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

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5
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1answer
263 views

differential form

one form $\alpha$ over a smooth manifold is non vanishing means for every $p\in M$, $\alpha_p\neq 0$. But $\alpha_p$ is linear map $T_M\to \mathbb R$, hence $\alpha_p(0)=0$. So confusion arises ...
3
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2answers
98 views

Proving that a particular submanifold of the cotangent space is Lagrangian

I have the following problem in my differential topology class: Let $M$ be an $n$-dimensional manifold and let $\omega$ denote the standard symplectic form on the cotangent space $T^*M$. Let $f \in ...
1
vote
1answer
139 views

Confused about Wikipedia page on differential forms

I know next to nothing about differential forms, but I saw them being mentioned repeatedly on this site, so I went to Wikipedia to try to understand what a differential form is. Most of it is going ...
0
votes
0answers
315 views

pullback of 1-differential form

Given a $ \phi_{0} \in \mathbb R$. Consider the function $ \displaystyle{\phi (t) = \phi_{0} + \int_0^t (ab'-a'b)(u) \text{du}}$ where $ a^2(t) + b^2 (t)=1$ Prove that: (i) $ \phi(0)= \phi_{0}$ ...
4
votes
1answer
240 views

$4$-form $ \omega \wedge \omega$ vanishes on $S^4$

If $\omega$ is a closed $2$-form on $S^4$, how can I show the $4$-form $ \omega \wedge \omega$ vanishes somewhere on $S^4$? I am guessing that the fact we're talking about the $2$-form being ...
14
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1answer
1k views

Geometric Explanation of Tamagawa Numbers

Sometimes in order to understand a concept thoroughly we need to have a algebraic view ( in terms of equations ) and corresponding geometric view. My interest always lies with understanding the ...
2
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1answer
89 views

How do you calculate an exterior derivative on forms in $\mathbb{R}^3$?

If we have a form, say, $\omega = f(x,y,z) \, dx + g(x,y,z) \, dy + h(x,y,z) \, dz$, what is the formula for the exterior derivative $d \omega$?
2
votes
1answer
291 views

Pullback on differential forms are linear

Let $T: \mathbb{R}^n \rightarrow \mathbb{R}^m$ be a smooth transformation. Define the pullback $T^*: C^k (\mathbb{R}^m) \rightarrow C^k (\mathbb{R}^n)$ (With $C^k(\mathbb{R}^n)$ being the set of ...
1
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1answer
134 views

What exactly are the definitions of $\varphi^*dz$ and $\varphi^*d\bar{z}$?

Suppose $\varphi$ is a smooth map on $\mathbb{C}$. For a function $f$, we define a $0$-form $\varphi^*f$ as $\varphi^*f=f\circ\varphi$. Also, $$ \varphi^*\,dx=\frac{\partial\varphi_1}{\partial ...
4
votes
2answers
85 views

For a $1$-form $h$, why does $\int_\Gamma \varphi^*h=\int_{\varphi\circ\Gamma}h$?

I'm trying to understand why for a differentiable arc $\Gamma:[a,b]\to\Omega$ and a $1$-form $h=fdx+gdy$, then $$ \int_\Gamma\varphi^*h=\int_{\varphi\circ\Gamma}h? $$ For background, $\Omega$ is an ...
2
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1answer
215 views

Relationship between Rotational Motion and Standard Linear Motion

I am looking to study elementary mechanics (physics) from the point of view of differential forms on manifolds, and moments, and am wondering if there are any texts that generalize the notions of ...
7
votes
1answer
98 views

Why does $d(\varphi^*f)=\varphi^*df$?

I'm trying to learn a bit about differential forms to supplement my study in analysis, but I'm having a hard time with some of the basic manipulations. Anyway, suppose $\Omega$ is an open set in ...
9
votes
1answer
442 views

Information captured by differential forms

My advanced calculus class is currently doing differential forms and I have a hard time really understanding what they are all about. I can read the proofs of the theorems given in Rudin's PMA chapter ...
5
votes
1answer
661 views

Relationship Between Differential Forms and Vector Fields

I am trying reach an understanding of precisely how the space of differential forms is related to the space of vector fields. These are the definitions that I understand and am using for these ...
1
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1answer
107 views

Is the differential form $\omega(x,y)=\frac{2x}{x^2+y^2-4}dx+\frac{2y}{x^2+y^2-4}dy$ exact on his natural domain?

The natural domain $D$ of the differential form $\omega$ is $D=D_1 \cup D_2$ where $D_1=B\bigl((0,0),2\bigr)$ is the open ball of center in the origin and radius $r=2$ and ...
2
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3answers
242 views

Differential forms and double improper integral

Can someone suggest me a list of book or workbook with examples and solutions on differential forms and double improper integral?
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1answer
201 views

If $S^{2n+1}$ is covering space of $X$, then $X$ is orientable.

Is there any direct way to prove that $n$-manifold is orientable? In AT we can just calculate $n$'th homology group and check whether it's $\mathbb Z$ or $0$. But I want a geometric method, using ...
3
votes
1answer
271 views

Intuitive interpretation of these differential forms

Let $\pi: S^2-\{N\}\to \mathbb R^2$ be the stereographic projection map. Let $\sigma:\mathbb R^2\to S^2-\{N\}$ be its inverse. Let $p\in S^2-\{N\}$ and $x_1,x_2\in$ the tangent space of $S^2$ Would ...
0
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1answer
263 views

Integrating factor and differential 1-forms

I am working on the following exercise: The function $f$ is called an integrating factor for the 1-form $\omega$ if $f({\bf x}) \neq 0$ for all $\bf x$ and $d(f\omega) = 0$. If the 1-form $\omega$ ...
3
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1answer
165 views

If $\omega$ is an $n$-form on a compact $n$-manifold $M$ without boundary, then $\omega $ is exact if and only if $\int\limits_{M}\omega=0$

If $\omega$ is an $n$-form on a compact $n$-dimensional manifold $M$ without boundary, then $\omega $ is exact if and only if $\int\limits_{M}{\omega }=0$. Maybe there are two ways - use de Rham ...
2
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1answer
95 views

Equality of integrals of differential forms

I have two $(n-1)$-forms $\omega_{1}$ and $\omega_{2}$ on $\mathbb{R}^n$ and a smooth function $g(x) \colon \mathbb{R}^n \to \mathbb{R}$ ($dg$ doesn't vanish anywhere) such that $dg \wedge \omega_1 = ...
2
votes
1answer
180 views

Differential forms on a $S^1$-manifold

I am reading about differential forms on manifolds with group actions and there is an 'obvious' formula which I don't quite understand. Let $X$ be a manifold with a smooth circle action, that is a ...
3
votes
1answer
200 views

Derivative of an integral of differential form

I have some smooth function $g(x) \colon \mathbb{R}^{n}_{+} \to \mathbb{R}_+$ such that $G_{t} = \{ x \in \mathbb{R}^n_+ \mid g(x) \leqslant t \}$ is compact. I consider a function $$ f(t) = ...
1
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1answer
227 views

Recover the differential form by its pullback

I have a smooth manifold $M$ in $\mathbb{R}^n$ given by $M = \{ x \in \mathbb{R}^n \mid g(x) = 0 \}$. Its atlas consists of a single chart $(M,\varphi)$, where $$ \begin{array}{rcl} ...
13
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1answer
473 views

What do $dz$ and $|dz|$ mean?

I'm having a hard time understanding complex differentials. I know that when I have a field $\mathbb K$ and a $\mathbb K-$vector space $\mathbb K^n,$ then we define $dx_i\in \mathrm{Lin}(\mathbb ...
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2answers
138 views

Closed differential form.

In the paper, page 5, Line 5, why the form $\eta_j$ is closed? We have to show that $d \eta_j=0$. Here $\eta_j = f(t)dt$ for some function $f$. Is it always true that $d\eta_j$ for any $f$? Thank you ...
1
vote
1answer
117 views

Prove that $f^*\omega=0$

I'd like some help to prove the following: $f\colon U \subset \mathbb{R}^m \to \mathbb{R}^n$, differentiable; $m<n$ ; $\omega$ is a $k$-form in $\mathbb{R}^n$, $k>m$. Show that $f^*\omega = 0$ ...
9
votes
2answers
237 views

checking if a 2-form is exact

Consider the 2-form $$\sigma=\frac{x_1 dx_2 \wedge dx_3 + x_2dx_3\wedge dx_1+ x_3 dx_1 \wedge dx_2}{(x_1^2+x_2^2+x_3^2)^{3/2}}.$$ I need to show if it is exact or not. Suppose it is exact, then there ...
5
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3answers
323 views

$\Omega=x\;dy \wedge dz+y\;dz\wedge dx+z\;dx \wedge dy$ is never zero when restricted to $\mathbb{S^2}$

We define a 2-form $\Omega$ on $\mathbb{R^3}$ by $\Omega=x\;dy \wedge dz+y\;dz\wedge dx+z\;dx \wedge dy$. How can I show that $\Omega|_\mathbb{S^2}$ is nowhere zero? Before proving that how can I ...
4
votes
1answer
97 views

Computing $d\omega$ and $g^\ast \omega$ when $(x,y)=g(s,t)=(st,e^t)$ and $\omega= xdy$

Define $g:\mathbb{R^2} \rightarrow \mathbb{R^2}$ by $(x,y)=g(s,t)=(st,e^t)$ and let $\omega= xdy$. How can I compute $d\omega$ and $g^\ast \omega$? Actually, I computed $d\omega =tds \wedge e^tdt$. ...
4
votes
1answer
248 views

Uniqueness of Winding Number

This is an exercise from Spivak's Calculus on Manifolds, problem 4-27. Define the singular 1-cube $c_{R,n}:[0,1]\rightarrow \mathbb{R}^2 - \{0\}$ to be $c_{R,n}=(R\cos(2\pi nt), R\sin(2\pi nt))$. ...
1
vote
1answer
194 views

Elementary symmetric polynomials and matrices of 1-forms

Let $A$ be a $n \times n$ matrix of 1-forms (for example, a connection form). Note that $A \wedge A$ is not $0$, but by using the anti-symmetry of the wedge product applied to the entries of $A$ we ...
0
votes
1answer
126 views

Statement of the $d d^c$-Lemma

I'm looking at the definition of Green's function $g_\mu$ for the Laplacian $\Delta_\mu$ associated to a positive $(1,1)$-form $\mu$ on a Riemann Surface $X$. In specific the main request that the ...
3
votes
2answers
154 views

An identity of differential forms

Let $\omega=\sum_j \omega_jdy^j$. We want to show that $d(F^{*}\omega)=F^{*}(d\omega)$, where $F: M \to N$ is a map from manifold $M$ to manifold $N$. Local coordinate systems of $M, N$ are $(y^1, ...
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2answers
161 views

Closed forms and a simple relation with Cauchy-Riemann

I have a very basic question, sorry for that )=. Let's fix some notation first. Let $ dz = dx + i \; dy $ . Given $f \in C^1$, $f : D \subset \mathbb C \to \mathbb C$, we define $df = f_x \; dx + ...
3
votes
1answer
163 views

Computing the restriction of a differential form

Define $\omega$ on $\mathbb{R}^3$ by $\omega = x\,dy\wedge dz + y\,dz\wedge dx + z\,dx\wedge dy$. Thus far I have computed $\omega$ in spherical coordinates $(\rho,\phi,\theta)$, as well as computed ...
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vote
1answer
208 views

The chain rule for a function to $\mathbf{C}$

Let $f:U\longrightarrow \mathbf{C}$ be a holomorphic function, where $U$ is a Riemann surface, e.g., $U=\mathbf{C}$, $U=B(0,1)$ or $U$ is the complex upper half plane, etc. For $a$ in $\mathbf{C}$, ...
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vote
0answers
59 views

derivatives of coordinates on a riemann surface

Let $X$ be a compact connected Riemann surface of genus $g>0$. Let $(\omega_1,\ldots,\omega_g)$ be a basis for $H^0(X,\Omega^1)$. Let $x$ be a point in $X$ and let $z:U\longrightarrow B(0,1)$ be a ...
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0answers
229 views

Differential forms and a chain rule

Let $U$ be a Riemann surface and let $z:U\longrightarrow B(0,1)$ be a diffeomorphism, where $B(0,1)$ is the open unit disc in $\mathbf{C}$. So $z$ is a coordinate around $P=z^{-1}(0)$. Let $Q\in U$ ...
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1answer
333 views

Illustration of vector calculus vs. differential forms

I am looking for a nice illustration of how vector calculus relates to differential forms. A demonstration that employs physics is appreciable (e.g. electromagnetism). In particular, while dualizing ...
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0answers
171 views

Two “different” adjoints of exterior derivative on manifolds with boundary in the $L^2$-setting

The follow problem appears in the setting of $L^2$-differential forms on manifolds with boundary. An abstracted operator theoretic problem is given below. Suppose $M$ is a smooth Riemannian manifold ...
4
votes
1answer
517 views

Bundle orientability vs manifold orientability

Given a vector bundle, I am a bit hazy about the difference between the notions of its orientability as a bundle and as a manifold. I think I know that the following are true, A tangent bundle of a ...
3
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0answers
75 views

The canonical (1,1)-form on a compact Riemann surface gives locally a subharmonic function

Let $X$ be a compact connected Riemann surface of genus $g>0$. We have the so called canonical (1,1)-form $\mu$ on $X$ defined as follows. Choose an orthonormal basis $(\omega_1,\ldots, \omega_g)$ ...
4
votes
1answer
213 views

Exercise concerning the Lefschetz fixed point number

I can't see a good approach to the third part of the following problem: Let $f: M \to M$ be a smooth map of a compact oriented manifold into itself. Denote by $H^q(f)$ the induced map on the ...
5
votes
1answer
503 views

How to find or guess the homotopy operator?

In the proof of the Poincare Lemma for compactly supported cohomology,the homotopy operator K suddenly appears and satisfies the equation 1-e*π*=±(dK-Kd),that is too lucky!I do not know how to find or ...
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2answers
401 views

Examples of Computations in Algebraic Topology

I have started reading "Differential Forms in Algebraic Topology" by Bott, Tu, recently. While I'm quite happy with the exposition of the theorems and explanation of theoretical results, I'm missing ...
11
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1answer
453 views

Coordinate free proof that curvature is the “square” of the connection

Here's the setup. Consider a vector bundle $E$ over a manifold $M$ and let $\Omega^*(M, E)$ denote the space of $E$-valued differential forms (i.e. the space of sections of the vector bundle ...
6
votes
1answer
469 views

Differential Form on a Riemann Surface

The following problem is basically from Miranda's "Algebraic Curves and Riemann Surfaces", which I am reading on my own; if there are any rules against posting textbook problems, my apologies! Let ...
14
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2answers
933 views

Intuition behind $dx \wedge dy=-dy \wedge dx$

I was re-reading this old book of mine; and I noticed that in defining the rules of differential forms, it "makes sense" that we have the rule $dx \wedge dx=0$ because if $dx$ is infinitesimal, then ...