Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [differential-forms]

For questions about differential forms, a class of objects in differential geometry and multivariable calculus that can be integrated.

4
votes
0answers
32 views

Computing $\int_{S^{1} \times S^{1}} d\theta_{1} \wedge d\theta_{2}$

Let $S^{1} \times S^{1}$ be the torus embedded in $\mathbb{R}^{4}$. I want to compute $\int_{S^{1} \times S^{1}} d\theta_{1} \wedge d\theta_{2}$ I believe this should be "essentially" $\int_{0}^{2\...
0
votes
0answers
27 views

Holomorphic forms: a very basic notion

Let $M$ be a holomorphic (complex) manifold with: $$(1)\quad dim_{C}(M)=m. $$ What I understand regarding a holomorphic form is that it is: $$(2)\quad \alpha^{(r,0)}=\frac{1}{r!}f_{\mu_1,\dots,\mu_r}...
0
votes
1answer
38 views

What sequence converges to f'''(x) [on hold]

We know that $f'(x)=\lim_{h\to0} \frac{f(x+h)-f(x-h)}{2h}$ and $f''(x)=\lim_{h\to0}\frac{f(x+h)+f(x-h)-2f(x)}{h^2}$. What about $f'''(x)$?
0
votes
0answers
16 views

Integration of a (0,1)-form on the boundary of a Riemann surface

In Simon Donaldson's book, he says that for any (0,1)-form $\theta$ on a compact connected Riemann surface $X$, the integral of $\partial\theta$ over $X$ is zero by Stokes' theorem - but that seems ...
0
votes
1answer
41 views

Compute this integral of a form

I need to compute the integral: $$ \int_{\mathbb{S}^3}\alpha, $$ where $\alpha\in \Omega^3(\mathbb{R}^4)$ is given by $\alpha=-4t\;dx\wedge dy\wedge dz+4z\;dx\wedge dy\wedge dt-4y\;dx\wedge dz\wedge ...
0
votes
0answers
14 views

Question about proof of n-1 form inducing normal unit vector field

Suppose we have a $n-1$ dimensional manifold $M \subset \mathbb{R}^n$ and a non-vanishing $n-1$ form $\omega$ on $M$. This implies the existence of a normal unit vector field on $M$. The proof of ...
2
votes
1answer
37 views

n-1 form inducing normal unit vector field

Suppose we have a $n-1$ dimensional manifold $M \subset \mathbb{R}^n$ and a non-vanishing $n-1$ form $\omega$ on $M$. How would this imply the existence of a normal unit vector field on $M$?
0
votes
2answers
36 views

Show that $i^*\omega=-d\alpha$ where $\omega$ is canonical symplectic structure.

I am stuck on the following problem. Let $\omega$ be the canonical symplectic structure. Let $\alpha$ be a differential 1-form on $M$, and let $$L=\{(p,\alpha(p)):p\in M\}\subseteq T^*M.$$ ...
0
votes
2answers
29 views

Moving frame method with non-matrix Lie group

I am trying to understand the modern formulation of the moving frame method for Lie group acting on a manifold. I know the following theorem Let be $M$ a manifold, $G$ a Lie group and $\omega$ the ...
0
votes
1answer
25 views

How/why does the contraction of standard volume form give the canonical form.

$M \subset \mathbb{R}^{N}$ is a (oriented) $n-1$ dimensional submanifold. Suppose $\nu \in T_{p}M^{\bot}$, of length one (a normal unit vector on $M$). How and why does the contraction $\nu_{\neg}(...
0
votes
0answers
30 views

integrate a differential form over the indicated smooth cube

I'm supposed to integrate the differential form over the smooth cube and I'm not sure if I'm on the right track. So suppose, $ \int_c x dy+ y dz$ where $c: [-1,1] \rightarrow \mathbb{R^{3}}$ is $c(...
0
votes
0answers
50 views

Joint probability distribution using differential algebra

I came across this and this answers explaning how to compute joint probability distribution by using differential forms, which I think it is an elegant method to deal with this kind of problem. I have ...
0
votes
2answers
50 views

Differential forms and order of integration

I don't understand how $$ \int_{a_2}^{b_2} \int_{a_1}^{b_1} f(t_1,t_2) dt_1 dt_2 = \int_{a_1}^{b_1} \int_{a_2}^{b_2} f(t_1,t_2) dt_2 dt_1 $$ can agree with the fact that $dt_1 \wedge dt_2 = -...
0
votes
0answers
16 views

Find the values $\omega_{X}(v)$ of the $1$-forms of $\omega$ at all possible locations $X$ and directions $V$

I just started reading from a differential forms book, and I've been struggling with the following problem: Find the values $\omega_{X}(v)$ of the $1$-forms of $\omega$ at all possible locations $...
0
votes
0answers
57 views

Differential Forms and Applications by do Carmo - Chapter $6$ - Lemma $1$.

Let be $M$ a compact manifold $M$ which has a finite number of singular isotaled points and $I$ the index of a vector field around a singular isolated point. The definition of the index $I$ of a ...
0
votes
1answer
48 views

Are these definitions of a differential form equivalent?

The definition from my notes says that a differential $k$-form is a section of $\bigwedge^k T^*X \rightarrow X$, so $\omega \in \Omega^k(X)$ would be a map $\omega : X \rightarrow \bigwedge^k T^*X$ ...
1
vote
0answers
38 views

Cohomology of n-sphere minus k discs

If $M=S_n \backslash K$, where $K$ is the union of $k\geq1$ disjoint disks $D_i$, how would you compute the de Rham cohomology of $M$?
0
votes
0answers
37 views

Differential 1-forms on an irreducible projective variety

Let $k$ be an algebraically closed field and let $X$ be an irreducible projective variety over $k$. I am wondering what the module of differential 1-forms on $X$ is. Since $X$ is a projective variety,...
0
votes
0answers
20 views

Expression of $n$-form with two charts.

Let $(U , \varphi)$ and $(V , \psi)$ be two charts on a $n$-dimensional differentiable manifold $M$, with $U \cap V \neq \emptyset$, such that $\varphi = (x_1 , \ldots , x_n)$ and $\psi = (y_1 , \...
0
votes
1answer
25 views

Double cross product in 2D

Hello I have a question about a double cross product, appearing in centrifugal force \begin{align*} \mathbf{F}_{centrifugal} = -m \boldsymbol{\omega} \times [\boldsymbol{\omega} \times \mathbf{r}] \, ....
1
vote
0answers
9 views

Locally closed in the sense of distributions implies closed?

Let $F \in L^{p}(\mathbb{(-1,1)^{n}}; \Lambda^{2}\mathbb{R}^{n})$ be an $L^{p}$ $2$-form on the open cube $(-1,1)^{n}.$ Suppose we know that for every $x \in (-1,1)^{n},$ that there exists $0 < r = ...
0
votes
0answers
19 views

Relationship between between “calculus differentials” and exterior differentials

I'm having a hard time understanding differentials. I know this topic has been covered here in StackExchange, but I really can't find an answer to my question here. So, I've been taught two different ...
3
votes
1answer
72 views

First de Rham Cohomology group of the 2-torus

Show that for the de Rham cohomology, $H^{1}_{dR}({T^{2}})$ is isomorphic to $\mathbb{R}^{2}$ by showing that the following map: $[\alpha] \to (\int_{S^{1}}f^{*}_{1}\alpha,\int_{S^{1}}f^{*}_{1}\alpha)$...
2
votes
0answers
45 views

Exterior Derivative over Quaternions

I was wondering if it is possible to define the exterior derivative of a quaternionic valued function. I am doing the quaternionic analogue of a previously complex valued computation, namely something ...
0
votes
0answers
21 views

How to calculate the wedge product of differential forms with arbitrary coefficients

I need to calculate the wedge product between some differential forms of the type:   $\omega=P_1(x_1, ..., x_n)dx_1+\cdots+P_n(x_1, ..., x_n) dx_n$ and $d\omega$, i-e, $\omega\wedge d\omega$. where ...
4
votes
0answers
37 views

Integrating on a Manifold

I'm new to working with differential forms and integrating over manifolds. I think that I have the following problem solved, but I'm not all that confident in my work. Let $D=\{(x,y,z)\in\mathbb{R}...
1
vote
0answers
33 views

Integration of the sphere

Given the unit sphere $S^1$, and a 1 form $\alpha$ on $S^1$, Let $F:(0,2\pi)\to S^1$ be $F(\theta) = (cos \theta, sin \theta)$ be a parametrization of the sphere. Then we know that $\int_{S^1} \alpha =...
0
votes
0answers
25 views

Calculating lie derivatives and change of coordinates

I have the following exercise questions from my differential geometry course: Let $X(r) = y \frac{\partial}{\partial x} + x \frac{\partial}{\partial y}$, $r = (x, y)$ be a vector field on $\mathbb{R}^...
4
votes
1answer
91 views

Covariant derivative: QFT vs. Math

In class, we have seen that the covariant derivative of some form $R$ can be written as: $$DR = dR + [A, R] = dR + A\wedge R - R\wedge A \tag1$$ Here, $d$ represents the external derivative over ...
0
votes
0answers
17 views

Evaluate integral of a differential-2 form over a rectangular plane S with given vertices

I am new to differential forms and I am currently working on this question Evaluate the integral I of the given 2-form over the rectangular plane S with the following vertices: (0,0,0),(0,1,1),(1,1,...
0
votes
1answer
25 views

Expressing the coordinate dependent and indepent forms of the $(0,1)$ tensor in different coordinate systems

I'm studying general relativity and and learning up on tensors through a lecture series. It says that $\omega_\mu$ represents a 1-form in the $x^\mu$ coordinate system. The coordinate independent 1-...
0
votes
1answer
45 views

Showing that a 2-form on an odd dimensional space is not degenerate

On an odd-dimensional space $\mathbb R^{2n+1}$ with coordinates $x_1...x_n;y_1...y_n;t$ consider the following 2-form: $$\omega^2=\sum dx_i \land dy_i-\omega^1 \land dt$$ where $\omega^1$ is any 1-...
0
votes
1answer
56 views

Differential Forms and Applications by do Carmo - Divergence theorem

I read the proof of Stokes theorem for manifolds by do Carmo's book and I'm trying understand an example (the Divergence theorem) given after the proof of Stoke's theorem, but I didn't understand. A ...
1
vote
0answers
34 views

Find differential forms invariant under local flow

Problem 6, UMD August 2018 Topology/Geometry exam Let $\xi = \frac{d}{dx}, \eta = x\frac{d}{dx}$ (on the real line). Find local flows for these vector fields. a) Prove or disprove: These integrate ...
1
vote
2answers
62 views

Fubini's Theorem, but $dxdy = -dydx$

As differential $2$-forms, clearly $dxdy=-dydx$ by alternation. Yet just as clearly, $\int_0^1 \int_0^1 dxdy \neq -\int_0^1 \int_0^1 dydx$. Where's the abuse of notation here?
15
votes
2answers
802 views

What is a form?

I have read about differential forms, bilinear forms, quadratic forms and some other r-linear forms but I still have this shred of doubt in my mind on what exactly is a form. I have an assumption that ...
0
votes
1answer
31 views

Lie derivative of differential form

Let $M$ be an $S^1$-manifold and let $\omega$ be a $k$ form on $M$. Let $X\in\mathfrak{X}(M)$ a vector field on $M$. Now my question is, If $\mathcal{L}_X(\omega)=0$, then $(\Phi_X^t)^*\omega=\omega.$...
2
votes
1answer
32 views

A basis for the holomorphic differentials of a hyper-elliptic Riemann surface

Bumped in to this problem while trying to understand hyper-elliptic Riemann surfaces, and being a bit new to the subject of Riemann surfaces, was not that much confident on a couple of things. ...
3
votes
3answers
79 views

Non-vanishing volume form on $S^2$

I have been given the form $\mu=x\,dy\wedge dz+y\,dz\wedge dx+z\,dx\wedge dy\in\Omega^2(S^2)$. I am asked to prove that this form is never vanishing. That is, $\nexists p\in S^2$ such that $\mu_p=0$. ...
0
votes
0answers
37 views

Diffeomorphisms preserving a family of 1-forms

Let $M$ be a manifold. Suppose I have a one parameter family of differential 1-forms $\eta_t$, $t \in [0,1]$, depending smoothly on the parameter $t$. Suppose there is a diffeomorphism $\psi_0$ of $M$ ...
0
votes
0answers
33 views

Product manifolds and exterior derivative with interior product

While studying differential geometry, I read this part of a proof and I didn't understand it. Given a $2$-manifold $\Omega$ and an interval $I=(-\epsilon, \epsilon)$, consider the cartesian product $M=...
0
votes
0answers
30 views

A question from Henri Cartan's complex variables textbook

I was thinking about one question from the book and got quite a bit of confusion and had no idea where to start. The text is Henri Cartan's "Elementary Theory of Analytic Functions of One or Several ...
1
vote
1answer
41 views

A question on Poincaré-Hopf Theorem for meromorphic forms.

In Griffiths' Introduction to Algebraic Curves. In the proof of the following statement, Let $\omega$ be a meromorphic 1-form on a compact Riemann surface $C$, then $\sum_{p\in C} \...
2
votes
0answers
25 views

About wedge product of differential forms.

When antisymmetrizing a tensor product, we usually define it as \begin{equation} T_{[i_1,i_2,\cdots,i_n]}:=\frac{1}{n^!}\sum_{\sigma\in S_n}\text{sign}(\sigma)T_{i_{\sigma(1)},i_{\sigma(2)},\cdots,...
4
votes
1answer
85 views

Pullback of differential forms and determinant

I'm studying differential geometry using the book "Godinho Natàrio - An introduction to Riemannian Geometry". These are the definitions and theorems I'm working with: Definition 1 (Pullback of a ...
1
vote
1answer
28 views

Connection on the cotangent bundle

I'm reading "Differential forms and connections" by R. Darling and I must have made a mistake in problem 4 in section 9.4. It states: Prove that $\nabla_X \omega := \iota_X\, d\omega$ is not a ...
1
vote
2answers
35 views

Integration along fibers is independent of the lift

According with Wikipedia https://en.wikipedia.org/wiki/Integration_along_fibers Let $\pi: E \to B$ be a fiber bundle over a manifold with compact oriented fibers. If $\alpha$ is a $k$-form on $E$, ...
3
votes
2answers
71 views

Difference between $dx \wedge dy$ and $dxdy = dA$.

(Beginner in differential forms) In $\mathbb{R}^2$, consider the differential form $\omega = dx \wedge dy$ and infinitesimal area element $dA = dxdy$. I already know that $$\int_{\mathbb{R}^2} w = \...
0
votes
1answer
38 views

Show that $d\omega=(-1)^{n-1} (divX)dx_1 \wedge\dots\wedge dx_n$

Let, $X = (X_1,\dots,X_n): D \subset \Bbb R^n \to \Bbb R^n$ be a $C^1$ vectr field on a domain $D$. Define $divX := \sum_{i=1}^n {\frac{\partial X_i}{\partial x_i}}$. Define an $(n-1)$ form $\omega$...
1
vote
0answers
27 views

Defining a connection in $\mathbb{R}^2$ using a connection $1$-form

I'm reading Hitchin's paper Self-duality Equations on a Riemann Surface (Hitchin, self duality). In the first chapter on pages 63/64 he considers a principal $G$-bundle $P$ over $\mathbb{R}^4$ and a ...