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

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Closed forms and rational numbers

Let $\omega$ be a 1-form defined in an open set $U\subset \mathbb{R}^{n}$. Assume that for each closed differentiable curve $c$ in $U$, $\int_{c} \omega$ is a rational number. Prove that $\omega$ is ...
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0answers
17 views

Coordinates of exterior derivative of dual basis of local frame for the tangent bundle

Let $M$ be an $n$-manifold. Let $E_1, E_2,\dots, E_n : U\subset M \to TM $ be a local frame for $TM$ with associated local dual frame $\epsilon^1, \epsilon^2,\dots, \epsilon^n : U\subset M \to T^*M $. ...
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0answers
12 views

Adjoint of the gauge covariant derivative

Suppose $A=A_1dx_1+A_2dx_2$ is a 1-form connection in $\mathbb{R}^2$ and $D_A \phi=d\phi-iA\phi$ is the gauge covariant derivative with $\phi=\phi_1+i\phi_2$ is a complex scalar field. May I ask what ...
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0answers
33 views

Function as a combination of 1-forms on a Riemann surface

My question is quite simple, I hope it's not also stupid.. Consider $R$ a Riemann surface and $\omega_1$, $\omega_2$ two $(1,0)$-forms (i.e. holomorphic forms) and $\varphi_1$, $\varphi_2$ two $(0,1)$-...
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37 views

Challenging calculation of a Jacobian for an unusual matrix coordinate transformation

I am studying a random matrix ensemble and I am having trouble performing a coordinate transformation. My question is very straightforward, but perhaps a bit technical. I have the following integral--...
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0answers
18 views

About the choice of a differential form such that the exterior derivative is a a determinant of a Jacobian of an application

If $D\subset \mathbb{R}^{2}$ is a compact domain with regular boundary and $F:D\to \mathbb{R}^{2}$ is such that $F(x)=(f(x),g(x))$ where $F \in C^{2}$ , then if i choose the 1-form $\omega=f.dg$, ...
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1answer
36 views

Let $ \omega = dx_1 \wedge dx_2 + dx_3 \wedge dx_4 + \cdots + dx_{2n-1} \wedge dx_{2n} \in \mathbb{R}^{2n}$. Find $\omega^n$ (in respect to $\wedge$) [duplicate]

Let $ \omega = dx_1 \wedge dx_2 + dx_3 \wedge dx_4 + \cdots + dx_{2n-1} \wedge dx_{2n} \in \mathbb{R}^{2n}$. Find $\omega^{n}$ (in respect to $\wedge$) When I say "$\omega^{n}$ (in respect to $\wedge$...
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51 views

Integrating factor for a non exact differential form

I can't find an integrating factor for the differential form $$ -b(x,y)\mathrm{d}x + a(x,y)\mathrm{d}y $$ where $$ a(x,y) = 5y^2 - 3x $$ and $$ b(x,y) = xy - y^3 + y $$ The problem has origin form ...
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0answers
30 views

Symmetric power bunde of half spinor representation

Can anyone give me a reference for understanding $$\Lambda^2_{+c}\cong S^2V_+$$ where $\Lambda^2_{+c}$ is the complexified bundle of self-dual two forms and $S^2V_+$ is the symmetric power bundle of ...
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1answer
47 views

Frobenius condition in terms of Lie brackets

Let $\alpha$ be a $1$-form and $\xi = \ker \alpha$. Frobenius theorem tells us that $\xi$ is integrable iff $\alpha\wedge{\rm d}\alpha = 0 .$ In the book "Introduction to Contact Topology" from ...
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1answer
46 views

How to evaluate a $1$-form on a vector field?

I have the one form : $dz + x\, dy$. If I want to evaluate this on a vector field, say $-\partial_{z}$, how do I do it? I don't understand what 1-forms defined with an exterior derivative do to ...
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0answers
39 views

Computing $\alpha\wedge d\alpha$ where $\alpha = dz + x\,dy$

Let $\alpha = dz + xdy$ be a one-form on $\mathbb R^3$. I would like to compute the wedge product $\alpha\wedge d\alpha$ as explicitly as possible but I am not sure if I am doing this correctly. $$d\...
2
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2answers
73 views

Finding the antiderivatives for the differential form $\alpha = x_2 d x_1 \wedge d x_2$

Consider the differential form $$\alpha = x_2 d x_1 \wedge d x_2$$ on $\mathbb{R}^3$. I first want to find a $1$-form $\beta \in \Omega^1(\mathbb{R}^3)$ that is an antiderivative for $\alpha$, i.e. ...
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2answers
110 views

The integral of a function on manifold and differential form

When we want to integrate a function f over a manifold M, we may meet some problems, for example, the problem showed in the picture below: Then people used differential form to integrate. But it ...
3
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0answers
46 views

How to define integration along surfaces in $\Bbb R^4$?

In $\Bbb R^4$ we have curves, bi-dimensional surfaces and hypersurfaces. We integrate vector fields along curves and in hypersurfaces using a normal direction. I know that the adequate object to ...
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1answer
64 views

A question about pullback of the K-form

Let $M$ be an oriented $m$-dimensional manifold. Suppose the support of $\omega$ is in an open subset $U$ of $M$, and $\phi \colon U \to R^m$, $\psi \colon U \to R^m$ are two different charts on $M$ ...
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1answer
25 views

Apply connections to a gauge transformation?

I'm reading Donaldson's book, Floer homology groups in Yang-Mills theory. On page 82, he considers a trivial bundle $P$ over a $4$-manifold $X$ with tubular ends which is equipped with a connection $...
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1answer
22 views

Proving a formula for the exterior derivative of a specific $k$-form, given in base representation

Let $U \subseteq\mathbb{R}^n$ be open, and let $\omega$ be an $(n-1)$-form that's given by $$\omega = \sum_{i=1}^n (-1)^{i-1} F_i\, dx_1 \wedge\dots\wedge dx_{i-1} \wedge dx_{i+1} \wedge\dots\wedge ...
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1answer
75 views
+100

Need help understanding local coordinates of differential forms

I was trying to check a variation of theorem 2.2: I did the first part of the proof using the standard contact structure on $\mathbb R^{2n + 1}$ which worked out as expected. Then I wanted to do it ...
2
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2answers
78 views

Help Debugging a Bogus Proof

We want to prove the standard fact that a smooth function $u :R^2 \to R$ with $ \nabla u = 0$ everywhere in some connected open set $ \Omega $ is constant in that set. I'm comfortable with the usual ...
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1answer
34 views

Finding a base representation for a differential form that's given through the determinant of a matrix

Let $U \subseteq \mathbb{R}^n$ be open. I first want to show/acknowledge that the mapping $$\omega_p(v_1, ..., v_{n-1}) := det(p, v_1, ..., v_{n-1}), p \in U, v_i \in \mathbb{R}^n$$ defines a ...
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1answer
67 views

Isomorphisms between $\mathbb{R}^3$ and $\left( \mathbb{R}^3\right)^*$

Consider the oriented euclidean space $\mathbb{R}^3$. Every vector $A \in \mathbb{R}^3$ determines a $1$-form $ω^1_A$ by $ω^1_A(ξ)=(A,ξ), ξ\in \mathbb{R}^3$ (scalar product) and a $2$-form $ω^2_A$ by \...
3
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2answers
49 views

Exterior Product of 2-Forms in $\mathbb{R}^n$

So I am studying the book of Arnold, "Mathematical Methods of Classical Mechanics" and I am trying to understand the differential forms. The exterior product is defined for $1$-forms $ω_1$ and $ω_2$ ...
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1answer
29 views

Domain simply connected for a differential form $\omega$

I have the domain $\Omega=\{(x,y): 2y+x>0\}$ and the differential form $$\omega=\frac{y}{2y+x}dx+\left(\log (2y+x)+\frac{2y}{2y+x}\right)dy.$$ I would like to evaluate $\int_{\gamma} \omega$ ...
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1answer
20 views

Differential forms vector space over function field?

Let $V$ be a vector space and $V^*$ be its dual space. Then I know that $V^*$ is considered a vector space because we can scale the basis covectors by real numbers and add them together and all of ...
3
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2answers
103 views

Kernel of $\omega^\#$ is $k$-dimensional

Let $M$ be a smooth manifold with coordinates $\{q^i\}_{i=1}^n$ .The variables $(q^i,p_i)$ are coordinates on the cotangent space $\Omega=T^*M$. Any cotangent space carries a natural one-form $\tilde{\...
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27 views

Inner products on $C_c^\infty$ corresponding to differential forms

Let $M$ be a compact, oriented Riemannian manifold of dimension $n$. We can locally identify smooth $k$-forms by smooth functions $\mathbb{R}^n \to \mathbb{C}^{m}$, where $m = \binom{n}{k}$ (via a ...
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0answers
60 views

Integral of an interior product is 0

Here is a question about problem 3.12 from Do Carmo's Riemannian Geometry. $M$ is a compact orientable and connected Riemannian $n$ manifold. $f$ is a differentiable function on $M$ such that $\Delta ...
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2answers
47 views

Stokes' Theorem and path integrals in $\mathbb{C}$

I have seen very short proofs of the Lemma of Goursat and Cauchy's Theorem using Stokes' Theorem. I have learnt Stokes' Theorem in the setting of (not complex) smooth manifolds as described in https://...
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1answer
48 views

Restriction of an $n$-form to a submanifold

I am currently studying exterior calculus on manifolds and am not sure if I understand things correctly. In my textbook (R.W.R. Darling, Differential Forms and Connections) there is an example of a $2$...
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1answer
43 views

What am I doing wrong when calculating this pullback?

Let $\omega = \sum_{j=1}^{n+1} x_j dy_j - y_j dx_j $ be a differential form on the sphere $S^{2n +1}$. Let $G = Z_2$ be the group acting on the sphere. I want to apply the following proposition to ...
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1answer
56 views

Exterior derivative of complex differential form

I have this question, from several complex variables: Start with the differential form: $$\omega(z)=\sum_{\nu=1}^{n} \frac{(-1)^{\nu-1}\bar{z}_{\nu}}{|z|^{2n}} d\bar{z}[\nu] \wedge dz, $$ where $dz=...
2
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1answer
38 views

Demonstration of a basic formula involving differential forms

I'm writing some notes on Lie Groups and I'm not sure if I should demonstrate this formula or not. Assume $\omega$ is a differencial form and $X,Y$ fields con a Manifold M, is there a simple way to ...
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2answers
32 views

Let $\theta$ a $1-$form. Why $d\theta(X,Y)=X\theta(y)-Y\theta(X)-\theta([X,Y])$?

Let $\theta$ a $1-$form. Why $$\mathrm d\theta(X,Y)=X\theta(Y)-Y\theta(X)-\theta([X,Y])\ \ ?$$ I know that $\theta=\sum_{i=1}^n a\mathrm d x^i$ where $a\in \mathcal C^\infty (U)$ and $U$ an open. ...
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0answers
24 views

Orientation form on manifold cut out by $m$ functions

Let $f: \mathbb{R}^n \to \mathbb{R}^m$ be a smooth function. If $a$ is such that $f$ has surjective derivative at all points in $f^{-1}(a)$ then this is an $n-m$ dimensional manifold $X$. I'm trying ...
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1answer
35 views

Radial differential form

Hi have to determinate a primitive of this differential form: $$\omega = \frac{xy}{\sqrt{(x^2+y^2)}}dx + \frac{x^2 + 2y^2}{\sqrt{(x^2+y^2)}}dy$$ As far as I know this should be a radial form which I ...
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25 views

Differentiable function f(x)

Let $f(x)$ is a differentiable function satisfying $f'(x) + 100 f(x) ≤ 1 $ Then $f(x) -1/k$ is a non increasing function of $x$ , then we have to find the value of $k $ I tried , but at last ...
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31 views

Condition for set of de Rham cohomology classes is linearly independent

If we have a set of 1-forms $w_1, ... w_n$ on a smooth manifold $X$ I can show that $w_i$ are linearly dependent if and only if $w_1 \wedge ... \wedge w_n = 0$. I wondered if this is also true in the ...
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1answer
25 views

Area form of $M^2 \subseteq \Bbb R^4$: does $(-1)^{i+j-1}\det_{i'j'}(n,\nu) = dx^i \wedge dx^j$?

This is a follow up question from this one. I'm asking separately since one way or another, that one is sort of answered. Now, we know that if $M^{n-1} \subseteq \Bbb R^n$, then the volume element $dM$...
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33 views

Do I understand the divergence theorem correctly?

Suppose the area, volume or hyper volume covered by a vector is $$ \mathrm{V}\left(\vec{u}\right) = u_x \times u_y \times \ldots $$ And the area, volume or hyper volume covered by a matrix is $$ \...
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1answer
89 views

Area form for $M^2 \subseteq \Bbb R^4$

We know that in general, given a orientable hypersurface $M^{n-1} \subseteq \Bbb R^n$, the volume form on $M$ is given by $$dM = \sum_{i=1}^n(-1)^{i-1}n_i\,dx^1 \wedge\cdots\wedge \widehat{dx^i}\wedge ...
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0answers
25 views

Is $\sigma_u \times \sigma_v \neq \vec{0}$ essential for $\int_{\alpha} \vec{F} \cdot d\vec{r} = \iint_\sigma \mathrm{curl}\,\vec{F} \cdot d\vec{S}$?

I think I have proved the following version of Stokes' theorem: Teorem 1: Let $\beta: [0,4] \to \mathbb{R}^2$ be the curve given by \begin{equation} \beta(t) = \begin{cases} (t,0) & \mbox{if }...
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28 views

Moduli space of differential forms

In this paper, Debarre-Voisin refer to the "moduli space" of differential 3-forms $\sigma \in \bigwedge^3(V_{10}^*)$ on a fixed vector space $V_{10}$ of dimension 10, and state that this space is $\...
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0answers
29 views

Integral of a k-form

Just got to the culminating chapter in Munkres’s Analysis on Manifolds and I’ve been thrown for a loop. The author is in the process of defining the integral of a k-form η on A, an open set of R$^k$. ...
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55 views

Exterior derivative on principal bundle

In Nakahara's Geometry, Topology and Physics on page 375, he constructs a Lie-algebra-valued one-form $\omega$ on a principal bundle $P$ by "lifting" a Lie-algebra-valued one-form $\mathcal A_i$ on an ...
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1answer
23 views

Degree of smooth map of manifolds depends on orientation choice?

I'm a little to confused as to why it appears that the degree of a smooth map $f: M \to N$ between smooth manifolds appears to only be defined up to sign - I'm not sure where my mistake is. By ...
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0answers
23 views

Connection planes associated to differential 1-forms

In textbook in differential geometry such an idea appears: there is a tangle bundle $\pi:TM\rightarrow M$ and we are actually looking at the trivialization $\pi^{-1}(U)=U\times \mathbb{R}^n$. We are ...
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3answers
67 views

Showing de Rham cohomology $H^1(S^n)$ is zero

I'm trying to find an elementary way to see that the 1st de Rham cohomology of the n-sphere is zero for $n>1$, $H^1(S^n) = 0$. This is part of an attempt to find the de Rham cohomology of the n ...
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0answers
46 views

Meaning of alternation in definition of wedge product

Spivak defines the wedge product as $\omega \wedge \nu = \frac{(k+l)!}{k!l!} \text{Alt}(\omega \otimes \nu)$ and I have been running into some conceptual issues here. The alternation is defined as $...
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1answer
43 views

Is there a matrix that converts the gradient of any function to gradient of other function?

The study of hamiltonian mechanics brought me to the following question. Let $n$ be a natural number ($n>1$). Let $A(\mathbf{x})$ be a $n\times n$ matrix consisting of functions $a_{ij}(\mathbf{x}...