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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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Poincare residue and trivializing section of canonical bundle of plane cubic.

I am trying to get my hands dirty and do the following computation, but I don't feel like I'm doing it right. Help would be very much appreciated! I will tell you the setup of the calculation, and ...
maxo's user avatar
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0 answers
64 views

Homological Algebra for Analysis

While studying differential forms, I encountered some concepts from homological algebra, such as (co)chain complexes, de Rham cohomology, pullbacks, and others. Is it reasonable to study the basics of ...
veirab's user avatar
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2 votes
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Relationship between the spectra of the zero and first Hodge Laplacian on 2 dimensional manifolds.

Considering an oriented and compact surface embedded in $\mathbb{R}^3$. I would like to know if there exisits a particular relationship between the spectrum of the Laplace-Beltrami operator $\Delta_0 =...
Ricardo Gloria's user avatar
3 votes
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50 views

Properties of the de Rham complex for $\mathbb{R}^{3}$

I wasn't able to find a construction of de Rham complex for $\mathbb{R}^{3}$ using de Rham theorem. This is my attempt, in which I have some uncertainties. Consider $$ \Omega^{0}(\mathbb{R}^{3},\...
Matthew Willow's user avatar
1 vote
1 answer
58 views

Čech cohomolog of a good cover of the real projective plane

I was reading the Bott/Tu book and trying compute the Čech cohomology of a good cover of the real projective plane (exercise 9.10). The nerve of the cover is dipicted as below: we glue the opposite ...
Xipan Xiao's user avatar
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2 votes
0 answers
46 views

Hodge star decomposition in non-diagonal manifold product

I'm studying differential forms and I came across the following problem. From what I learnt in another question, when a manifold can be decomposed as $X \times Y$, then the formula found there works ...
Fredrigo6's user avatar
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0 answers
79 views

Why isn't there a nice vector integration theory on smooth manifolds?

I'm very unfamiliar with differential geometry, but recently I'm forced to learn it. One thing I've noticed is it seems differential forms only works nicely for real valued functions defined on smooth ...
user760's user avatar
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2 votes
1 answer
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$\Lambda^n(M)$ is not isomorphic to $C^{\infty}(M)$ if M is not orientable

Let $M$ be a differentiable Manifold of dimension n. If $M$ is orientable, then there exists an $\omega \in \Omega^n(M)$ (top-degree differential form) such that $\omega(p) \neq 0$ $ \forall p \in M$. ...
Jahi02's user avatar
  • 301
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0 answers
40 views

Question about moving Hodge $\star$ to the argument of a 1-form

Let $\alpha$ be a 1-form on an $n$-dimensional vector space $V$ and $v_1,...,v_{n-1}$ (1-)vectors in $V$. Is it true that $$ \star \alpha(v_1,...,v_{n-1}) = \alpha\big(\star(v_1 \wedge ... \wedge v_{n-...
Niels Slotboom's user avatar
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65 views

Partial Derivative of Pullback of Differential form

I'm new to differential forms and the book I'm reading contains a part I don't understand. It states the following: Let $k\geq 1$ and assume that $D \subset \mathbb{R}^k$ and $U \subset \mathbb{R}^n$ ...
Josef K.'s user avatar
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Lie derivative of 1 form

I want to derive $\mathcal{L}_X (\omega) = (i_x d + d i_x) \omega$ for 1-form. I read Lie derivative of 1-form and it makes sense but I don't kmeow why my thing is not working. What I tried was $$ (\...
池田隼's user avatar
1 vote
0 answers
35 views

Explicitly understanding forms vs densities on a Moebius strip

I encountered a post John Baez made on some old Usenet thread in which he talked about how in order to integrate things on a non-orientable manifold you need pseudoforms/densities rather than forms, ...
Andreas Christophilopoulos's user avatar
-1 votes
2 answers
110 views

Confusion about differential forms and integration

I'm self-studying general relativity using Sean M. Carroll's textbook. I recently made it to sections 2.9 and 2.10, which talk about differential forms and integration of functions on manifolds. I ...
Aidan Beecher's user avatar
2 votes
1 answer
92 views

Can this closed form not exist on the sphere?

This is part (b) to a problem that I solved part (a) for a while back. I finally had time to complete part (b), and I wanted to know if this is a valid answer (proved correctly): Problem Statement: ...
Nate's user avatar
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2 votes
1 answer
64 views

Showing a form is closed using integrals

Let $M$ be a manifold and $\omega \in \Omega^r(M)$ be an $r$-form. Suppose $$\int_N \omega = 0$$ for all submanifolds $N$ of dimension $r$ that are diffeomorphic to a sphere. Show that $\omega$ is ...
Nancium's user avatar
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Positivity of the first fundamental form

my question is how to prove that the first fundamental form is positive definite. I know what the first fundamental form is, that is the restriction of the scalar product to all tangent spaces, that ...
Vuk Stojiljkovic's user avatar
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45 views

Differential Forms and Path Integrals

How to prove this problem from Fulton Book (Algebraic Topology) Show that an open set U in the plane is connected if and only if there is a segmented path between any two pints of U. Can you show that ...
Haxhi Dacaj's user avatar
1 vote
0 answers
55 views

Is this the correct value for integrating this differential form over a sphere?

I posted this problem but made a mistake. I redid my work and got another sensible looking answer (the volume of the unit sphere). Does this look like a correct solution? This isn't an assigned ...
Nate's user avatar
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1 vote
0 answers
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Error in "Principles of Algebraic Geometry" by Griffiths and Harris

At page $148$ of "Introduction to Algebraic Geometry", Griffiths and Harris define a positive line bundle as a line bundle $L\to M$ with a metric such that $(i/2\pi)\Theta$ is a positive $(1,...
Temoi's user avatar
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1 answer
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$L^2(\mathbb{R})$ 1-form not differential of $L^2(\mathbb{R})$.

In my attempt to solve following exercise: Construct a smooth 1-form on $\mathbb{R}$ with $\int |\psi|^2 dt < \infty$ for which there exists no function $f$ on $\mathbb{R}$ such that $\int |f|^2 ...
Pastudent's user avatar
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2 votes
1 answer
35 views

Is the Hodge dual operation metric independent?

I'm trying to find out if the Hodge star operation is metric-independent, and I've come across two definitions of it. One, from Wikipedia, states that you can define it by the property $$\alpha \wedge ...
Pau Bañón Pérez's user avatar
0 votes
0 answers
60 views

Fourier transform of differential form

I'm trying to make sense of how to take the Fourier transform(FT) of the Yang-Mills gauge field $A=A_\mu dx^\mu \in \Omega^1(M)$, where $M$ is $\mathbb{R}^4$, let's say. A shortcut to this problem ...
Richard122's user avatar
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0 answers
23 views

Induced action on vector valued differential forms

Let $F(X)$ be the frame bundle of a smooth $n$-dimensional manifold $X$. Let $G$ be a finite subgroup of $GL_n(\mathbb C)$ acting by automorphisms on $X$ and let $V$ be a $GL_n(\mathbb C)$-...
Flavius Aetius's user avatar
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0 answers
65 views

Wedge Product and Differential Forms, example

Let $x=id_{\mathbb{R}^4}$, $\alpha=dx^1+x_2dx^2\in \Omega^1\mathbb{R}^4$, $\beta=\sin(x_2)dx^1\wedge dx^3+\cos(x_3)dx^2\wedge dx^4\in \Omega^2\mathbb{R}^4$, $h(x_1, x_2, x_3, x_4)=(x_1, x_2, x_3x_4, ...
Lu1998's user avatar
  • 27
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0 answers
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Tu, Exercise 18.5

In Exercise 18.5 Tu asks the reader to find the transition formula of a 2-form on an n-manifold. He provides the formula below before asking the reader to find $a_ij$ in terms of $b_{kl}$ $\omega = \...
EEH's user avatar
  • 83
1 vote
1 answer
94 views

Equivalent forms of second Bianchi identity on $TM$

$\DeclareMathOperator{End}{\mathrm{End}}$ This question is already asked here Second Bianchi identity on tangent bundle but with no answer. Let $M$ be a smooth manifold, and $E \to M$ a smooth vector ...
Alex Pawelko's user avatar
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49 views

A Question Regarding Cartan’s Absorption Method

I want to ask a question from the book named “ Cartan for Beginners : Differential Geometry via Moving Frames and Exterior Differential Systems” as to how one can absorb an apparent torsion. Suppose ...
iliTheFallen's user avatar
3 votes
1 answer
176 views

Understanding Newtonian mechanics using concepts from differential geometry

In a book I'm reading (Friedrich and Agricola), I encountered the following definition of a "Newtonian system": An autonomous Newtonian system is a triple ($M^m$, $g$, $X$) consiting of a ...
guibor's user avatar
  • 135
3 votes
2 answers
83 views

Computing pullback of the one form $dx$

I have the following 1-form in $\mathbb{R}^2$ which is $\omega=dx$, and it is given the map $$\varphi: \mathbb{R}^2 \to S^2 \setminus \lbrace N \rbrace $$ $$ (x,y)\mapsto \left( \frac{2x}{x^2+y^2+1}, ...
Daniel R.S's user avatar
1 vote
1 answer
69 views

What does it mean to integrate with respect to the complex conjugate.

I am an undergraduate mathematical physics student doing a summer project. I am very familiar with "mathematicicans complex analysis", but am having difficulty with how fast and loose with ...
Jack's user avatar
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4 votes
1 answer
287 views

Linearity of differential forms

I put my hands on "Linear Algebra" by Serge Lang, second edition, and I noticed that it contains some sections that were later removed in the following third one. In one of the removed parts ...
Andreas Compagnoni's user avatar
0 votes
1 answer
44 views

Why $\theta ([\xi_f, \xi_g])=\frac{1}{2}\xi_f \langle \theta ,\xi_g\rangle -\frac{1}{2}\xi_g \langle \theta ,\xi_f\rangle$?

Let $(M,\omega)$ be a symplextic manifold with $\omega =-d\theta$. For $f,g\in C^{\infty}(M)$, we can define a Poisson bracket $\{ f,g\}=\omega (\xi_{f},\xi_f)$ where $i_{\xi_f}\omega=\omega(\xi_f,\...
Mahtab's user avatar
  • 769
0 votes
0 answers
35 views

Poincare's Lemma for 1-form on star-shaped domain in $\mathbb{R}^n$ [duplicate]

There are many proofs available in textbooks and online for Poincare's lemma for $k$-forms, but I am having trouble understanding them and hoping a simple case will help. For example, The wikipedia ...
utx7563yu's user avatar
1 vote
0 answers
32 views

Sign of the permutation when I show that $\ast \ast w= (-1)^{n(n-k)} w$ for the Hodge operator

Let $w=\sum_{I} a_{I}dx_{I}$ be a $k$-form in $\Bbb R ^n$. Let us consider the Hodge operator in a combinatorial form, i.e. as an $(n-k)$ form such that $$\ast(dx_{i_{1}} \wedge \cdots \wedge dx_{i_{...
Wrloord's user avatar
  • 1,810
0 votes
0 answers
63 views

How to show that a differential form is not exact?

I want to show that the differential form $w = x^2\sin(y) dx \wedge dy + 2x \sqrt{1+y^4} dx \wedge dz \in \Omega(\mathbb{R}^3)$ is not exact. Would it be enough to show that $w$ is not closed? Or does ...
seitanist.snail's user avatar
0 votes
1 answer
70 views

Why don't I see "vector-valued vector fields"?

Let $P\xrightarrow{\pi}M$ be a principal $G$-bundle with connection $\omega\in \Omega^1(P;\mathfrak{g})$. When I am studying these things, there are Lie-algebra valued differential forms all over. ...
Wyatt Kuehster's user avatar
2 votes
1 answer
79 views

The natural map from compact vertical cohomology to de Rham cohomology is not injective

Let $\pi:E\to M$ be an oriented vector bundle. In Bott-Tu's book Differential Forms in Algebraic Topology, the compact vertical cohomology $H^*_{cv}(E)$ is defined by using differential forms $\omega$ ...
user302934's user avatar
  • 1,620
0 votes
0 answers
17 views

Holomorphic one form and its association to Minimal surface

I am new to studying holomorphic one form. While studying minimal surface, it is written that in The Weierstrass-Enneper Representation $\mathbf{X}(z)=\Re\int \left( \frac{1}{2} (1 - g^2) \, dh, \frac{...
bikram poddar's user avatar
1 vote
0 answers
49 views

Compute d$\omega$ in Cartestian coordinates for a given $\omega$

Define a $2$-form $\omega$ on $\mathbb{R}^3$ by $$\omega=x\text{d}y\wedge\text{d}z+y\text{d}z\wedge\text{d}x+z\text{d}x\wedge\text{d}y$$ Compute d$\omega$. Using the formula for $$\text{d}\omega=\text{...
Superunknown's user avatar
  • 2,971
2 votes
4 answers
152 views

Understanding standard basis of the tangent space

Reading Do Carmo, differential forms, when the tangent space is defined there it says that the vectors $e_i$ form a canonical basis for $\mathbb{R}^n_0$, and then it considers the "translates&...
Daniel R.S's user avatar
0 votes
0 answers
20 views

Help in proving statements related with the gradient.

I'm given a differentiable function $f: \mathbb{R}^n \to \mathbb{R}$, (i.e scalar field), and we define $$ \langle grad \hspace{0.5mm} f(p), u \rangle = df_p(u) $$ for all $p \in \mathbb{R}^n$ and all ...
Daniel R.S's user avatar
1 vote
0 answers
56 views

Decomposition of Differential Forms

Bott-Tu's Differential forms in algebraic topology says every differential form over $\mathbb R^n\times \mathbb R=\{(x,t)\}$ uniquely decomposed into two types of forms, one type with $d t$ and one ...
Eric Ley's user avatar
  • 768
0 votes
1 answer
43 views

Solve the system of differential equations associated with an ellipsoid.

Consider the ellipsoid, given by $$ f(u, v) = \bigl( a\sin(u)\cos(v), b\sin(u)\sin(v), c\cos(u) \bigr), \quad 0 \leq u \leq \pi, \quad 0 \leq v \leq 2\pi, $$ and consider the $1$-form in $\Bbb R^2$ ...
Wrloord's user avatar
  • 1,810
0 votes
0 answers
39 views

Applications of Signature of 4k Dimensional Manifold?

Algebraic topology finds many applications outside of pure math. For instance, differential forms (at least 0- and 1-forms) find applications in engineering thermodynamics. The state of a ...
Jeffrey Rolland's user avatar
0 votes
0 answers
62 views

Abuse of Leibniz notation on differential n-forms

Leibniz notation is notorious for being abused. e.g. Chain rule: $$\frac{dy}{dt}= \frac{dy}{dx}\frac{dx}{dt}$$ Inverse Function Theorem: $$\frac{dy}{dx} = \left(\frac{dx}{dy}\right)^{-1}$$ Separation ...
Leon Kim's user avatar
  • 525
2 votes
2 answers
80 views

Do functions distribute over the wedge product?

I am wrapping my head around the arithmetic properties of the wedge product. I understand that constants do distribute over the wedge product, i.e. for $c_1,c_2\in\mathbb{R}$, it holds \begin{equation}...
seitanist.snail's user avatar
0 votes
0 answers
38 views

Let $\omega =\rho d\theta$ be a volue form of the circle $S^1$. Who are the diffeomorphisms of the circle that let $\omega$ invariant?

Consider the parametrization $\phi:]0,2\pi[\to S^1$ given by $\phi(\theta)=e^{i\theta}$. So the Lebesgue measure is given in local coordinates by the form $d\theta_z(\partial_z)=1$. I know that the ...
Gomes93's user avatar
  • 2,155
6 votes
0 answers
68 views

Do differential forms fully describe the geometry?

It's clear to me that differential forms contain a lot of geometric information about a differential manifold, they describe volume elements, determinants, orientation, connections, curvature, de Rham ...
Diana's user avatar
  • 77
3 votes
1 answer
55 views

Applying fundamental theorem of calculus where upper integration bound for F's associated integral is not equal to F's input

Normally, the FTC is stated like this: Fundamental Theorem of Calculus: Let $f$ be Riemann integrable on $[a, b]$. For $x \in$ $[a, b]$, define $F(x)=\int_a^x f$. If $f$ is continuous at a point $x \...
Charlie Weiler's user avatar
0 votes
1 answer
30 views

calculating over the Hodge operator in a two form in $\Bbb R ^3$

If $w$ is a $k$-form in $\Bbb R ^n$, we define a $n-k$ form by $$\ast(w)=\sum a_{i_1, \dots, i_k} \ast(dx_{i_{1}} \wedge \dots \wedge dx_{i_{k}})=\sum a_{i_1, \dots, i_k} (-1)^{\sigma}(dx_{j_{1}} \...
Wrloord's user avatar
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