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

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Show $H_{dR}^1(S^n)=0$ for $n>1$ without de-Rham Theorem, and some similar questions.

Question Without using de Rham's theorem, prove: (1) Show $H_{dR}^1(S^n)=0$ for $n>1$. (2) Use (1) to show $H_{dR}^1(RP^n)=0$ (3) a n-form $\Omega$ is exact on $S^n$ if and only if $$ ...
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1answer
29 views

Notation for the set of holomorphic differential forms

I used to write $\Omega^p(M)$ for the set of $p$-forms on $M$ and $\Omega^{p,q}(M)$ for the set of $(p,q)$-forms on a complex manifold $M$. Now some authors use $\Omega^p(M)$ to denote the set ...
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1answer
237 views

Elementary symmetric polynomials and matrices of 1-forms

Let $A$ be a $n \times n$ matrix of 1-forms (for example, a connection form). Note that $A \wedge A$ is not $0$, but by using the anti-symmetry of the wedge product applied to the entries of $A$ we ...
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0answers
41 views

Cohomology classes of the DeRham cohomology

May be $TM$ a tangent bundle of the manifold $M$ and $\wedge^n TM$ the set of all $n$-forms. The map $d: \bigwedge^n TM \rightarrow \bigwedge^{n+1}TM$ is called the exterior derivative and it holds ...
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0answers
27 views

Integral Over the N-Sphere in the framework of chains:

Integration over manifolds is commonly defined with object called chains. What about if I want to integrate the exterior derivative of a $k-form$ over the n-sphere and use Stokes theorem: ...
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2answers
67 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
61 views

Poincaré Lemma, differential forms and I do have troubles

I think I need some hints about a proof I am currently reading in order to understand it. This question is similar to the construction used in Lemma 17.9 in the book "Introduction to smooth manifolds" ...
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1answer
47 views

The adjoint of left exterior multiplication by $\xi$ for hodge star operator

As we know, for $V$ vectoral space and a orientation $\mathcal{O}$ on $V$ and $e_{1},...,e_{n}$, the hodge star operator $\ast:\wedge V^*\rightarrow\wedge V^*$ is defined for ...
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3answers
672 views

Poincaré Lemma Contractible Hypothesis

Poincaré's Lemma is often stated as saying that a closed differential form on a star-shaped domain is exact. More generally, it is true that a closed differential form on a contractible domain is ...
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1answer
40 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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1answer
30 views

Pullback Solid Angle, Stereographic projection

I have an issue with a differential geometry task. Given is the solid angle form: $$\omega = \frac{\epsilon_{ijk} x^i dx^j \wedge dx^k}{2 [ (x^1)^2 + (x^2)^2 +(x^3)^2]^{3/2}}$$ The aim of the task ...
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1answer
39 views

Lie derivative for a wedge product $\omega_{1}\wedge\omega_{2}$

I have to prove that $L_X\omega_{1}\wedge\omega_{2}=(L_X\omega_{1})\wedge\omega_{2}+\omega_{1}\wedge(L_X\omega_{2})$ using the definition ...
2
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1answer
335 views

Pullback distributes over wedge product

I'm looking to prove that the pullback of a smooth function distributes over wedge product, i.e. $$\varphi^*(\omega \wedge \eta) = \varphi^* \omega \wedge \varphi^* \eta. $$ Here the question is ...
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0answers
37 views

Volume form on $(n-1)$-sphere $S^{n-1}$

Let $\omega$ the (n-1) form on $\mathbb{R}^n$ $$\omega=\sum_{j=1}^{n}(-1)^{j-1}x_{j}dx_{1}\wedge\cdots\wedge \hat{dx_{j}}\wedge\cdots dx_{n}$$ Show that the restriction of $\omega$ to $S^{n-1}$ in ...
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1answer
91 views

Is $\omega=\sin\varphi\,\mathrm{d}\theta\wedge\mathrm{d}\varphi$ an exact form?

Let $\omega=\sin\varphi\,\mathrm{d}\theta\wedge\mathrm{d}\varphi$ be a $2$-form on $\mathbb{R}^3\setminus\{0\}$. Then ...
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1answer
69 views

An interpretation of $ \frac{\partial^{2}}{\partial x^{2}} $.

$ \left( \dfrac{\partial}{\partial x} \right)_{p} $ is both an element of the tangent space $ {T_{p}}(M) $ and a linear functional on $ {C^{1}}(M) $, while $ (\mathrm{d}{x})_{p} $ is an element of the ...
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1answer
49 views

For $(n-1)$-form $\omega$ on $M^{n}$ compact, orientable without boundary, then $d\omega$ vanish for some point

Let $M^{n}$ manifold compact, orientable without boundary and $\omega$ $(n-1)$-form then there is $p\in M$ such that $d\omega(p)=0$. This is for my homework of integration on manifolds & Stokes ...
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40 views

Is this differential 2-form closed

Consider a unit sphere $S^2 \subset R^3$ and a map $\omega_p : T_pS^2 \times T_pS^2 \to \mathbb{R}$ defined by $$\omega_p(u,v) = (u \times v) \cdot p$$ How do I know is this 2-form (on $S^2$) closed ...
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1answer
35 views

Poincare: Change of form to a primitive 1-form

Given a 2-form w on $R^3$, such that dw=0; how does one find all 1-forms k such that dk=w? Provided a homotopy H(t) one can pull back w and apply the Poincare operator on it. But what if one does not ...
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1answer
20 views

Deduce that there is a cup product that is well-defined

I have showed that if $\alpha$ and $\beta$ are closed forms on a smooth manifold $M$, then $\alpha \wedge \beta$ also be closed. Further, if one of $\alpha$ or $\beta$ is exact, than $\alpha \wedge ...
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0answers
32 views

Does the wedge product of bundle-valued forms induce a universal object?

Given a smooth manifold $M$ and a vector bundle $E$ over $M$, the $C^\infty(M)$-module of $E$-valued $p$-forms on $M$ is defined to be $$\Omega^p(M; E) := \Gamma_M\left( \bigwedge^p T^*M \otimes E ...
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Wedge product of Lie algebra valued differential forms [duplicate]

Let $\mathfrak{g}$ be the Lie algebra of a matrix Lie group. Furthermore, let us consider the following $\mathfrak{g}$-valued $p$-form and $\mathfrak{g}$-valued $q$-form: \begin{equation} ...
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2answers
55 views

Differential forms in Physics

In the context of physics, I just read about the the symplectic 2-form $\omega$ on a symplectic manifold $M$ of dimension $2n$. Unfortunately, I could not follow a few arguments. I.e. it was said ...
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1answer
87 views

Closed 1-forms implies exact 1-forms

I have two problems, the first one I think I´ve prove it, but I have problems on the second one. Let $\omega$ a closed $1$-form defined on a open $U\subset \mathbb{R^2}$ and let ...
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1answer
56 views

Geometric intuition about the exterior derivative

Let $M$ be a smooth manifold. One $k$-form is a section of the bundle $\bigwedge^k T^\ast M$, that is, if $p\in M$ and $\omega$ is a $k$-form then $\omega_p$ is one $k$-linear alternating real ...
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1answer
69 views

What makes differential forms special

There are two natural structures defined on differential manifolds, namely tangent bundle and its dual cotangent bundle. An element of tangent bundle $TM$ at a base point $p\in M$ can be described by ...
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1answer
31 views

Express a quadratic form as a sum of squares using Schur complements

So I was able to figure out the first part of this problem, but I have no concept of how it relates to Schur complements, so I'm not sure (no pun intended) how to proceed. The question is as follows: ...
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34 views

1-form integration

Let $\alpha:[-1,1]\rightarrow R^2$ be the curve segment given by $\alpha=(t,t^2)$. If $\phi=v^2du+2uvdv$, (the fist component of $R^2$ is $u$ and the second one is $v$) I have $$\int_\alpha ...
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2answers
645 views

Symmetric and wedge product in algebra and differential geometry

I have been struggling with this issue for a while (and asked a similar question here), but still not found a satisfying answer. The question boils down to: which is the correct identity? $dx \, dy ...
3
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1answer
45 views

Integration with 2-forms

Wikipedia says: Let $$ \omega=f_{z}\, \mathrm dx \wedge \mathrm dy + f_{x}\, \mathrm dy \wedge \mathrm dz + f_{y}\, \mathrm dz \wedge \mathrm dx $$ be a 2-form on a surface with parametrization ...
8
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2answers
35 views

$\text{Alt}\,(\phi_1 \otimes \phi_2 \otimes \phi_3)$

How do I write out $\text{Alt}(\phi_1 \otimes \phi_2 \otimes \phi_3)$ for $\phi_1, \phi_2, \phi_3 \in V^*$?
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0answers
21 views

Proof of a result in differential forms

I want to prove the following result: Let $w = w_1 dx + w_2 dy + w_3 dz$ a 1-differential form, such that $ w_1,w_2,w_3$ are homogenous of order $\alpha$ show that if $w $ is closed then $ w = df$ ...
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1answer
46 views

Is $\omega = x^2\,dy\,dz +y^2\,dz\,dx + z^2\,dx\,dy$ exact?

this isn't a homework problem or anything. Basically is $\omega = x^2 \,dy\,dz +y^2\,dz\,dx + z^2\,dx\,dy$ exact? That is, is there a $\lambda$ such that $\omega=d\lambda$, if so what is it? I think ...
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47 views

Basic Question on Mayer-Vietoris Sequence

On Pg 449 of Lee's Introduction to SMooth Manifolds (2nd Edition), the Mayer-Vietoris Theorem is given: Let $M$ be a smooth manifold. Let $U$ and $V$ be open in $M$ such that $U\cup V=M$. Then ...
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1answer
46 views

Area form a differential form?

I just read that $\omega_x( \eta, \zeta) := \langle x, \eta \times \zeta \rangle $ is the area form on the sphere, where $x \in \mathbb{S}^2$ and $\eta,\zeta \in T_xM.$ All I see is that this is ...
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1answer
35 views

Derivative for numerical models.

I am working in Mechanical engineering and Computer vision, in which I use a matlab code (or codes) to represent a specific system or process. Of course such model has an input , an implimented ...
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1answer
24 views

rank of the symplectic form

This is a general question about ranks of differential forms. I read in a book the phrase "symplectic form has constant rank..." I understand that the symplectic form is a nondegenerate differential ...
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1answer
29 views

Compute the wedge product n times

Let $\omega$ be a 2-differential form in $\mathbb{R}^{2n}$ given by $$\displaystyle \omega=dx^1\wedge dx^2+dx^3\wedge dx^4 + \cdots + dx^{2n-1}\wedge dx^{2n}$$ Compute: $$\displaystyle ...
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1answer
34 views

how to find points where a k-form is nonvanishing.

for example, if we are given 2-form $\omega=2xdx\wedge dy+2ydy\wedge dz$, what are the points where the form vanishes? I can only think of points $(0,0,z)$, is it all? Additionally, if we have a form ...
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0answers
65 views

Integration over a manifold with boundary (Check).

Assume that $ f: \Bbb{R}^{3} \to \Bbb{R} $ is a smooth function such that $ M \stackrel{\text{df}}{=} \left\{ \mathbf{x} \in \Bbb{R}^{3} ~ \middle| ~ f(\mathbf{x}) \ge 0 \right\} $ is a non-empty ...
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1answer
35 views

An Application of Stokes's Theorem

Let $D^2=\{(x,y)\in \mathbf R^2: x^2+y^2\leq 1\}$ be the unit disc in $\mathbf R^2$, and $D^3=\{(x,y,z)\in \mathbf R^3: x^2+y^2+y^2\leq 1\}$ be the unit disc in $\mathbf R^3$. Let $i_{\pm}:D^2\to ...
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1answer
43 views

Is the form closed?

$S$ is an n dimensional unit sphere such that $S^n=(x\in \Bbb R^{n+1}: |x|=1)$ with some fixed orientation and $\omega$ is a volume form on $S$. Prove that $\omega$ is closed. Prove that $\omega$ ...
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0answers
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Compute the double Hodge star operator

I am taking a course in Multivariable Analysis and I am asked to do the following problem: Show that $\ast\ast\omega = (-1)^{k(n-k)}\omega$ So I start as follows: We know that $\displaystyle ...
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10 views

Prove a certain property of the Hodge double star operator

I want to solve the following problem Show that $\ast\ast\omega = (-1)^{k(n-k)}\omega$ where $ \displaystyle \ast\omega =\sum_I \text{sgn}(I,J)\omega_I dx^J$ and $\omega$ is a k-form in ...
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15 views

Use a Lie series in order to find the solution to initial value problem

We were presented with a fairly difficult bonus question on my multivariable calculus exam today. I was hoping you all could hope me crack it. The question is as follows: Use a Lie series to find, ...
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59 views

How to define the boundary operator using the exterior derivative?

I am looking for a way to define the boundary operator $\partial : M^n \to N^{n-1}$ from an $n$-dimensional manifold $M$ to its boundary $N$ using the the expression \begin{equation*} \int_M d \alpha ...
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1answer
26 views

Direct differentaition

In a STEP problem I found this: the vector "$\vec r$" is given as below ($a$ and $L$ are constants) and then it said to perform direct differentation to get the second equation. $$\vec r = a (\sin ...
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1answer
21 views

Basic question that has to do with exterior derivative.

Basic question: If we have $$Y=d\left(\frac{1}{\alpha} +\frac{1}{\bar{\alpha}}\right)$$ where $d$ is exterior derivative, i.e, $Y$ is a $1$-form. Now we could write that as ...
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0answers
48 views

Relationship between Cartan and Fréchet derivative

Let $f: X \rightarrow \mathbb{R}$ be smooth, then the Fréchet derivative is a map $Df: X \rightarrow L(X, \mathbb{R}).$ But if $f: M \rightarrow \mathbb{R}$ is smooth and $M$ a manifold, then the ...
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0answers
61 views

Integrability problem in Cauchy Integral Formula

This is problem on the integrability of a 2-form in the nhbd of its singularity. I was looking at the general Cauchy formula (general because it works for $\mathcal C^1$ function, and makes the case ...