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

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2
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34 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
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
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97 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 ...
2
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200 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. ...
2
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0answers
116 views

Pullback of a 1 form on the circle

Q: Let $M$ be a smooth compact manifold, and suppose there is a smooth map $F:M \rightarrow S^{1}$ whose derivative is non-zero at every point. Prove that the de Rham cohomology space $H^{1}(M)$ is ...
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77 views

Homotopy, Stokes Theorem and Orientation

I have a problem in which the theory and the computation disagree about a minus sign. My question requires a little setting up. I have a complex valued 2-form $$ \omega = \alpha(\xi_1,\xi_2)\, ...
2
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66 views

An error applying the Stokes theorem?

M is the surface $z=x^2+y^2$ with standard orientation for $x^2+y^2\leq 1$ and $\varphi = 4x^2ydy+z^2dz$ I'd like to verify that $\int_Md\varphi=\int_{\partial M}\varphi$, which I did, but ...
2
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116 views

Prove Green's theorem for circles

So the problem is in the title. The rules are that I can't split the circle into "rectangles" and I can't use pull-back. I tried to do something similar to the proof on unit squares. The problem is ...
2
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0answers
75 views

A few identities on differential forms

I just started studying differentiable manifolds, and I've encountered a few allegedly simple questions that aren't really simple to me. If possible, I'd be very happy to see full proofs (without ...
2
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0answers
97 views

The first proof for Poincare lemma in history

How can I get a reference about the first proof of Poincare lemma in history? I already know some methods of proof, but I do want to know the original approach. Thanks for your help!
2
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210 views

Volume form on $\mathbb{S}^2$

Let $\omega = x_1 \,dx_2 \wedge dx_3 + x_2 \,dx_3 \wedge dx_1 + x_3\, dx_1 \wedge dx_2$, with $(x_1,x_2,x_3) \in \mathbb{S}^2$, be a volume form on $\mathbb{S}^2$, and let $f : \mathbb{S}^1 \times ...
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60 views

Help with local coordinates computations-differential forms

I am reading a lecture notes on differential forms and tangent bundles on complex manifolds and I got stuck on one line where the author does not explain how he did the computation: Let ...
2
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0answers
50 views

Laplacian on a warped product.

Let $(M, g)$, $(N, h)$ be complete Riemannian manifolds (not necessarily compact). Let $f : M \rightarrow (0, \infty)$ be a smooth function, and finally let $$\overline{M} = M \times_f N$$ be the ...
2
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0answers
301 views

$\alpha \wedge \beta = 0$ iff $\beta = \alpha \wedge \gamma$

I have been given the following problem: Let $\alpha$ be a nowhere-zero 1-form. Prove that for a (p+1)-form $\beta$ $(p\geq0)$, one has $\alpha \wedge \beta = 0$ if and only if $\beta = \alpha ...
2
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0answers
77 views

What is the differential of a function?

I'm reading Do Carmo's Differential Forms and Applications (1st ed) and on page 6 he takes a differentiable map $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$, a point $p \in \mathbb{R}^{n}$ and a ...
2
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0answers
212 views

Exact and Closed forms on Manifolds with Boundary

Let $\bar{M}$ be a manifold with boundary and let $M$ be its interior. Is this statement correct? A smooth k-form $\alpha$ on $\bar{M}$ is closed (exact) if and only if its restriction to $M$, i.e. ...
2
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742 views

Top deRham cohomology group of a compact orientable manifold is 1-dimensional

Let $M$ be a compact smooth orientable manifold of dimension $n$. I am looking for a simple proof that $H_{dR}^n(M) \cong \mathbb R$. Equivalently, an $n$-form which integrates to 0 is exact. I can ...
2
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68 views

How to apply Gauss's theorem when the metric is unknown

Let $f:U \to \mathbb{R}^3$ be a surface, where $U=\{(u^1,u^2)\in \mathbb{R}^2:|u^1|<3, |u^2|<3\}.$ Consider the two closed square regions $F_1=\{(u^1,u^2)\in \mathbb{R}^2:|u^1|\leq1, ...
2
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0answers
172 views

Calulation of pullback of form

If $M$ is $2n+1$ dimensional manifold, and $M'= M\times \mathbb R$ Let $x_1,y_1,... x_n, y_n,t', t$ be coordiante of $M'$. With $t$ for coordinate for $\mathbb R$. Let $$ \omega= \sum_{i=1}^n ...
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0answers
27 views

What is an integral differential form and how do we recognize it as such?

I am reading about embedding theorems of various types of manifolds (Kodaira's embedding theorem being one of them), and one condition that repeats in all of them is that the manifolds should be ...
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0answers
29 views

Is the curl of a vectorfield expressable in terms of a Lie derivative?

I am loving differential forms and the general version of Stokes theorem. However, I am also fond of the intuitive pictures of my older physics days of $grad$, $div$ and $curl$. I recently stumbled ...
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0answers
52 views

Formula for the curvature $2$-form.

I'm currently reading a textbook to do with curvature and $k$-forms. It says that the curvature $2$-form given connection $1$-form, $A$, is $$F =d^A A = dA+A \wedge A$$ It then goes on to say that ...
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0answers
25 views

Application of stoke's theorem - Doubt on calculation of integral

This is an exercise question from Spivak's calculus on manifolds chapter number 4 question 26. Show that $\int_{C_{R,n}}d\theta=2\pi n$, and use stoke's theorem to conclude that $C_{R,n}\neq \partial ...
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0answers
37 views

Show that if $\omega$ is a 1-form differential, then $\left\vert\int_{C}\omega\right\vert\leq ML$

Show that if $\omega$ is a 1-form differential define on $U\subset\mathbb{R}^{n}$, $c:[a,b]\to U$ is a differentiable curve and $\vert\omega(c(t))\vert\leq M$, for all $t\in [a,b]$, then ...
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0answers
31 views

Is there a way to compute the Poincaré dual of the following type of degree $(2n-2)$ de Rham class?

Given a closed, connected, symplectic manifold $(X^{2n},\omega)$, is there a systematic method to computing the Poincaré dual surface to degree $(2n-2)$ classes of the form $$[\omega]^{n-2}\cup B + ...
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37 views

Let $f\colon \Bbb R^n \to \Bbb R^n$ be the translation, $f(x)=x+a$. Show that $\deg(f)=1$

Let $U,V$ be connected open subsets of $\Bbb R^n$ and let $f\colon U \to V$ be a $C^{\infty}$ proper map. For all $w \in \Omega_c^n(V)$, $\int_{U}f^*(w)=\gamma \int_Vw$. Now define $\deg(f)=\gamma$, ...
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0answers
16 views

Building Non-vanishing sections in a certain way

Assume you have a vector bundle $\Pi: E \rightarrow M$ where $E$ is the total space, $M$ is a compact manifold. Assume you know it is parallelizable. Let $\psi_i : \Pi^{-1}U_i \rightarrow U_i \times ...
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0answers
62 views

Flow into a tetrahedron

This problem is from Harold Edwards' Advanced Calculus: A Differental Forms Approach. It is exercise $4c$ in section $1.3$. For a unit flow in the $z$-direction find the total flow into the ...
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0answers
23 views

Change of the sequence of differentation in physics?

Assume to have a quantity A which is calculated from the formula $A=\frac{dB}{dC}$. dC can be written as dC=dEdF so $A=\frac{dB}{dEdF}$. I assume that the differential of A is also ...
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0answers
38 views

On closed, symplectically embedded surfaces of ambient compact symplectic manifolds, how does one avoid pulling back a 2-form to an exact 2-form?

Obviously, if one pulls back an exact (with respect to the de Rham d) differential form by any map, then one obtains an exact form on the submanifold. But if one starts out with a form that isn't ...
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0answers
50 views

Identity regarding the components of a dual basis

This problem is from Robert Wald's "General Relativity." The problem is 4(b) from chapter 2. Let $Y_1\cdots Y_n$ be smooth vector fields on an $n$-dimensional manifold $M$ such that at each $p\in M$ ...
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0answers
41 views

Show that $1$-form has particular coordinate representation.

I want to show that for a nowhere-vanishing $1$-form $\alpha$ on an $n$-dimensional manifold $M$ we have that $\alpha \wedge d\alpha = 0$ if and only if around every point $p$ there exists a ...
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0answers
30 views

Zeroes of differential form embedded

Let us consider $S^1$ as a manifold embedded in $\mathbb{R}^2$. Let $dx_1\in\Omega^1(\mathbb{R^2})$. $$ Z_{\mathbb{R}^2}:=\{p\in\mathbb{R}^2:(dx_1)_p=0\}\\ ...
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0answers
52 views

Is there a deRham (co)homology for vector-valued differential forms?

Is there a deRham (co)homology for vector-valued differential forms? The deRham (co)homology of differential forms has been well-discussed and well-founded, along with the fact that for the exterior ...
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0answers
45 views

Line integral and differential forms

Let $df = \frac{\partial f}{\partial x_1 } dx_1 + \frac{\partial f }{\partial x_2 } dx_2$ be a $1-form.$ I know that that the line integral (along a curve $\gamma:[0,t] \rightarrow \mathbb{R}^2$) is ...
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0answers
27 views

Show that $d^2(w)=0$ for every $k$-form in $\mathbb{R}^n$

The question says to start showing this for 0-forms, and I did it. But how can I extend it for any k-form?
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0answers
66 views

Wedge product equality of 2n-forms in 2n+2 dimension

I have a question that seems to me clear but i couldn't prove it. I have a diffeomorphism between the cotangent bundle of the n-sphere and the complex quadric as $$T^*(S^n)\to (S^n)^{\mathbb{C}},\ ...
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0answers
38 views

Integral Over the N-Sphere in the framework of chains:

Integration over manifolds is commonly defined with object called chains. What about if I want to integrate the exterior derivative of a $k-form$ over the n-sphere and use Stokes theorem: ...
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0answers
44 views

1-form integration

Let $\alpha:[-1,1]\rightarrow R^2$ be the curve segment given by $\alpha=(t,t^2)$. If $\phi=v^2du+2uvdv$, (the fist component of $R^2$ is $u$ and the second one is $v$) I have $$\int_\alpha ...
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0answers
55 views

Compute the double Hodge star operator

I am taking a course in Multivariable Analysis and I am asked to do the following problem: Show that $\ast\ast\omega = (-1)^{k(n-k)}\omega$ So I start as follows: We know that $\displaystyle ...
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0answers
37 views

An expression of covectors acting on vectors on the tangent space of a manifold

Let $M$ be a smooth manifold. Take $p\in M$ and $(U,\varphi)$, $\varphi:U\rightarrow \mathbb{R^n}$, a chart around $p$. Let $\mathbb{R}^n\left[\frac{\partial}{\partial x_i}\right]$ and ...
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0answers
80 views

Integrability problem in Cauchy Integral Formula

This is problem on the integrability of a 2-form in the nhbd of its singularity. I was looking at the general Cauchy formula (general because it works for $\mathcal C^1$ function, and makes the case ...
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0answers
30 views

How to obtain the line element in cylindrical coordinates, using definition of differential forms

In general, a volume element is a k-form on an K-dimensional manifold. a k-form w on $\mathbb{R}^{n}$ is defined as $w(x) = \sum_{i_{1}<i_{2}<...<i_{k}} w_{i_{1}i_{2}...i_{k}}(x) ...
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0answers
45 views

Cohomology of two pieces of torus

In an exercise from an old exam, I found myself confronted with $M=\{(\sqrt{x^2+y^2}-2)^2+z^2=1\}$, $U=M\cap\{x\neq0\vee y>0\}$, $V=M\cap\{x\neq0\vee y<0\}$ and $U\cap V$, all subsets of ...
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0answers
56 views

Pullback 1-differential form

Let $(x_0,x'_0) \in \mathbb{R}^2$ be initial data for the Euler Lagrange equation with some given Lagrangian $L: \mathbb{R}^2 \rightarrow \mathbb{R}.$ Then $F^t$ is the flow that maps the initial ...
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0answers
44 views

Bianchi Identity - Gauge Theory

I am reading through some lecture notes (found here) and following a proof of the Bianchi identity in the context of principal bundles. That is, $h^*\Omega = 0$, where $\Omega$ is the curvature ...
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0answers
39 views

One form and Vector fields on a manifold in terms of local coordinates.

Prove : $d$$\omega$$(V,W)$=$V \omega (W) - W \omega(V) -\omega([V,W])$ in local coordinates where $\omega$ is a one-form and V,W are vector fields on a smooth manifold M. I do not know how to ...
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0answers
65 views

Connection on Tangent Bundle of Group Manifolds

The thing puzzling me is about a transformation rule of connection on tangent bundle of group manifolds: Assume one has a compact and simply connected Lie group $K$. One can give a metric $g$ on $K$ ...
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0answers
51 views

Integrating factor for non-exact differential

Given $\frac{xdy+ydx}{x^2+y^2}$ I took partials of either coefficient wrt to other variable I get $\partial P/\partial y$ as $\frac{x^2-y^2}{(x^2+y^2)^2}$ and $\partial Q/\partial x$ as ...
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0answers
86 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