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

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Derivations of the algebra of differential forms

It is well known that the interior product, the Lie derivative, and the De Rham differential are derivations of the algebra of differential forms. Does there exist other derivations of this algebra ...
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2answers
146 views

Pullback of a Volume Form Under a Diffeomorphism.

I have an exercise here, which I have no idea how to do. Problem: Let $ U $ and $ V $ be open sets in $ \mathbb{R}^{n} $ and $ f: U \to V $ an orientation-preserving diffeomorphism. Then show that ...
2
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0answers
47 views

What is the differential of a function?

I'm reading Do Carmo's Differential Forms and Applications (1st ed) and on page 6 he takes a differentiable map $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$, a point $p \in \mathbb{R}^{n}$ and a ...
0
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1answer
134 views

Finding Reeb Vector Field Associated with a Contact Form

I would greatly appreciate it if you could help me with the following: I'm curious as to how to find the Reeb field $R_w$ associated to a specific contact form $w$; does one actually find $R_w$ as ...
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1answer
49 views

Is $\omega = dU = sin(x+y)dx+cos(x+y)dy$ an exact form?

In my thermodynamics homework I should prove that $dU = sin(x+y)dx+cos(x+y)dy$ is a function of state. Which means it's integration over any path be constant or in other word $dU$ should be an exact ...
6
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3answers
496 views

Differential Forms and Vector Fields correspondence

Barrett O'Neill's Differential Geometry book says that Classical vector analysis avoids the use of differential forms on $\mathbb{R}^3$ by converting 1-forms and 2-forms into vector fields via the ...
2
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1answer
45 views

Cohomology of volume forms

If g and h are Riemannian metrics on the same manifold, say both of volume 1, then it follows (I guess from Poincaré duality) that their volume forms dvol_g and dvol_h are cohomologous. Question: is ...
5
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2answers
579 views

Cartan's magic formula

A possible proof of Cartan's magic formula $$L_X = i_X \circ d+d \circ i_X$$ is to follow the steps: Show that two derivations on $\Omega^{\bullet}(M)$ commuting with $d$ are equal iff they agree on ...
5
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1answer
152 views

Is $ds$ a differential form?

I am somewhat confused as to whether $ds$ (line element) is actually a differential form... we have (in $\mathbb{R}^2$): $$ds^2 = dx^2 + dy^2$$ Differential 1-forms are supposed to be linear ...
4
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5answers
980 views

How to calculate gradient of $x^TAx$

I am watching the following video lecture: https://www.youtube.com/watch?v=G_p4QJrjdOw In there, he talks about calculating gradient of $ x^{T}Ax $ and he does that using the concept of exterior ...
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1answer
61 views

How to prove matrix differential rules using differential forms rules

I am learning differential forms on my own through lectures on youtube and one of the things that I am attempting to do is to check if I can derive matrix differential rules using the rules for ...
4
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2answers
163 views

Why do all solutions to this equation have the same form?

In this paper on page 45, the authors state that Let's assume we know that $$ w \times dw = d\varphi + \sum_{j=1}^3\alpha_jdx_j, \tag{1}$$ where $\varphi \in H^1(\mathbb{R}^3, \mathbb{R})$, ...
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1answer
45 views

Proof and counter-example that a chain $c_{R, n} \ne \partial c$. Where is the error?

If $R > 0$ and $n \in \mathbb{Z}$ we can define the singular 1-cube $c_{R, n}\colon [0, 1] \rightarrow \mathbb{R}^2$ by $$c_{R, n}(t) = (R\cos(2\pi n t), R\sin(2\pi n t))$$ We know that $c_{R, n} ...
5
votes
1answer
102 views

A form problem between $S^3$ and $S^2$.

Let $\phi: S^3 \rightarrow S^2$ be an smooth map. a) Suppose that $\omega$ is a 2-form on $S^2$ with $\int_{S^2} \omega =1$. Show that there exists a 1-form $\alpha$ on $S^3$ with ...
4
votes
1answer
159 views

The homology of the torus.

I am reading "Riemann surface" by Donaldson. On page 68, the calculation of the first homology of the torus $T$ is given but there are several steps that I don't understand. Here is the calculation. ...
1
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1answer
70 views

How to show $[\omega]=0$ implies $[\omega^n]=0$?

I'm trying to prove the following: If $(M, \omega)$ is a symplectic manifold and $[\omega]=0$ then $[\omega^n]=0$, where $[\omega]$ is the De Rham cohomology class of $\omega$. Well what I've done ...
5
votes
1answer
170 views

1- forms on a torus

I think this is a very simple question but I'm not really confident in mathematics (even if I like it very much) Let's fix a cube $[0,1]^3$ in $R^3$ and identify opposite sides, so as to construct a ...
9
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1answer
240 views

Homework: closed 1-forms on $S^2$ are exact.

From the 2008 UCLA Geometry-Topology qualifying exam: let $\theta$ be a $1$-form on $S^2$ with $d \theta = 0$. Construct a function $f$ on $S^2$ with $d f = \theta$. I'm not very confident in my ...
3
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1answer
249 views

A $2$-form on $S^2$ is exact if it integrates to zero.

I'm trying to show that a $2$-form on $S^2$ is exact if and only if it integrates to zero, without appealing to de Rham's theorem (basically only using the Poincaré lemma [that every closed form on a ...
4
votes
1answer
93 views

Integrating 2-form

In $\mathbb{R}^3$ I consider the compact 2-dimensional manifold $$ M=\left\{(x,y,z)\in\mathbb{R}^2: z=xy\right\} $$ which is orientated by the (global) map ...
5
votes
2answers
279 views

Why is arc length not a differential form?

I read that the arc length is not a differential form. But I don't understand why it isn't. I understand that differential forms are integrands and arc length is an expression which is integrable. ...
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0answers
60 views

Show that $\int_Md\omega=0$.

Let $\omega$ be a continiously differentiable $(k-1)$-form in the open set $U\subset\mathbb{R}^n$ and $M\subseteq U$ an orientated compact k-dimensional manifold. Show that $$ ...
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1answer
109 views

Relation between de Rham cohomology and integration

This question is a follow-up to When does a null integral implies that a form is exact? . As mentionned in the selected answer, given certain conditions it is possible to find an isomorphism between ...
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3answers
540 views

What is the motivation for differential forms?

I am that point in my mathematical career where I am learning differential forms. I am reading from M.Spivak's Calculus on Manifolds. So far I have gone over the tensor and wedge products and their ...
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votes
3answers
105 views

Inner product of De Rham cohomology classes

Is there a well-defined inner product between cohomology classes? In particular, is it possible to extend the Hodge inner product? If I try, I obtain this: $$\int *(\omega + d\lambda)\wedge (\sigma + ...
3
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1answer
110 views

What do we mean when we say a differential form “descends to the quotient”?

Let $S$ be a surface and let $f:S\rightarrow S$ be a diffeomorphism. We define the mapping torus $M_f$ of the pair $(S,f)$ to be the quotient $$(S\times I) /\sim \quad \text{ where } \ (1,x) \sim ...
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votes
3answers
432 views

When does a null integral implies that a form is exact?

It is trivial to prove that the integration of a $(n-1)$ exact form on the boundary of a $n$-manifold is 0. What about the contraposative ? If the integration of a $(n-1)$-form on the boundary of a ...
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2answers
126 views

“Ordinary” and “polar” vector fields in Euclidean $3$-space

In his book Differential Forms with Applications to the Physical Sciences, on pages 19--20, Harley Flanders writes: "a one-form $$ \omega = P\,dx+Q\,dy+R\,dz$$ may be identified with an ...
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1answer
100 views

symplectic strucutre

Suppose $\omega$ is symplectic structure on $\mathbb R^n$. Let $\omega_0:=\omega|_{x=0}$. Let $\overline{\omega}= \omega_0-\omega$ and for $t\in[0,1]; \omega_t:= \omega+ t\overline{\omega}$. How ...
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0answers
134 views

Intuitive meaning of $L_X\omega$

Suppose $\alpha$ is a one form and $\omega$ is two form such that $$\alpha= \omega(X,.)\text{ for some } X$$ Then what does intutive meaning of the following expression $$L_X\omega= \omega$$
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1answer
206 views

Lie derivative along time-dependent vector fields

In "Lectures on Symplectic Geometry" by A. C. da Silva (http://www.math.ist.utl.pt/~acannas/Books/lsg.pdf) the author gives the following definition: $$ \mathcal{L}_{v_t} := \frac{\mathrm d }{\mathrm ...
11
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3answers
683 views

Geometric understanding of differential forms.

I would like to understand differential forms more intuitively. I have yet to find a book which explains how the use of the exterior product in differential forms ties into the geometrical ...
3
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1answer
132 views

De Rham cohomology notation

According to http://en.wikipedia.org/wiki/De_Rham_cohomology, one defines the $k$-th de Rham cohomology group $H^{k}_{\mathrm{dR}}(M)$ to be the set of equivalence classes, that is, the set ...
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1answer
370 views

Writing Integrals using Differential Forms

Consider some smooth curve $C \subset \mathbb{R^n}$ and $\gamma:[a,b] \subset\mathbb{R}\rightarrow C$ a parametrisation of $C$ and a continuous vector field $K:\mathbb{R^n} \rightarrow \mathbb{R^n}$. ...
2
votes
1answer
82 views

Question on Differential forms

Let $\Omega\subseteq \mathbb{R}^2$ be an open set and $\omega=\omega^1dx_1+\omega^2dx_2$ an $1$-form on $\Omega$ and $$L=\omega^2\frac{\partial}{\partial x_1}-\omega_1\frac{\partial}{\partial ...
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1answer
28 views

Question on 2-chain on $\mathbb{R}^3$

Let $\gamma:[0,1]\to\mathbb{R}^3\setminus\{0\}$ be a simplex, with $\gamma(0)=\gamma(1)$. How can I show that exists a $2$-chain $\sigma$ on $\mathbb{R}^3\setminus\{0\}$ such that ...
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vote
1answer
181 views

Locally exact differential in a disk is exact

I'm reading through Ahlfors' Complex Analysis text for self study, and I found difficulty with a proof. In chapter 4 he defines a locally exact differential as a differential who is exact in some ...
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0answers
51 views

What exactly does it mean to say that “functions cannot be integrated on Riemann surfaces”?

I've seen statements of this sort used to motivate the introduction of differential forms, and I'm not sure exactly what's meant. Obviously if you start by defining differentiation as an operation ...
2
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0answers
165 views

Exact and Closed forms on Manifolds with Boundary

Let $\bar{M}$ be a manifold with boundary and let $M$ be its interior. Is this statement correct? A smooth k-form $\alpha$ on $\bar{M}$ is closed (exact) if and only if its restriction to $M$, i.e. ...
3
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2answers
126 views

Prove that $\frac{1}{2\pi}\frac{xdy-ydx}{x^2+y^2}$ is closed

I would like to prove that $\alpha = \frac{1}{2\pi} \frac{xdy-ydx}{x^2+y^2}$ is a closed differential form on $\mathbb{R}^2-\{0\}$ . However when I apply the external derivative to this expression ...
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1answer
202 views

Clarification about differential forms in polar coordinates

In my course about differential forms, we define 1-forms as follows: If $(e_1,..,e_m)$ is the standard basis of $\mathbb{R}^m$ and $\sigma$ a chart around $p\in M$ on the m-dimensional manifold $M$, ...
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1answer
94 views

A differential form to compute the k-volume of a k-parallelogram in n dimensions

Computing the k-volume of a k-parallelogram (i.e. a parallelogram spanned by k n-dimensional vectors) in n dimensions is straightforward: Let $P=[\overrightarrow{v_1},...,\overrightarrow{v_k}]$, then ...
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2answers
235 views

Integral of wedge product of two one forms on a Riemann surface

I'm having trouble verifying an elementary assertion made in this answer on MathOverflow. It seems more like a math.stackexchange question, so I'm asking it here. Anyway, the assertion is as follows ...
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1answer
132 views

Existence of a left-invariant $n$-form on a Lie group of dimension $n$

This Do Carmo, Riemannian Geometry, Chapter 1, Exercise 7: Show that there exists a left invariant differential $n$-form $\omega$ on $G$ ($G$ is a compact connected lie group and $\dim G=n$). ...
3
votes
1answer
89 views

Algorithm/Procedure for finding $\sigma$ such that $\omega=d\sigma$

I know that the Poincare's lemma asserts that under certain conditions a differential form $\omega$ is exact, i.e. it possesses an antiderivative $\sigma$, such that $\omega=d\sigma$. But as ...
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vote
2answers
70 views

Equality involving exterior product..

suppose you have a differential form $\omega$ writting in local coordinates as $$\omega=\sum_{i=1}^ndx_i\wedge dy_i.$$ Can anyone help me showing the following equality: $$\omega^n=n!(dx_1\wedge ...
4
votes
1answer
57 views

Smooth homotopic maps and closed forms..

does anyone have any idea for showing the following: Let $f_0, f_1:M\rightarrow N$ smooth homotopic maps between the manifolds $M$ and $N$. Suppose $M$ is compact with no boundary. Show that for every ...
8
votes
1answer
211 views

Maurer-Cartan 1-form

Can anyone help me with the following? Let $\rho$ be the right-invariant Maurer-Cartan 1-form $$\rho = dg\ g^{-1}$$ I want to show that the MC equation $$d\rho - \rho \wedge\rho = 0$$ holds. So ...
3
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1answer
251 views

Curvature tensor of 2-sphere using exterior differential forms (tetrads)

$ds^2= r^2 (d\theta^2 + \sin^2{\theta}d\phi^2)$ The following is the tetrad basis $e^{\theta}=r d{\theta} \,\,\,\,\,\,\,\,\,\, e^{\phi}=r \sin{\theta} d{\phi}$ Hence, $de^{\theta}=0 ...
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votes
4answers
475 views

Classic example of a non exact form

Let $\dfrac{xdy-ydx}{x^2+y^2}$ be a 1-form defined in $\mathbb{R}^2\backslash\{0\}$. Where can I find a detailed proof that it is not exact? I would prefer a proof that doesn't use results about ...