c(s,t)=(\frac{1}{2}s^2,st,\frac{1}{2}t^2)$$ Let $$\omega=xy^2dz$$ Questions: i) Compute $c^*\omega$ ii) Compute $c^*d\omega$ ...
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0answers
26 views

Show system has a solution using Frobenius method

System: $$\frac{\partial u}{\partial x}=v,\frac{\partial u}{\partial t}=w$$$$\frac{\partial v}{\partial x}=G(x,t),\frac{\partial v}{\partial t}=-\dot{a}(t)G(x,t)$$$$\frac{\partial w}{\partial ...
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1answer
60 views

Left-invariant differential form on homogeneous space

I'm trying to learn by working through an example but I got stuck with the line of reasoning provided in the text. Let $G$ be the group of matrices $$g= \begin{bmatrix} 1 &0 &\log{x} \\ y ...
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0answers
20 views

Why do entries of Maurer-Cartan 1-form span space of left-invariant 1-forms

Suppose $G$ is a $k$-dimensional Lie group of $n\times n$ matrices of the form $[g_{ij}(x_1,...,x_k)]$ where $g_{ij}$ are smooth functions. As a follow-up to the question I posted recently, I now ...
2
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0answers
32 views

Differentiability of a map into $\wedge T^*M$

Let $M$ a differentiable manifold and $p \in M$. Consider $(U;x_1, \dots, x_n)$ a coordinated system around $p$. $\{dx_\phi\}_\phi$ with $\phi \in \cal{P}(\{1, \dots n\})$, $dx_\phi=dx_{i1} \wedge ...
2
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1answer
43 views

A question about a part of the proof of the existence of the exterior derivate.

Let $M$ a differentiable manifold, $(U;x_1,\ldots,x_n)$ a coordinated system and $\omega$ a differential form which domain intersects $U$. In $U \cap \operatorname{Dom} \omega$, $\omega$ can be ...
0
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0answers
27 views

Time Derivative of the integral over a singular k-cube

I am stuck on this question, and was wondering if someone could provide a hint of where to start? I can't see the first step.
0
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1answer
31 views

First order differential equation problem

Suppose we have $$ \frac{dy}{dx} +f(x)y = r(x) $$ and it has two solutions $y_1(x)$ and $y_2(x)$ then how to prove that solution of differential equation $$ \frac{dy}{dx} +f(x)y = 2r(x) $$ Will be ...
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0answers
61 views

Sheaf of differential p-forms

Shafarevich defines the cotangent bundle at page 60 of "Basic Algebraic Geometry 2". Now he says that: 1) $\mathcal{F}_x=\mathcal{O}_x dt_1 + \dots + \mathcal{O}_x dt_n$, where $\mathcal{F}_x$ is the ...
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2answers
55 views

Given a nowhere zero vector field $Z$, does there exist a one-form $\gamma$ such that $\gamma(Z) = 1$?

Take $M$ a smooth manifold, and $Z$ a vector field on $M$ such that $Z(p)\neq0$ for all $p\in M$. Is there a one form $\gamma \in \Omega^1(M)$ such that $\gamma(Z)=1$? I started to work locally, but ...
2
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0answers
66 views

Differential one-forms and change of coordinates

Consider two differential one forms: $$\omega=\sum_{i=1}^N \omega_i dx^i$$ $$\omega'=\sum_{i=1}^N \omega'_i dx'^i$$ As I recall from my analysis courses, the symbols $dx$ are a particular notation ...
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0answers
56 views

The induced map on the de Rahm cohomology of a surjective submersion.

Let $M,N$ be two smooth manifolds and $f: M \rightarrow N$ a surjective submersion (so $f$ and $f_*$ both surjective everwhere). It is straightforward to show that then the pullback of $k$-forms: ...
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1answer
61 views

What does $d\zeta_1\wedge\cdots\wedge d\zeta_n$ mean in the context of Cauchy formula (on polydiscs)?

A Polydisc of center $z^o=(z_1^o,\dots,z_n^o)\in\Bbb C^n$ and multiradius $r=(r_1,\dots,r_n)\in(\Bbb R^+)^n$ is defined as $$ P_{z^o,r}:=\prod_{j=1}^n\Delta_{z_j^o,r_j} $$ where ...
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0answers
19 views

$k$-cube definition clarrification

If $c:I^k\rightarrow\mathbb{R}^n$ is a singular $k$-cube. What will $\partial c$ be? Will it be a singular $(k-1)$-cube or a $(k+1)$-cube or something else? Definition - A singular $k$-cube on $U$ ...
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1answer
27 views

Compute $L_\mathbb{X}\beta$

Given information: $$\alpha+(x+y)dy+(x^2-y^2)dz$$$$\beta=zdx\wedge dy+xzdx\wedge dz$$$\mathbb{X}$ is the vector field given by $$\mathbb{X}=(0,-x,-1)$$ I have found $i_\mathbb{X}\beta=2xzdx$ ...
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1answer
35 views

Compute the contraction of $i_\mathbb{X}\beta$

Question: Let $\beta=zdx\wedge dy+xzdx\wedge dz$, and let $\mathbb{X}$ be the vector field on $\mathbb{R}^3$ given by $\mathbb{X}=(0,-x,-1)$. Compute $i_\mathbb{X}\beta$, combining terms where ...
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1answer
42 views

Real/Complex differentials forms

Given $f:\Bbb C^{n}\to\Bbb C$ identified with $f:\Bbb R^{2n}\to\Bbb C$, in a book I read that $$ \partial_x f\,dx+\partial_yf\,dy=\partial_zf\,dz+\partial_{\bar z}f\,d\bar z $$ and that this could be ...
1
vote
1answer
53 views

Compute $\int_cd\omega$ and $\int_{\partial c}\omega$

Question: Let $c:I^2\rightarrow\mathbb{R}^3$ be the singular $2$-cube given by $$c(s,t)=\left(\frac{1}{2}s^2,st,\frac{1}{2}t^2\right)$$Let $x=(x,y,z)$ denote the cartesian coordinates on ...
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0answers
10 views

Obtain first-order system

Question: Let $G(x,t)=g''(x-a(t))$, where $g$ and $a$ are smooth. Combine equations: $$\frac{\partial u}{\partial x}=v;\frac{\partial u}{\partial t}=w$$$$\frac{\partial v}{\partial t}=\frac{\partial ...
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1answer
13 views

Derive necessary and sufficient conditions meaning

Question: Let $\mathbb{X}$ and $\mathbb{Y}$ be vector fields on $\mathbb{R}^3$ given by $$\mathbb{X}(x,y,z)=(1,0,p(x,y)r(z))$$ $$\mathbb{Y}(x,y,z)=(0,1,q(x,y)r(z))$$ where $p,q$ and $r$ are smooth, ...
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vote
2answers
76 views

Exactness of $dx,dy$

Let $dx,dy$ denote differential $1$-forms. It is easy to verify that they are closed. My question is: Does there exist a space such that either $dx$ or $dy$ or both are exact? (A ...
2
votes
3answers
94 views

Divergence theorem in complex analysis

I am revisiting my understanding of integration by parts in several complex variables, but I have run into an apparent contradiction. This shows my understanding is flawed, which is somewhat ...