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

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37 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 ...
2
votes
1answer
61 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
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1answer
50 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
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1answer
30 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
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2answers
49 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
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1answer
69 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
95 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
48 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
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2answers
34 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) = ...
0
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0answers
19 views

Finding closed non-exact differential forms

I have been thinking about this for quite some time now. I worked on the example of the torus and determined the generators of $H^n_{dR}(T)$ by trial and error. It worked but it wasn't satisfying. I ...
0
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1answer
36 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
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1answer
65 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
99 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
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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 $c^*d\omega$ ...
0
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0answers
19 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 ...
2
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0answers
36 views

Laplacian on sphere with differential forms [closed]

I want to express the Laplacian on the 2-sphere in terms of differential forms. Does anybody know how this can be done? I am not so familiar with submanifolds, thus I would appreciate help very much. ...
0
votes
1answer
37 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
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0answers
14 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
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0answers
31 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
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1answer
41 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 ...
0
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0answers
23 views

Time Derivative of the integral over a singular k-cube

I am stuck on this question, and was wondering if someone could provide a hint of where to start? I can't see the first step.
0
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1answer
28 views

First order differential equation problem

Suppose we have $$ \frac{dy}{dx} +f(x)y = r(x) $$ and it has two solutions $y_1(x)$ and $y_2(x)$ then how to prove that solution of differential equation $$ \frac{dy}{dx} +f(x)y = 2r(x) $$ Will be ...
0
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0answers
45 views

Sheaf of differential p-forms

Shafarevich defines the cotangent bundle at page 60 of "Basic Algebraic Geometry 2". Now he says that: 1) $\mathcal{F}_x=\mathcal{O}_x dt_1 + \dots + \mathcal{O}_x dt_n$, where $\mathcal{F}_x$ is the ...
1
vote
2answers
51 views

Given a nowhere zero vector field $Z$, does there exist a one-form $\gamma$ such that $\gamma(Z) = 1$?

Take $M$ a smooth manifold, and $Z$ a vector field on $M$ such that $Z(p)\neq0$ for all $p\in M$. Is there a one form $\gamma \in \Omega^1(M)$ such that $\gamma(Z)=1$? I started to work locally, but ...
2
votes
0answers
48 views

Differential one-forms and change of coordinates

Consider two differential one forms: $$\omega=\sum_{i=1}^N \omega_i dx^i$$ $$\omega'=\sum_{i=1}^N \omega'_i dx'^i$$ As I recall from my analysis courses, the symbols $dx$ are a particular notation ...
0
votes
0answers
32 views

The induced map on the de Rahm cohomology of a surjective submersion.

Let $M,N$ be two smooth manifolds and $f: M \rightarrow N$ a surjective submersion (so $f$ and $f_*$ both surjective everwhere). It is straightforward to show that then the pullback of $k$-forms: ...
1
vote
1answer
55 views

What does $d\zeta_1\wedge\cdots\wedge d\zeta_n$ mean in the context of Cauchy formula (on polydiscs)?

A Polydisc of center $z^o=(z_1^o,\dots,z_n^o)\in\Bbb C^n$ and multiradius $r=(r_1,\dots,r_n)\in(\Bbb R^+)^n$ is defined as $$ P_{z^o,r}:=\prod_{j=1}^n\Delta_{z_j^o,r_j} $$ where ...
0
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0answers
14 views

$k$-cube definition clarrification

If $c:I^k\rightarrow\mathbb{R}^n$ is a singular $k$-cube. What will $\partial c$ be? Will it be a singular $(k-1)$-cube or a $(k+1)$-cube or something else? Definition - A singular $k$-cube on $U$ ...
0
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1answer
26 views

Compute $L_\mathbb{X}\beta$

Given information: $$\alpha+(x+y)dy+(x^2-y^2)dz$$$$\beta=zdx\wedge dy+xzdx\wedge dz$$$\mathbb{X}$ is the vector field given by $$\mathbb{X}=(0,-x,-1)$$ I have found $i_\mathbb{X}\beta=2xzdx$ ...
1
vote
1answer
32 views

Compute the contraction of $i_\mathbb{X}\beta$

Question: Let $\beta=zdx\wedge dy+xzdx\wedge dz$, and let $\mathbb{X}$ be the vector field on $\mathbb{R}^3$ given by $\mathbb{X}=(0,-x,-1)$. Compute $i_\mathbb{X}\beta$, combining terms where ...
1
vote
1answer
37 views

Real/Complex differentials forms

Given $f:\Bbb C^{n}\to\Bbb C$ identified with $f:\Bbb R^{2n}\to\Bbb C$, in a book I read that $$ \partial_x f\,dx+\partial_yf\,dy=\partial_zf\,dz+\partial_{\bar z}f\,d\bar z $$ and that this could be ...
1
vote
1answer
50 views

Compute $\int_cd\omega$ and $\int_{\partial c}\omega$

Question: Let $c:I^2\rightarrow\mathbb{R}^3$ be the singular $2$-cube given by $$c(s,t)=\left(\frac{1}{2}s^2,st,\frac{1}{2}t^2\right)$$Let $x=(x,y,z)$ denote the cartesian coordinates on ...
0
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0answers
9 views

Obtain first-order system

Question: Let $G(x,t)=g''(x-a(t))$, where $g$ and $a$ are smooth. Combine equations: $$\frac{\partial u}{\partial x}=v;\frac{\partial u}{\partial t}=w$$$$\frac{\partial v}{\partial t}=\frac{\partial ...
0
votes
1answer
10 views

Derive necessary and sufficient conditions meaning

Question: Let $\mathbb{X}$ and $\mathbb{Y}$ be vector fields on $\mathbb{R}^3$ given by $$\mathbb{X}(x,y,z)=(1,0,p(x,y)r(z))$$ $$\mathbb{Y}(x,y,z)=(0,1,q(x,y)r(z))$$ where $p,q$ and $r$ are smooth, ...
1
vote
2answers
72 views

Exactness of $dx,dy$

Let $dx,dy$ denote differential $1$-forms. It is easy to verify that they are closed. My question is: Does there exist a space such that either $dx$ or $dy$ or both are exact? (A ...
2
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3answers
75 views

Divergence theorem in complex analysis

I am revisiting my understanding of integration by parts in several complex variables, but I have run into an apparent contradiction. This shows my understanding is flawed, which is somewhat ...
0
votes
1answer
68 views

Deriving generators for $H^1(T)$: what are $dx$ and $dy$?

By trial and error I found that $dx,dy$ are generators of $H^1_{dR}$ of $T=S^1\times S^1$. Verifying that they generate the first cohomology group is not difficult. My problem is: I found them by ...
3
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1answer
41 views

Find the Poincare Dual of a ray in $\mathbb{R}^2-\{0\}$

This is example-exercise 5.16 in Bott and Tu (which I'm independently reading through.) The problem states: Let $M=\mathbb{R}^2-\{0\}$, and $X\subseteq M$ be the closed submanifold ...
1
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1answer
50 views

Generators of $H^1 (T)$: take two

Previously, I asked about how to prove that $dx + dy$ is a generator of the de Rham cohomology group of the torus. Now it occurred to me that $dx$ and $dy$ are both also generators of $H^1(T)$. ...
1
vote
2answers
78 views

Relation between volume form and cross product

Euclidean three-dimensional space (it's simpler). Defining $\eta={e^*}^1 \wedge {e^*}^2 \wedge {e^*}^3$, with $\{{e^*}^1,{e^*}^2,{e^*}^3\}$ dual of the orthonormal basis, and indicating the classic ...
0
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1answer
36 views

Find a one-form $\alpha_0$ such that $\lim_{t\to 0}\hat\Phi_t^*\beta=d\alpha_0$

Given before question: $\hat{\mathbb{X}}_t\circ\hat\Phi_t=\frac{\partial}{\partial t}\hat\Phi_t$ and $$\beta=\frac{(x^2-y^2)dy\wedge dz+2xydz\wedge dx}{(x^2+y^2)^2}$$ I have shown that $d\beta=0$ ...
1
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0answers
77 views

Frobenius theorem method

Question: Let $A(z)$ be smooth with nonvanishing derivative $A'(z)$, and consider the system with first-order PDE's, $$\frac{\partial u}{\partial x}=\frac{\partial K}{\partial ...
3
votes
1answer
36 views

Finding the Lie derivative of a complex valued function

Question: Let vector field $\mathbb{X}$ be given by $\mathbb{X}(x,y)=(-y,x)$ Let $f(x,y)$ be the complex-valued function given by $f(x,y)=(x+iy)^m$ where $m>0$ Show that $L_\mathbb{X}f=imf$ My ...
1
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1answer
29 views

Poles of Complex Functions or One-Forms?

The function $f(z) = \frac{1}{\sqrt{z}\sqrt{1-z}}$ with branch cuts chosen so that $f$ is analytic on $\mathbb C-[0,1]$ has a pole at infinity according to this walkthrough of a branch cut contour ...
1
vote
2answers
49 views

Compute the contraction of a 1-form with a vector field

Question: Let $\alpha$ be the $1$-form on $\mathbb{R}^3$ given by $\alpha=zdy-ydz$ and let $\mathbb{X}$ be the vector field on $\mathbb{R}^3$ given by $\mathbb{X}=(0,y,-z)$. Compute ...
4
votes
1answer
48 views

What does $\alpha$ be the $1$-form and $\beta$ the $2$-form on $\mathbb{R}^3$ mean?

In an earlier post to math.stackexchange I asked a question beginning with: Let $\alpha$ be the $1$-form and $\beta$ the $2$-form on $\mathbb{R}^3$ given by $$\alpha=(x+y)\,dy+(x^2-y^2)\,dz$$ ...
2
votes
0answers
52 views

Compute the wedge product

This is my first time computing the wedge product, I am not sure if I have done it correctly as I do not have solutions, if I have gotten the answer right or am doing the right method please say. ...
3
votes
1answer
57 views

How to determine whether a differential $1$-form is globally welldefined?

This is a question that occurred after working on finding a generator of the first de Rham cohomology group of the torus. It was pointed out to me that the differential $1$-form $$ dx + dy$$ was ...
3
votes
2answers
91 views

Applying the Frobenius theorem to a decomposable 2-form

So I have the following problem: Suppose $\omega=\phi \wedge \theta$ is a closed decomposable 2-form on $M$ a manifold (decomposable just means it can be written as a wedge of 1-forms). Suppose $p\in ...
2
votes
3answers
69 views

Generators of $H^1(T)$

Let $T$ denote the torus. I am working towards an understanding of de Rham cohomology. I previously worked on finding generators for $H^1(\mathbb R^2 - \{(0,0)\})$ but then realised that for better ...