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

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3
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1answer
47 views

Ways of thinking about vector-valued differential forms

I am trying to get a better intuition of vector-valued differential forms. Let $V$ be a vector space and $M$ a smooth manifold. Consider the space $\Omega^k(M;V)=\Gamma((M\times V)\otimes ...
0
votes
1answer
54 views

Differential operator computation

I need to show that if $\omega=xydx+3dy-yzdz$ $\nu=xdx-yz^2dy+2xdz$ then $d(\omega\land\nu)=(d\omega)\land\nu-\omega\land(d\nu)$ I really do not understand how this differential operators work. I ...
1
vote
2answers
171 views

Sign problems in complex computations

$\newcommand{\dd}{\mathrm{d}} \newcommand{\eg}{\epsilon} \newcommand{\mg}{\mu} \newcommand{\ng}{\nu} \newcommand{\rg}{\rho} \newcommand{\et}{\wedge} \newcommand{\lbar}{\overline} ...
3
votes
1answer
53 views

Associated bundles: isomorphism between spaces of differential forms.

I think this will be an easy question for numerous people. Let $\pi:P\rightarrow M$ be a principal bundle and $\rho:G\rightarrow GL(V)$ a representation. The space of $k$ forms on $M$ with values in ...
3
votes
1answer
40 views

Volume form for a product manifold.

Given two Riemannian manifolds $(M,g^M)$ and $(N,g^N)$, we can construct a product Riemannian manifold $(M\times N,g^{M \times N})$ as described in Product of Riemannian manifolds? . Is there a ...
2
votes
2answers
73 views

Notations of the coordinate functions in differential forms

I'm reading a book on differential forms, and it says: A basis for $(\mathbb{R}_p^3)^*$ is obtained by taking $(dx_i)_p, i=1,2,3$, where $x_i:\mathbb{R^3}\to\mathbb{R}$ is the map which assigns to ...
0
votes
1answer
42 views

2-forms, Edwards, Differential forms approach

Find the value of the 2-form $dxdy+3dxdz$ on the oriented triangle with $(0,0,0)$ $(1,2,3)$ $(1,4,0)$ in that order. I have tried various subtractions and plugging in values and have been unable to ...
1
vote
0answers
45 views

Clarification of vector valued forms (sections and tensor products)

I am seeking a bit of clarification on vector valued forms. Intuitively, a vector valued differential form is a differential form on $M$ whose values lie in some vector space. More accurately, Let ...
0
votes
1answer
39 views

Global differentials on projective line

Given a field $k$, it is well known that the global differentials $\Omega_{\mathbb{P}_k^1/k}$ of the projective line $\mathbb{P}_{k}^1$ are $\mathcal{O}(-2)$. This is usually proved by observing that ...
0
votes
1answer
22 views

Relationship between $\operatorname{supp} w$ and $\operatorname{supp} dw$

I would like to show that $\operatorname{supp} dw\subset \operatorname{supp}w$, where differential $m$-form in a $\Sigma$ surface of dimension $m+1$. If $$w(u)=\Sigma_i (-1)^ia_i(u) du_1\wedge ...
1
vote
1answer
32 views

If differential forms agree on one chart do they agree everywhere?

Let $\alpha,\beta$ be two $k$-forms on a manifold $M$. If there exists some chart $(U,x^1,\dots,x^n)$ on which $\alpha=\beta$ does it follow that $\alpha$ and $\beta$ are the same forms? In different ...
0
votes
0answers
6 views

Model of continuum mechanics or thermodynamics derived from potential and paradoxe on state variables

(1) In material sciences based on thermodynamics and continuum mechanics, one usually introduces potentials to summarize the constitutive model. All constitutive relations derive from the introduced ...
10
votes
2answers
1k views

Apparent counter example to Stoke's theorem?

I think I found an apparent contradiction to Stoke's theorem with this 2-differential form $M= \overline{B^{2}}- \{ 0 \}$, $\partial M = S^1$, $$\omega = \frac{x~dy-y~dx}{x^2+y^2}$$ defined in ...
0
votes
0answers
29 views

Hodge star on this expression

if we have $\alpha$ a complex function, and we want to take the Hodge dual of $$d\bigg(\frac{1}{|\alpha|^2}\bigg)d(\bar{\alpha})$$ what will that give us? Can we take ...
1
vote
1answer
93 views

Exterior derivative of forms derived from a metric

Let $(M,g)$ be a Riemannian manifold. From $g$ and a fixed vector field $V$ we can derive the following two differential forms: A $1$- form $\alpha(X) = g(V,X)$, i.e. $\alpha = \iota_Vg$. A $2$-form ...
0
votes
0answers
53 views

Closed 1-forms on Simply Connected Manifold

Is it true that closed 1-forms on a simply connected differentiable manifold are exact. If so, could you explain why? Thanks very much
0
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0answers
22 views

Help in the proof of Poincaré Theorem to differential forms

I'm revising the proof of Poincaré Theorem, but I don't understand a pass of proof. Let be $E$ and $F$, Banach spaces and $U\subset E$ open set. Consider $\omega\in\Omega_p^n(U;F)$ a p-differential ...
0
votes
2answers
59 views

Self-studying differential forms and tensors

I am interested in understanding the generalized Stokes' Theorem. From my understanding, this theorem involves differential forms and exterior algebra, and tensors (to some extent). I'm not ...
2
votes
1answer
59 views

Tangent space and differential forms of quotient groups

I have difficulty understanding the following argument because it seems that many details are swept under the rug and I am looking for a detailed rigorous exposition on this (I'll try to make clear ...
5
votes
1answer
51 views

divisor of a section of the sheaf of logarithmic differentials

Let $S=\{0, 1, \infty\} \subset \mathbb{P}^1$ and let $\Omega^1_{\mathbb{P}^1}(\log S)$ be the line bundle of logarithmic differentials along $S$. Consider the form $$ ...
0
votes
1answer
32 views

$\bigwedge^nT^∗M$ is trivial $\Leftrightarrow M$ is orientable

I can't figure out how to prove the following: Let $M$ be an $n$-manifold. Then $\bigwedge^nT^∗M$ is trivial $\Leftrightarrow M$ is orientable Thank you
0
votes
2answers
66 views

Prove $\omega\wedge\eta=0$ iff there exist a $\theta\in A^3(\mathbb{R}^5)$ such that $\eta=\omega\wedge\theta?$ [closed]

Suppose $\omega\in A^1(\mathbb{R}^5)$ and $\eta\in A^4(\mathbb{R}^5)$. prove $\omega\wedge\eta=0$ iff there exist a $\theta\in A^3(\mathbb{R}^5)$ such that $\eta=\omega\wedge\theta?$
1
vote
1answer
44 views

for a vector field $v$, what is the differential 1-form obtained from $v$ by the canonical isomorphism induced by the inner product.

I am studying analysis on manifolds and am trying some problems and I am conceptually stuck on one. It is from Do Carmo's book Differential Forms and Applications, chapter one problem 11 b. I'll ...
1
vote
0answers
34 views

Locally Hamiltonian vector fields

Consider the following definitions (taken from [1]) Definition. Let $E$ be a Banach space and $B: E \times E \to \mathbb R$ a continuous bilinear mapping. Then $B$ induces a map $B^\natural: E \to ...
1
vote
0answers
26 views

Given a smooth 2-form $\varphi$ on a punctured Riemann surface, is there $\nu$ of type (1,0) such that $\varphi=d\nu$?

Let $X$ be a compact Riemann surface, $p\in X$, and $\varphi$ be a smooth 2-form on a $X-\{p\}$, and hence exact. I'm wondering if it is possible to find a form of type (1,0) whose differential is ...
4
votes
1answer
61 views

Second Chern class of $TS^2$

The induced metric on a sphere may be given by, $$ds^2 = d\theta^2 + \sin^2\theta\, d\phi^2$$ By using Cartan's method of moving frames, one can compute the curvature 2-form in an orthonormal basis ...
5
votes
2answers
102 views

Problem on Integration: $\Bbb R-\Bbb C$ split and pull back of forms

This post is not short. However I'm sure that a guy who good handle these concepts, could read and answer in five minutes. I only want to write my attempt, in order to understand where I'm wrong. Let ...
5
votes
1answer
39 views

Cotangent Fields: Exactness vs. Conservation

Given a smooth manifold. Then a cotangent field is exact iff conservative: $$\alpha\in\mathcal{X}^*(M):\quad\alpha=\mathrm{d}h\iff\oint\alpha=0$$ How to prove this properly?
0
votes
0answers
36 views

Linear independence of differential 1-forms

Let be $(E,\mathbb{K}),(F,\mathbb{K})$ Banach space and $U\subset E$ open set. If $f_1,...,f_n\in\Omega_1(U;F)$ differential 1-forms are linearly independents, where $$\Omega_1(U;F)=\{f:U\rightarrow ...
1
vote
1answer
22 views

How to prove that angular velocity is not a derivative of angular displacement?

The angular velocity $\omega$ of a two-dimensional solid body is given by $$\omega = \hat{z} \cdot \frac{\vec{r} \times \vec{v}}{r^2},$$ where $\vec{r}$ and $\vec{v}$ are the position and the velocity ...
1
vote
1answer
72 views

A proof of exactness of closed 1-forms on the two-sphere

Remark. I'm aware that here the same problem is solved. I'm in trouble with the proof I'm going to quote. Consider the following Claim. Every closed 1-form $\beta$ on $S^2$ is exact. This is an ...
1
vote
0answers
56 views

What is this inner product on differential forms?

I am trying to understand the definition of $d^\ast$ of $d$ where $d$ denotes the exterior derivative as given in these lecture notes. (please see page 3) Here are my thoughts so far: Let us ...
-1
votes
2answers
59 views

Notation of coordinate representation in Lee

In Lee's Introduction to Smooth Manifolds he writes $$ \omega = \omega_i dx^i$$ where $\omega$ is a differential form. See for example page 293. What does $\omega_i dx^i$ stand for? According ...
3
votes
1answer
75 views

Cauchy integral formula: can it be proved like this?

Consider the Cauchy theorem: Let $D\subset \mathbb C$ be a domain such that $\partial D$ is smooth and $\overline{D}$ is compact. Let $f$ be holomorphic on $D$ and continuous on the closure. Then ...
1
vote
1answer
55 views

Closed non-exact $2$-form on $T = S^1 \times S^1$

I would like to find a closed non-exact $2$-form on $T = S^1 \times S^1$. Here are my thoughts: Since $d\theta$ and $d\varphi$ are closed non-exact $1$-forms an obvious candidate is $d\theta \wedge ...
0
votes
1answer
35 views

Finding two inequivalent closed, non-exact $1$-forms on $T = S^1 \times S^1$: second check

This is a follow up on my previous question. I would like to test whether I understand the first part of the answer given to me there by rewriting it in my own words. Please could someone tell me ...
1
vote
2answers
65 views

Stokes or homotopy?

The problem states Show that if $X$ is a simply connected manifold, then $\oint_{\gamma}\omega=0$ for all closed 1-forms $\omega$ on X and all closed curves $\gamma$ in $X$. However I have ...
6
votes
1answer
80 views

Finding two inequivalent closed, non-exact $1$-forms on $T = S^1 \times S^1$

I've been studying the torus and the first cohomology group $H^1_{dR}(T)$ for a couple of weeks now. I finally had a breakthrough of understanding and would like to kindly request the community to ...
4
votes
2answers
104 views

How to evaluate this integral: $\oint dx$?

I am trying to understand differential forms. Now I tried to evaluate $$ \oint_{S^1}dx$$ I should get anything non-zero but I don't know how to do it (even though I know the result). If $S^1$ in ...
2
votes
1answer
55 views

Relationship between differential forms and coordinates

For the purpose of this question let us restrict our considerations to smooth $3$-manifolds. So the manifold $M$ we consider here is endowed with smooth coordinate charts $(x,y,z)$. What I have ...
0
votes
2answers
40 views

$H^1_{dR}(S^1 \times S^1) = H^1_{dR}(S^1) \oplus H^1_{dR}(S^1)$ without Künneth?

As in the title: I am trying to derive $H^1_{dR}(S^1 \times S^1) = H^1_{dR}(S^1) \oplus H^1_{dR}(S^1)$. First let me share my thoughts: I am trying to derive that $$ H^1_{dR}(S^1 \times S^1) = ...
-1
votes
1answer
47 views

$(2n-1)$-form is closed [closed]

Consider in $\mathbb{R}^{2n}$ differential forms$$\omega = dx_1 \wedge dx_2 \wedge \dots \wedge dx_n\text{ and }\theta = \sum_{n+1}^{2n} (-1)^{j-1}x_jdx_{n+1} \wedge \dots \wedge dx_{j-1} \wedge ...
3
votes
1answer
81 views

Exterior derivative of a 2-form

I want to prove that the exterior derivative of a 2-form in $\mathbb{R}^n$: $${\alpha = \sum a_{ij} dx_i \wedge dx_j}$$ is: $${d \alpha = \sum (\frac{\partial a_{ij}}{\partial x_k} + \frac{\partial ...
3
votes
1answer
110 views

What does a standalone $dx$ mean?

Some literature uses $dx$, in the context of differential equations, in a confusing way without defining what it really stands for: $Mdx + Ndy = 0$ Does it mean one of the following or something ...
2
votes
0answers
23 views

Verify stokes theorem example [duplicate]

Let $c:I^2\rightarrow\mathbb{R}^3$ be the singular $2$-cube given by $$c(s,t)=(\frac{1}{2}s^2,st,\frac{1}{2}t^2)$$ Let $$\omega=xy^2dz$$ Questions: i) Compute $c^*\omega$ ii) Compute $c^*d\omega$ ...
1
vote
0answers
26 views

Show system has a solution using Frobenius method

System: $$\frac{\partial u}{\partial x}=v,\frac{\partial u}{\partial t}=w$$$$\frac{\partial v}{\partial x}=G(x,t),\frac{\partial v}{\partial t}=-\dot{a}(t)G(x,t)$$$$\frac{\partial w}{\partial ...
0
votes
1answer
65 views

Left-invariant differential form on homogeneous space

I'm trying to learn by working through an example but I got stuck with the line of reasoning provided in the text. Let $G$ be the group of matrices $$g= \begin{bmatrix} 1 &0 &\log{x} \\ y ...
1
vote
0answers
21 views

Why do entries of Maurer-Cartan 1-form span space of left-invariant 1-forms

Suppose $G$ is a $k$-dimensional Lie group of $n\times n$ matrices of the form $[g_{ij}(x_1,...,x_k)]$ where $g_{ij}$ are smooth functions. As a follow-up to the question I posted recently, I now ...
2
votes
0answers
32 views

Differentiability of a map into $\wedge T^*M$

Let $M$ a differentiable manifold and $p \in M$. Consider $(U;x_1, \dots, x_n)$ a coordinated system around $p$. $\{dx_\phi\}_\phi$ with $\phi \in \cal{P}(\{1, \dots n\})$, $dx_\phi=dx_{i1} \wedge ...
2
votes
1answer
43 views

A question about a part of the proof of the existence of the exterior derivate.

Let $M$ a differentiable manifold, $(U;x_1,\ldots,x_n)$ a coordinated system and $\omega$ a differential form which domain intersects $U$. In $U \cap \operatorname{Dom} \omega$, $\omega$ can be ...