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

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Symplectic form and wedge sum

The wedge sum of $k$ symplectic 2-forms is given by ( if $\omega = \sum_i e_i^* \wedge f_i^*$) $$ \omega^k = k! \sum_{1\le i_1 <...<i_k \le n} (e_{i_1}^* \wedge f_{i_1}^*) \wedge ... \wedge ...
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2answers
55 views

Definition of Cech-De Rham complex. Can't understand its definition!!!!

Let $M$ be a manifold and let $\mathcal{U}:=\{U_\alpha\}_{\alpha\in I}$ be an open covering, $I$ be a totally ordered set. For every $p$ and for every $\alpha_0<\dots<\alpha_p$, $$ ...
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1answer
36 views

Locally flat manifold from Frobenius, differential forms approach

It is a known fact that a Manifold $M$ is locally flat if and only if the Riemann curvature tensor (Levi-Civita connection) vanishes. To show the if part is easy, but I am trying to follow the details ...
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2answers
784 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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1answer
402 views

Pullback distributes over wedge product

I'm looking to prove that the pullback of a smooth function distributes over wedge product, i.e. $$\varphi^*(\omega \wedge \eta) = \varphi^* \omega \wedge \varphi^* \eta. $$ Here the question is ...
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2answers
43 views

2-form on a smooth manifold

Let $M$ be a smooth manifold, $f:M$ $\rightarrow \mathbb{R}$ differentiable and $p\in M$ with $df(p)=0$. I am trying to show that the application, $$\begin{matrix}\mathfrak{X}(M)\times ...
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3answers
523 views

The algebraic de Rham complex

Let $A$ be a commutative $R$-algebra (or more generally a morphism of ringed spaces). Then there is an "algebraic de Rham complex" of $R$-linear maps $A=\Omega^0_{A/R} \xrightarrow{d^0} \Omega^1_{A/R} ...
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31 views

Integrate symplectic two-form

I am supposed to show the following (maybe falsely stated) theorem Let $\Phi$ be a symplectic group action $\Phi: G \times M \rightarrow M$ on a manifold $M$. Assume that the symplectic form $\omega$ ...
2
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1answer
81 views

Manifold is not orientable

Let $M$ be a manifold of dimension $n$ such that there exist two charts $(U_a,\phi_a)$ and $(U_b,\phi_b)$ such that $U_a,U_b$ are connected and $U_a\cap U_b\ne\emptyset$. Moreover the ...
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1answer
26 views

Analytic expression of a 1-form

Let $M$ be a differentiable manifold, $V\in\mathcal{X}(M)$ a vector field on $M$ and $\alpha\in\mathcal{X}^*(M)$ a 1-form. Let $L_{V\alpha}$ be another 1-form defined by: ...
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1answer
18 views

Restriction of differential $1$-forms to open subsets?

A vector field on a manifold $M$ is a linear map $X:C^\infty(M)\longrightarrow C^\infty(M)$ with an additional property. The set $\mathfrak{X}(M)$ of all vector fields on $M$ is a ...
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2answers
39 views

Volume forms and volume of a smooth manifold

Choose a volume form $\omega$ on $M$, oriented manifold. For every $F\in C^{\infty}_c(M)$, we define $$ \int_M F:=\int_M F\omega $$ where in the right hand term $M$ is taken wit positive orientation ...
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2answers
32 views

2-form whose self-wedge does not vanish?

I know that any 2-form is decomposable if and only if its self-wedge vanishes. Is there an element $β ∈ A_2(R^n)$ such that $β ∧ β \neq 0$. Obviously, this $\beta $ must be indecomposable, but I ...
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1answer
55 views

Good book about differential forms

I'm a looking for a good book to self-study differential forms. Particularly, I'm looking for a book that is as similar as possible to Bert Mendelson's "Introduction to topology" (i.e. a book that ...
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1answer
57 views

Expressing a differential form in terms of a scalar function

We can express every k-form in the form $ \omega(x) = \sum_Id_Idx_I $ where $ I$ is k-tuple and $ d_I$ is just some scalar function of x. That's entirely understandable for me. But while reading ...
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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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17 views

Order in integration of differential forms

Suppose $$G = dC + (b-1)\sum_{i=1}^{2} \delta_i \wedge \omega_i - \frac{b}{2\pi}d\phi \wedge \sum_{i=1}^{2}\omega_i$$ defines a 4-form in terms of a 3-form $C$, two forms $\omega_1$ and $\omega_2$ ...
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4answers
518 views

Intuition behind an integral identity

A proof for the identity $$\int_{-\infty}^{\infty} f(x)\, dx=\int_{-\infty}^{\infty} f\left(x-\frac{1}{x}\right)\, dx,$$ has been asked before (for example, here), and one answer to that question ...
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1answer
67 views

Trouble understanding proof of this proposition on contact type hypersurfaces

I have finally started working on my thesis. I am reading the first reference book, which is Dusa McDuff, D. Salamon, Introduction to Symplectic Topology. It's a tough struggle, given my not-too-great ...
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1answer
45 views

Basis free way to show $\alpha(V,-)=\alpha(W,-)=0$ implies $\alpha([V,W],-)=0$?

Given a 2-form $\alpha$, I want to show that, if $X$ and $Y$ are vector fields such that $\alpha(X,Z)=\alpha(Y,Z)=0$ for all vector fields $Z$, then $\alpha([X,Y],Z)=0$ for all vector fields $Z$. ...
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1answer
31 views

Relearning multivariable calculus through differential forms

While I learned multivariable calculus a few years ago, I have never felt I understand it well enough. Now I have time to go back and correct this. Since I have been through subjects like real ...
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2answers
15 views

If $σ$ is an exact differential $1$-form on the plane, then the form $ω=σ+xdy$ is not exact

If $σ$ is an exact differential $1$-form on the plane, then prove that the form $ω=σ+xdy$ is not exact. In the previous part of the question we have calculated the integral of the differential ...
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2answers
61 views

Non-integrability of distribution arising from 1-form and condition on 1-form

Suppose $M$ is a $(2k+1)$-dimensional manifold on which a 1-form $\alpha$ is defined. $M$ is termed as a contact manifold if the distribution arising from $\alpha$ is nowhere integrable, i.e. if: ...
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3answers
74 views

Compute $\int_M \omega$

Let $M=\{(x,y,z): z=x^2+y^2, z<1\}$ be a smooth 2-manifold in $\Bbb{R}^3$. Let $\omega=xdy\wedge dz+ydz\wedge dx+zdx\wedge dy\in \Omega^2(\Bbb{R}^3)$. Compute $$\int_M \omega.$$ I parametrised ...
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0answers
42 views

Differential forms defined by integration

Let $\omega_1,\omega_2$ be two n-forms on a $n$-dimensional manifold $M$. Now, imagine we have for every open $N \subset M$ that $$\int_{N}\omega_1 = \int_N \omega_2.$$ Can anybody show me how to ...
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1answer
44 views

The explicit expression for integral of forms

Could anyone please help me with the following three questions? They are simple questions, but I am confused. With a $2$-form $F=\frac{1}{2}F_{\mu\nu}dx^{\mu}\wedge dx^{\nu}$ in 4 dimension, what is ...
3
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1answer
68 views

Integrating the Hopf invariant for $\pi:S^3\to S^2$

I've been working on the last part of problem 9., chapter 9 in Nakahara's Geometry, Topology and Physics all day, with no success, and am in need of some assistance. We are asked to compute the Hopf ...
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2answers
40 views

Pullback of a linear map on a 2-form.

I am having a bit of trouble understanding a homework question and was seeking some clarification. Note, I have edited this question after I worked a couple of things out. Given a 2-form $v=dx_1 ...
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1answer
32 views

Can every 2 form be represented as a linear combination of these specific two forms?

This question is Question 2 from Ilka's book on page 8. The first part is to prove that every $\omega^2\in \Lambda^2(V^{\ast})$ can be represented as \begin{equation*}\tag{1} ...
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1answer
32 views

Prove that this $1$-differential form on $S^1$ is well-defined

Let $U_i:=\{p\in S^1:x_i\ne 0\}$, $i=1,2$, be two open sets of $S^1$. Define $$ \omega_p := \begin{cases} \Bigl(\bigl(-\frac{dx_2}{x_1}\bigr)|_{U_1}\Bigr)_p\,\,\,\;\text{if}\,\,p\in U_1\\ ...
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1answer
97 views

How can I prove $dz=dx+idy$?

Let's see $\Bbb C$ as an $\Bbb R$-vector space. Hence it is isomorphic to $\Bbb R^2$ and it has dimension $2$. If $v_1,v_2$ is a basis for $\Bbb R^2$, every its element can be written as $xv_1+yv_2$; ...
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2answers
38 views

Axioms for 2-forms, why should $\omega_x(\Delta x \wedge \Delta x) = 0$

I am having trouble understanding this piece about 2-forms from Terence Tao's "Differential Forms and Integration". I understand the bilinearity requirement in analogy to the one-dimensional case. ...
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0answers
27 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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1answer
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Differential Forms: Show that $d^2 = 0$ by explicit computation [closed]

Show that $d^2 = 0$ by explicitly computing $d^2\omega$ for $\omega$ a $1$-form in $\mathbb{R}^3$.
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1answer
36 views

Proving wedge product is associative

Fix a real vector space $V$ of finite dimension. Let's denote by $\Lambda^p(V)$ the vector space of $p$-forms on $V$ (i.e. alternating $p$-tensors). Then we have the product $\wedge : \Lambda^p(V) ...
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2answers
31 views

Exercise about differential forms and pull-back: $\omega_p=0$ if and only if $(\omega|_U)_p=0$ [duplicate]

Let $M$ be a manifold, $\omega\in\Omega^q(M)$. Let $U\subset M$ be an open subset embedded as manifold in $M$. Let $p\in U$. Show that $\omega_p=0$ if and only if $(\omega|_U)_p=0$. Notation: Let ...
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2answers
50 views

Under what conditions can a general 2-form be written as a wedge product of two 1-form

Assume we have a 2-form $\omega \in \Lambda^2\mathbb{R}^n$. It is usually stated one can write $$\omega = \alpha \wedge \beta,$$ with $\alpha, \beta \in \Lambda^1\mathbb{R}^n$ only for $n < 4$. How ...
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1answer
53 views

Choice of order in the Leibniz rule is arbitrary?

One of the rules which characterizes the exterior derivative is that, for $\varphi$ a real-valued function and $\omega$ a $k$-form, we have $$d(\varphi \cdot \omega) = d\varphi \wedge \omega + ...
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2answers
32 views

Line integrals in differential form

I'm a bit confused as to the format of line integrals in differential form (i.e. the form in which Green's theorem is often presented). For example: $$ \oint\limits_\mathcal{C} \left( y^2 \mathrm{d}x ...
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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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1answer
103 views

Closed 1-forms implies exact 1-forms

I have two problems, the first one I think I've proved, but I have problems on the second one. Let $\omega$ a closed $1$-form defined on a open $U\subset \mathbb{R^2}$ and let $\gamma:[0,1]\rightarrow ...
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1answer
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Differential forms- riddle me this.

I stumbled over this answer on math.stackexchange Let $x$ be a point in $\rm M$. Then because $\omega$ is non degenerate at $x$, the antisymmetric matrix $\omega_x$ has full rank on the tangent ...
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2answers
134 views

Why is the exterior derivative called exterior derivative

I am studying exterior calculus, and I think I have some grasp of what is the exterior derivative. However its name still eludes me - why is it called a derivative? Is it just because the operator $d$ ...
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1answer
67 views

What is the first arrow in the Mayer-Vietoris sequence?

On page 449 of Lee's Introduction to Smooth Manifolds (2nd Edition), the Mayer-Vietoris Theorem is given: Let $M$ be a smooth manifold. Let $U$ and $V$ be open in $M$ such that $U\cup V=M$. Then ...
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3answers
76 views

Exterior derivative

Let $$\omega = \frac{1}{2} \sum_{i,j} \omega_{i,j} dx_i \wedge dx_j$$ be an antisymmetric form, so i.e. $\omega_{i,j} = - \omega_{j,i}.$ Now, in some lecture notes I found that $$d \omega ...
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2answers
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Dimension of the $0^{\text{th}}$ de Rham cohomology group of $U$

Let $U\subset\mathbb{R}^n$ be an open set. I proved that $$ H^0_{DR}(U):=\frac{\text{closed forms}}{\text{exact forms}}=\{f\in C^{\infty}(U):\,f\,\,\text{is locally constant} \} $$ I have to show ...
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2answers
36 views

Finite universal covering induces injective maps on cohomology

I am trying to prove the following: Suppose $M$ is a smooth, connected manifold with finite fundamental group and $f : \widetilde{M} \rightarrow M$ is its (smooth) universal cover. Show that $f^* : ...
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1answer
27 views

Hamiltonian vector field and symplectic geometry

I want to show the following theorem: For any Hamilton function $H : M \rightarrow \mathbb{R}$ on some symplectic manifold $M$ and symplectomorphism $f : M \rightarrow M$ we have $X_{H \circ f} = ...
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Integration by parts in the $x_i$ direction in the integral which uses differential forms

Let $\Omega \subset \mathbb{R}^n$ be open and bounded and let $K \subset \Omega $ be compact. Let $\xi \in C_0^\infty (\Omega )$ with $0\leq \xi \leq 1$ in $\Omega $ and $\xi \equiv 1 $ on $K$. For ...
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1answer
58 views

Stokes' theorem and symplectic geometry

Let $V = \mathbb{R}^2,$ as a vector space then the Poincaré invariant is an integral $\int_{\gamma} \theta$ where $\theta = p dx $ is the symplectic 1-form and $\gamma$ a closed curve. Now, it is ...