Questions tagged [differential-forms]

For questions about differential forms, a class of objects in differential geometry and multivariable calculus that can be integrated.

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22 views

Explicit evaluation of symplectic form on cotangent bundle

Let $M$ be a manifold and denote by $T^*M$ its cotangent bundle. Let $(x,U)$ be a coordinate chart so that $x: U\to \mathbb{R}^{n}$. Let $p\in U$ and $v\in T^*_pM$, then we can write $v = v_i dx^i$ ...
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1answer
14 views

Can someone explain me the notation of k-forms (Differentialforms)?

I'm looking at K-forms (Differential forms) and I somehow struggle a bit to understand the notation and the meaning of it. To be more precise I have problems in understanding the indexing. Let me ...
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45 views

In which way do the differential forms give a different view on multidimensional integration?

I am reading up on differential forms calculus, so more specifically we started with pfaff-forms and curve integrals. The multidimensional forms come later. Our prof. said at the time that you can ...
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27 views

Is the existence of a continuous top form sufficient for orientability of a smooth manifold?

I'm currently working through Tu's Introduction to Manifolds. He first defines orientability in terms of the existence of orientations on the tangent spaces which can be locally represented by a ...
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66 views

What is the definition of the integral of a differential form on a submanifold

Let $M$ be a smooth oriented manifold of dimension $n$. Let $\alpha = \alpha_{[0]} + \alpha_{[1]} + ...+ \alpha_{[n]}$ be smooth differential form on $M$ such that $\alpha_{[I]} \in \mathcal{A}^i(M)$....
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42 views

Inverse exterior derivative

The exterior derivative is defined as the unique $\mathbb{R}$-linear mapping, such that $df$ is a differential one-form for a zero-form $ f $, $d d\alpha = 0$ for any $\alpha$, and that it is an ...
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52 views

Volume form on the sphere $S^2$ [closed]

I know the volume form (in french "forme d'aire") on the sphere $S^2$ is given by, $$ \nu_{S^2} = z(dx \wedge dy) + x(dy \wedge dz) + y(dz \wedge dx) $$ I'm wondering how can I compute it ...
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75 views

Geometric Wedge Products of Two Currents and Push Forward a Current by an Automorphism of $\mathbb{P}^2$

Harmonic Currents of Finite Energy and Laminations, J. E. Fornaess and N. Sibony, GAFA, Geom, Funct, Anal, $2005$, $962-1003$. Page $993$. Let $\mathbb{P}^2$ be the Complex Projective Space and $$\...
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1answer
91 views

Understanding integration along fibers

I understand the definition of integration of a differential form along fibers as it is stated in Wikipedia article as follows: Let $\pi :E \rightarrow B$ be a fiber bundle over a manifold with ...
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1answer
50 views

How does a differential form looks like on $\mathbb{R}^n \times [0,1]$?

I have a question in the subsection of examples in Wikipedia article on integration along fibers which says the following: Let $\pi :M\times [0,1] \rightarrow M$ be an obvious projection. First ...
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27 views

Prove $\text{div(rot}X)=0$

Prove $\text{div(rot}X)=0$, $\forall X\in \mathcal{X(\mathbb{R^n})}$ $X$ is a vector field on $\mathbb{R^n}$, $\omega$ it's a form $X\mapsto \omega \mapsto d\omega \mapsto *(d\omega)=\text{rot}X$ (...
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1answer
68 views

question about this line integral whether it is exact

this form $\omega=-\frac y{x^2 + y^2}dx + \frac x{x^2 + y^2}dy$ ,which is not defined at origin .it is not exact on the whole x-y plane . but when y is not zero , $\omega=d(-\arctan(x/y))$, my ...
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1answer
63 views

volume preserving version of Moser's theorem

There exists an well-known theorem of Moser : Thereom(Moser) Let $M$ be a compact oriented smooth manifold and $\alpha,\beta$ be volume forms whose total volumes are the same. Then, there exists a ...
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76 views

Is there an analog for the "derivative as a function" in multiple variables?

Given the plane curve $y=f(x)=\cos(x)$, the function $f'(x)=-\sin(x)$ models the slope of the tangent line to $f$ at each point. Given the surface $z=f(x,y)=x^2+y^2$, the functions $z_x=2x$ and $z_y=...
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32 views

Integral of a 2-form with compact support on the cylinder. [closed]

Prove that the integral of a 2-form with compact support on the cylinder $S^1 \times \mathbb{R}$ is equal to 0. I came upon this simple exercise in my exercise list, I think this should follow rather ...
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1answer
53 views

Exterior derivative leibniz rule geometric

I‘ve watching the video of Keenan Crane on the exterior derivative https://youtu.be/jeiDXhCiF44 where he explains that the product rule of differential forms just describes how little volumes change. ...
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1answer
61 views

If $M$ is a $m$-dimensional smooth manifold, what is the rank of $\Omega^{m}(M)$ as a $\mathcal{C}^{\infty}(M)$-module?

It is clear to me that if $(U,\mathtt{x})$ is a chart of $M$, then $\Omega^{m}(U)$ is a free $\mathcal{C}^{\infty}(U)$-module of rank one. Is $\Omega^{m}(M)$ a free $\mathcal{C}^{\infty}(M)$-module of ...
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44 views

Maximize in the direction $\frac{\nabla f}{|| \nabla f||}$

Let $ p\in \mathbb{R^n}$ and $S(p,1)$ be the unit sphere, prove that $df\restriction_{S(p,1)}:S(p,1)\mapsto \mathbb{R}$ is maximized in the direction $\frac{\nabla f}{|| \nabla f||}$ $df(v)=\nabla f\...
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1answer
42 views

What are irreducible $n$-times antisymmetrized representations?

I came across the following statement while reading a string theory textbook: if $\boldsymbol{16}_s$ and $\boldsymbol{16}_c$ are the are respectively spinor and conjugate spinor representationsof $...
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1answer
78 views

Products $s\wedge t$ generate $C_.^\infty(M,E\otimes F)$?

I am reading Demially and came upon the claim that for every $s\in C^\infty_.(M,E)$, $t\in C^\infty_.(M,F)$, the wedge product $s\wedge t$ can be combined with the bilinear map $E\times F\to E\otimes ...
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81 views

Prove that $\text{rot(grad}f)=0$

Prove that $\text{rot(grad}f)=0$ $X$ is a vector field on $\mathbb{R^n}$, $\omega$ it's a form $X\mapsto \omega \mapsto d\omega \mapsto *(d\omega)=\text{rot}X$ (This is a problem from do Carmo's ...
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48 views

Why does the Hodge-star of d$x$ equal d$y$ and not -d$y$?

Wikipedia defines $$\star e_I = (-1)^{\sigma(I)} e_{\overline{I}}$$ where $\overline{I} = [n]\setminus I = \{\overline{i}_1< \cdots < \overline{i}_{n-k}\}$ and $\sigma(I)$ is the permutation $...
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2answers
98 views

Why is the space of differential forms $\bigoplus_{p=0}^n \Lambda_x^p$?

In Wald's book "General Relativity", the space $\Lambda_x$ of differential forms at a point $x$ is worked out in the following manner: Let $M$ be an $n$-dimensional manifold. The vector ...
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31 views

A $1$-form is closed if and only if $\frac{∂a_i}{∂x^j} =\frac{∂a_j}{∂x^i}, \forall\:i,j =1,2,...,n, i\neq j$

Prove that a smooth $1$-form $\omega=\sum_{i=1}^na_idx^i$ on $\mathbb{R^n}$ is closed if and only if $\frac{∂a_i}{∂x^j} =\frac{∂a_j}{∂x^i}, \forall\:i,j =1,2,...,n$ with $i\neq j$ $d\omega =\bigg(\...
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22 views

Integrability Conditions for a System of Differential Equations in the Language of Forms

Which Integrability Conditions for a System of Differential Equations in the Language of Differential Forms?
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1answer
115 views

Question on baby Rudin examples 10.12

I have two questions: $(a)$. Fix $a > 0$ and $b > 0$ and define $\gamma(t) = (a\cos(t),b\sin(t))$ where $(0\leq t \leq 2\pi)$. So that $\gamma$ is a closed curve in $R^2$. (Its range is an ...
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43 views

Different definitions on the tensor and wedge products

I'm studying differential forms and some linear algebra doubts popped up: Given $V$ a vector space, we define $A_k(V)$ as the space of alternating $k$-linear maps $V^k \to \mathbb{R}$. The tensor ...
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1answer
66 views

If $\omega \wedge d\omega = 0$ then $\omega = \lambda\,df$ for some real functions $\lambda, f$ [closed]

I found the following problem on an old qualifying exam and wasn't able to solve it, and I was wondering if anyone could help: Let $\omega$ be a smooth 1-form on a smooth manifold $M$. Suppose $\omega ...
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25 views

de Rham cohomology on $\mathbb R^n$ is equal to $\{0 \}$ for $k>0$, $\mathbb R$ for $k=0$.

Let $\Omega^k (U)$ be a set of $k-$ diffrential form on $U$, where $U \subset \mathbb R^n$ is region, and define $ Z^k (U)$ be a set of $k-$ closed form, $B^k (U)$ be a set of $k-$ exact form. Then, ...
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1answer
46 views

Inferring behavior of a function on the plane from curve integral

Let $P,Q:\mathbb{R}^2\rightarrow\mathbb{R}$ be two $C^1$ functions on the plane. Denote by $\Gamma$ the unit circle. $(P^2+Q^2)|_\Gamma>0$ and $\oint_\Gamma \frac{PdQ-QdP}{P^2+Q^2}\neq0$. Prove ...
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1answer
143 views

Baby Rudin 10.11 and 10.12

These are definitions which we need for the question let $y$ be a $1$-surface ( a curve of class $\mathscr C'$) in $R^3$, with parameter domain $[0,1]$. write ($x,y,z$)in place of ($x_1,x_2,x_3$), ...
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2answers
58 views

Evaluate $\int_cydx+zdy+zdz$ if c is intersection of upper hemisphere $x^2+y^2+z^2=4$ $z \geq0$ and $x^2+y^2=2x$

Evaluate $\int_cydx+zdy+zdz$ if c is intersection of upper hemisphere $x^2+y^2+z^2=4$ $z \geq0$ and $x^2+y^2=2x$ oriented counter clockwise from xy plane First, I made the following parametricisation: ...
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1answer
74 views

Show that $h$ is a smooth function with $dh = \omega$

I don't understand a few parts of Professor Nelson's proof of problem 9.3(b) (link). I'll provide the proof, then ask questions at the end. Lemma and Proof. Lemma. Let $\omega \in \Omega^1(\mathbb{R}^...
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1answer
83 views

Computing the wedge product of some differentials

I am trying to compute the wedge product of \begin{align*} &2x_p \mathrm{d}x_p \bigwedge_{i=1,\cdots,p-1}2(x_i \mathrm{d}x_i +y_i\mathrm{d}y_i)\bigwedge_{i=1,\cdots,p-1}x_{i+1}\mathrm{d}y_i+y_i\...
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1answer
56 views

Relation between $\eta\wedge \bar{\eta}\wedge \omega^{n-1}$ and $||\eta||^2\omega^n$

Let $X$ bet a compact Kahler manifold of dimension $\dim_{\mathbb{C}}(X)=n$ with Kahler form $\omega$. Let $\eta\in\Omega^{1,0}(X)$ be a differential form. I would like to know if there is a way to ...
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2answers
186 views

Proof of Cartan lemmma : why are the coefficients smooth?

I am new to differential geometry and I am struggling on the proof of the Cartan's lemma. The version I am trying to prove is the following. Let $M$ be a smooth $n$-manifold, $\omega^1,...,\omega^k,\...
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66 views

Leibniz rule Exterior Covariant Derivative

I'm reading Riemannian Geometry and Geometric Analysis by Jost and the way he defines the exterior covariant derivative comes naturally using the definition of the induced connection on the tensor ...
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31 views

What is the most natural ordering of basis 2-forms

When expanding 2-forms in $(\mathbb {R}^3 )^{\wedge 2}$ into a basis, you can use a cyclic trick to get the “most natural” basis set. $\omega = a_{1,2} dx^1\wedge dx^2+ a_{2,3} dx^2 \wedge dx^3 + a_{3,...
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27 views

Changing coordinates of given differential form

I've been given the following differential form: $$\eta = -\frac{y}{x^2+y^2}dx + \frac{x}{x^2+y^2}dy$$ and need to express it in the polar coordinates $(x,y) = (r\cos\theta, r\sin\theta)$. To do so I ...
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1answer
93 views

Differential form on $S^1$

I am a beginner at Manifolds. I tried the following problem, but not sure whether I got it correct not: Find a differential 1-form on $S^1$ My attempt: Define $f: S^1\to \mathbb{R}$ given by $f(x,y)=...
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40 views

Conventions regarding the de Rham complex

Let $M$ be a smooth $m$ dimensional manifold, $\Omega^k(M)$ the $\mathbb R$-vector space of smooth $k$-forms on $M$ and consider the two cochain complexes $$ 0\rightarrow \Omega^0(M)\xrightarrow{\...
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42 views

Showing a specific $p$-form on a star-shaped domain is the exterior derivative of a specific $(p-1)$-form.

The folloing is a problem from Chapter 17 of Lee's Introduction to Smooth Manifolds: Suppose $U\subseteq \mathbb{R}^n$ is open and star shaped with respect to $0$, and $\omega=\sum'\omega_Idx^I$ is a ...
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60 views

2-Form on $\mathbb{R}^3$ That Restricts to Surface Area 2-Form on Torus?

I would like to find a 2-form on $\mathbb{R}^3$ that 1) restricts to the torus to give a generator of the top-dimensional de Rham cohomology of the torus and 2) restricts to the torus to give the ...
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56 views

Curl of a Lie bracket of two vector fields

I am wondering if there is a nice (ideally coordinate free, something that holds in manifolds) identity of the form $\nabla \times [X,Y] = [\nabla \times X,Y] + [X, \nabla \times Y] + ...$, and if ...
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2answers
71 views

Existence of $k$-form with nonzero integral

Say that $N$ is an oriented, compact, connected manifold without border. If $\operatorname{dim}(N) = k$, does it always exists some $k$-form $\omega$ such that $\int_N \omega \neq 0$? I know how to ...
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23 views

Verification of this integration on forms by two different methods

Let $\Omega\subset\mathbb{R}^3$ be the upper half ball with radius $a$, i.e. the region bounded below by $z=0$ and above by $x^2+y^2+z^2=a^2$. Compute $$ \int_{\partial\Omega} xz\,dy\wedge dz + yz\,dz\...
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47 views

Interchange of limits of differential operators on differential forms

Let $M$ be a complex 2-dimensional manifold with the additional property that a smooth function $h:M \to \mathbb C$ is constant if and only if $\partial\overline\partial h = 0$ (which is equivalent (...
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1answer
92 views

Integration of differential form inclusion of $S^2$

Take this example. We have the natural inclusion $i : S^2 \rightarrow R^3$ and the differential form: $\omega = x dy \wedge dx + y dz \wedge dx + z dx \wedge dy $. How can we say that that the ...
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47 views

Understanding a definition of the exterior derivative

I'm trying to prove the exact same formula as in this question: $U \subset \mathbb{R^n}$ an open set, $\omega$ a k-differential form on $U$, $X_0, \dots, X_k$ vector fields on U, i. e. elements of $C^{...
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42 views

Baby Rudin 10.1

I want to be sure that I understand it correctly. As I understand it $\int_{I^k} f(x) dx $ = $\int_{a_{k-1}}^{b_{k-1}}$ ($\int_{a_k}^{b_k} f_k(x_1,...x_{k-1},x_k)dx_k$)$dx_{k-1}$ and to continue this ...

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