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

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1answer
45 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 ...
2
votes
1answer
58 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
votes
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 ...
1
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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. ...
0
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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 ...
1
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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 ...
0
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0answers
33 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 ...
1
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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 $dM$...
1
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0answers
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 $$ \...
7
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1answer
92 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 ...
0
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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 }...
1
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0answers
33 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 $\...
2
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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$^k$. ...
2
votes
1answer
62 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 ...
0
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1answer
24 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 ...
0
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0answers
24 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 ...
2
votes
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 ...
1
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0answers
49 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 $...
1
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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}...
1
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2answers
42 views

Invariant forms on a manifold

Probably a silly question - still, it's been bugging me for some time now. Say that we have an invariant $1-$form $\omega$ on a smooth manifold $M$, acted on by a group $G$. Then \begin{equation} \...
3
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0answers
46 views

Does integration wrt to a differential form always come from a measure?

More precisely, is there an $n$-manifold $M$ with an $n$-form $\omega$ such that there is no measure $\nu$ on $M$ satisfying $$\int f \omega = \int f d\mu $$ for all compactly supported smooth ...
2
votes
1answer
19 views

How do you compute a complex exterior derivative?

The context is deriving cauchy riemann equations using green's/stoke's theorem. The function is the complex function $f(x,y)=u(x,y)+iv(x,y)$ with associated one form $u(x,y)dx+iv(x,y)dy$. Here is my ...
0
votes
1answer
21 views

Finding the expression of a one form in a chart.

Given a one form on a manifold the formula I was given for finding its expression in a given coordinate chart is very strange and I dont understand it. I would appreciate if someone could give me a ...
4
votes
1answer
102 views

Solution to Cauchy-Riemann Differential Equation of Compact Support

I'm working through Forster's $\textit{Lectures on Riemann Surfaces}$ and am struggling with the following problem: Suppose $g \in \mathcal{E}(\mathbb{C})$ is of compact support. Prove there is a ...
2
votes
2answers
53 views

Formula of a two form for a parallelizable manifold

Let $M^n$ be a parallelizable manifold with the nowhere dependent vector fields $X_1,\ldots, X_n$ forming a basis for the tangent space at each point of $M$. The Lie brackets of these fields are ...
6
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1answer
63 views

Deciding whether a form in the exterior power $\bigwedge^k V$ is decomposable

Let $V$ be a vector space and $\bigwedge^kV$ be the $k$th exterior power. I'm trying to find a condition that characterizes when an element $\omega \in \bigwedge^kV$ is decomposable in the sense that $...
2
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1answer
39 views

Show that a k-form can be expressed as wedge product

I'm trying to show that given a $1$-form $\omega$ and a $k$-form $\alpha$ such that $\alpha \wedge \omega = 0$ then there exists a $(k-1)$-form $\beta$ such that $\alpha = \omega \wedge \beta$. I'm ...
3
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1answer
57 views

Differentiating the pull-back of a one-form

Let $\Omega$ be an open subset of a vector space $V$ and let $\alpha\colon \Omega\to V^*$ be a one-form on $\Omega$. Assuming that $\alpha$ is differentiable, then for any $x\in \Omega$, $D\alpha (x)\...
0
votes
1answer
62 views

Poincare's lemma from PMA Rudin

It's the so called Poincare's theorem from Rudin's book. I read this theorem fully but I have 2 questions: 1) How did he conclude that equations (120) holds? What did he use in his reasoning? This ...
1
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1answer
39 views

Why is the $1$-form ${dz\over z}$ on $\mathbb{C}^*$ closed?

Why is the $1$-form $\displaystyle\frac{dz}{z}$ closed in $\mathbb{C}^*$? In general, how to compute a complex one form's derivative? Thank you!
1
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1answer
48 views

Differential forms on a point

For the proof of Poincaré lemma, it's essential to evaluate $\Omega^p(*)$ where $*$ is zero dimensional manifold and $\Omega^p$ is a collection of all $p$-forms on given manifold. Clearly, $\Omega^0 (*...
1
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1answer
36 views

Intersection of kernels of linearly independent smooth 1-forms on $\mathbb R^n$

I'm trying to solve the following problem: Let $\omega^1,\dots,\omega^k$ be smooth $1$-forms on $\mathbb R^n$ that are linearly independent at each point of $\mathbb R^n$. For $p\in\mathbb R^n$, ...
0
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0answers
28 views

Definition of exact form

That's the definition of exact form in $E$. But if we look at the definition 10.18 we see that $\lambda \in C'$ but Rudin skip this condition. Can anyone please explain this moment
1
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0answers
38 views

Stokes' Theorem from PMA Rudin. Confusing moment with simplex

I am reading the proof of Stokes' theorem from PMA Rudin but one moment seems to very weird. Why Rudin considers the case when $\sigma=[0,\mathbf{e}_1,\dots, \mathbf{e}_k]$? After all $\sigma$ may ...
2
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0answers
39 views

Showing that $\|.\|$ is a norm of the space of 1-forms $\Omega^1(U)$, where $U\subset\mathbb{R}^n$.

Let $U\subset\mathbb{R}^n$ and let $\Omega^p(U)$ denote the vector space of $p$-forms ($p\in\mathbb{N}$). Define the isomorphism $\Phi:\Omega^{1}(U)\to\Omega^{n-1}(U)$ as $$\Phi\left(\omega=\sum_{i=1}^...
0
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2answers
68 views

$\mathbb{P}_{\mathbb{C}}^3$ is not isomorphic to $S^2 \times S^4$

I have been trying to solve this exercise given by my prof. The hint is to show that every $2$-form $w$ on $S^2 \times S^4$ is s.t. $w \wedge w = 0$, while this is not true in case of $\mathbb{P}_{\...
1
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1answer
34 views

Compact Poincaré dual of $S^{n-1}$ in $\mathbb{R}^n \backslash \{0\}$

I have been asked to find a sub manifold $S \subset M := \mathbb{R}^n \backslash \{0\} $ s.t. its compact Poincaré dual is a basis of the first cohomology group $H^1_c(M) = \mathbb{R}$. Now, $S$ must ...
0
votes
1answer
30 views

Symplectic form on $T^ ∗X$

If $\mu$ is any closed 2-form on a manifold $X$, then $d\alpha_X + π^∗\mu$ is also a symplectic form on $T^ ∗X$, where $\alpha_X$ is tautological one-form and $\pi:T^*X\to X$ is projection. In fact, ...
3
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2answers
98 views

How to understand the notion of a differential of a function

In elementary calculus (and often in courses beyond) we are taught that a differential of a function, $df$ quantifies an infinitesimal change in that function. However, the notion of an infinitesimal ...
4
votes
1answer
37 views

Finding all $2$-forms in the right half-plane that are invariant under glide transformations

I'm trying to find all 2-forms $\omega$ that are invariant under glide transformations in the right half-plane, $X = \{ (x,y) \in \mathbb{R}^2 : x > 0\}$. To do this, we can write the vector field ...
1
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1answer
55 views

Unit square as union of two simplexes

? If Rudin regarded $\Phi$ as a function of $2$-forms and suppose that $\omega=f(\mathbf{x})dx_{i_1}\land dx_{i_2}$ is $2$-form on $\mathbb{R}^m$ then $$I_{\Phi}(\omega)=\int \limits_{\Phi}\omega=\...
2
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0answers
32 views

Boundary of oriented $k$-simplex from PMA Rudin

But paragraph which I marked by red line seems to me confusing. Let $k=3$ then $\sigma=[\mathbf{p}_0,\mathbf{p}_1,\mathbf{p}_2,\mathbf{p}_3]$ and $\partial \sigma=[\mathbf{p}_1,\mathbf{p}_2,\mathbf{p}...
0
votes
1answer
63 views

Finding A 1-form on $R^2 - {\{(0,0)}\}$

I want to find a 1-form on $R^2 - {\{(0,0)}\}$ such that $w(Y) = 0$ and $w(X) = 1$. Here, $$X = -y\frac{\partial }{\partial x} + x\frac{\partial}{\partial y}\ \text{and}\ Y = x\frac{\partial }{\...
2
votes
1answer
61 views

A necessary and sufficient condition for the admittance of integrating factor

Let $\omega$ be a smooth 1-form on a smooth manifold $M$. A smooth positive function $\mu$ on some open subset $U\subset M$ is called an integrating factor for $\omega$ if $\mu\omega$ is exact on U. ...
1
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2answers
42 views

Specific example of integrating a 1-form over a curve

I was given the following definition in my course but no corresponding examples: Supppose $\gamma:[a,b]\rightarrow{M}$ is a smooth curve and $\omega$ a 1-form on $M$ (so $\omega:M\rightarrow{T^*M}$). ...
1
vote
1answer
60 views

Affine chains from PMA Rudin. Confusing examples

I understood the definition of affine $k$-chain and that he defines $\int \limits_{\Gamma} \omega$ as $(82)$. But I can't understand the last two above examples. What does they mean? Can anyone ...
0
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0answers
26 views

Equivalent conditions for $\mathfrak{F}$ to be a differential ideal

Heres the question: Let $\mathfrak{F}$ be an ideal of forms on a manifold $M$ locally generated by $r$ independent $1$-forms. Say $\mathfrak{F}$ is generated by $\omega_1,\ldots, \omega_r$ on $U$. ...
0
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0answers
29 views

Let $f : \mathbb{R}^n \to \mathbb{R}^n$, suppose that $\mu = dx^1\wedge\ldots\wedge dx^n$, then $f^{\ast}\mu = \det (df)\mu$

Let $f : \mathbb{R}^n \to \mathbb{R}^n$, suppose that $\mu = dx^1\wedge\ldots\wedge dx^n$, then $f^{\ast}\mu = \det (df)\mu$. I am trying to prove this. I made several low dimensional cases and ...
2
votes
1answer
41 views

Prove the exterior derivative of the following (n-1) form is zero

Let $\omega(x)=\frac{1}{{\parallel x \parallel}^n}\displaystyle\sum_{i=1}^{n}(-1)^{i-1}x_{i} dx_{1} \wedge \dots \wedge \widehat{dx_{i}} \wedge \dots \wedge dx_{n}$ be a differential $(n-1)$ form on $\...
1
vote
1answer
56 views

Confusing moment in Theorem 10.27 from PMA Rudin

Theorem 10.27 If $\sigma$ is an oriented rectilinear $k$-simplex in an open set $E\subset \mathbb{R}^n$ then $$\int \limits_{\overline{\sigma}}\omega=\varepsilon\int \limits_{\sigma}\omega \qquad (81)$...