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

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2
votes
1answer
70 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
41 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) = ...
3
votes
1answer
104 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
118 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
25 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
29 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
94 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
28 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
44 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
votes
0answers
28 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
votes
1answer
34 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
votes
0answers
74 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 ...
2
votes
2answers
61 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
84 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
67 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
63 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 ...
1
vote
0answers
26 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
votes
1answer
28 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
41 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
43 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 ...
2
votes
1answer
59 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
votes
0answers
10 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
18 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
85 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
votes
3answers
116 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
76 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 ...
4
votes
1answer
70 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
vote
1answer
54 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
91 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
votes
1answer
40 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
vote
0answers
107 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
46 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
vote
1answer
35 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
130 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 ...
5
votes
1answer
70 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
139 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
69 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
163 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
75 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 ...
2
votes
1answer
79 views

Constructing functions such that integral along any closed curve is non-zero

Consider smooth maps $f: \mathbb R^2 \setminus \{(0,0)\} \to \mathbb R$. How can I construct such an $f$ with the property that $$ \oint_C f \neq 0$$ for any closed curve $C$ around the origin? ...
1
vote
1answer
68 views

Using Poincaré lemma to find the generator $H^1$ of $\mathbb R^2 -\{(0,0)\}$?

I am working towards an undertanding of de Rham cohomology. For this reason I am trying to find generator(s) of $H^n_{dR}(\mathbb R^2 -\{(0,0)\})$ and currently I am working on the case $n=1$. I ...
4
votes
1answer
121 views

Understanding de Rham cohomology: geometrically speaking, when is a smooth function closed

On Wikipedia the de Rham cohomology groups are defined to be the cohomology groups of the de Rham cochain complex (equivalence classes of differential $k$-forms). By this definition the zeroth de ...
0
votes
0answers
47 views

Show that $\int_b d\omega=0$ where $b(s,t)=\Phi_s(c(t)) $

Let $c:[0,1]^k \rightarrow \mathbb{R}^n$; $t \mapsto c(t)$ be k-cell with $k < n$. Let $\mathbb{Y}$ denote a vector field on $\mathbb{R}^n$ with flow $\Psi_s$. Define a $(k+1)$-cell $b:[0,1]^{k+1} ...
2
votes
1answer
161 views

A non-vanishing one form on a manifold of arbitrary dimension

So the problem I have is: Let $\theta$ be a closed 1-form on a compact Manifold M without boundary. Further suppose that $\theta \neq 0$ at each point of M. Prove that $H^{1}_{dR}(M)\neq 0$. The ...
1
vote
1answer
70 views

Show $D^2=0$ iff $D=e^{-f}de^{f}$ for some function $f$ , where $D\omega := d\omega+\alpha \wedge \omega $

Let $\alpha$ be a $1$-form on $\mathbb{R}^n$. Define the following which takes $k$-forms to $(k+1)$-forms. $$D\omega := d\omega+\alpha \wedge \omega $$ Show that $D^2=0$ iff $D=e^{-f}de^{f}$ for ...
1
vote
0answers
35 views

Proving that $\frac{d}{dt}\int \Phi_t^*\omega=\int_{\Phi_t \circ \partial c} i_{\mathbb{X}}\omega$

Let $\omega$ be a closed $k$-form on $\mathbb{R}^n$ and $c:I^k \rightarrow \mathbb{R}^n$ a $k$-cube on $\mathbb{R}^n$. Let $\mathbb{X}$ be a vector field on $\mathbb{R}^n$ with flow $\Phi_t$. Show ...
1
vote
1answer
83 views

Understanding the definition of a pullback of a differential $k$-form and applying it in $1-d$

I am having trouble understanding the definition of a pullback of a differential k-form in a basic course in differentiable geometry. This is the definition I am given. I believe it is easier to ...
1
vote
1answer
78 views

Show that $(\textbf{S}^*\textbf{B})(u,v)=\textbf{B}(\textbf{S}(u,v))\cdot \textbf{N}(u,v) \ du \wedge dv$

Let $\textbf{S}(u,v):[0,1]^2 \rightarrow \mathbb{R}^3$ be a singular $2$-cube which is smooth. Note that $0 \leq u,v \leq 1$. Let $B(\textbf{r})=B_x \ dy \wedge dz + B_y \ dz \wedge dx + B_z \ ...
2
votes
2answers
161 views

Showing that $\int_{c} \omega =0$ when $\partial c =0$

Let $\omega$ be a $k$-form on $\mathbb{R}^n$ and suppose that $\omega=d\alpha$ for some $(k-1)$-form $\alpha$. Show that, for any singular $k$-cube $c$ on $\mathbb{R}^n$ with $\partial c=0$, ...