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

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565 views

A $2$-form on $S^2$ is exact if it integrates to zero.

I'm trying to show that a $2$-form on $S^2$ is exact if and only if it integrates to zero, without appealing to de Rham's theorem (basically only using the Poincaré lemma [that every closed form on a ...
3
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1answer
194 views

De Rham cohomology notation

According to http://en.wikipedia.org/wiki/De_Rham_cohomology, one defines the $k$-th de Rham cohomology group $H^{k}_{\mathrm{dR}}(M)$ to be the set of equivalence classes, that is, the set ...
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1answer
564 views

Curvature tensor of 2-sphere using exterior differential forms (tetrads)

$ds^2= r^2 (d\theta^2 + \sin^2{\theta}d\phi^2)$ The following is the tetrad basis $e^{\theta}=r d{\theta} \,\,\,\,\,\,\,\,\,\, e^{\phi}=r \sin{\theta} d{\phi}$ Hence, $de^{\theta}=0 ...
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1answer
125 views

differential form is exact on $ U \cup V$

Let $ U ,V \subset \mathbb R ^n$ two simply connected open sets such that $ \displaystyle{ U \cap V}$ is a connected set. If $\omega$ is a closed 1-form wich is exact in $U$ and $V$ prove that: ...
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1answer
246 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 ...
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1answer
349 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 ...
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1answer
239 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) = ...
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297 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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1answer
256 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 ...
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1answer
56 views

Differential forms on a projective curve: two constructions

Let $X$ be a projective curve on a perfect field $k$ ($k$-scheme integral, separated, of finite type) and let $K$ be the function field of $X$. Let's compare the following two constructions: 1. ...
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76 views

Intrinsic definition of differential k-form on smooth manifold

Suppose I have a $k$-dimensional manifold embedded in $\mathbb{R}^n$. Munkres defines a $k$-form on $M$ as a a function $\omega$ that assigns an alternating tensor at each point $p \in M$ that acts on ...
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1answer
59 views

Poincare's lemma for 1-form

Let $\omega=f(x,y,z)dx+g(x,y,z)dy+h(x,y,z)dz$ be a differentiable 1-form in $\mathbb{R}^{3}$ such that $d\omega=0$. Define $\hat{f}:\mathbb{R}^{3}\to\mathbb{R}$ by ...
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1answer
115 views

Spivak Calculus on Manifolds - Theorem 4-10

Part (4) of Theorem 4-10 in Spivak's Calculus on Manifolds says the following: If $\omega$ is a $k$-form on $\mathbb{R}^m$ and $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is differentiable, then ...
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1answer
44 views

$d(\iota_v\rho)=0 \implies d(\phi\iota_v\rho)=d\phi(v)\rho$?

My motivation is physical, but my question is purely mathematical. Everybody knows, that the power of the electric current in a piece of wire is $$P=UI$$ where the wire is regular domain $V$ in a ...
3
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1answer
40 views

There is no 2-form on $\mathbb R^3$ whose restriction to every surface gives its volume form

Prove that there isn't such a 2-form $\omega$ on $\mathbb{R}^3$ that $\omega$ restricted to any surface $\Sigma$ gives its volume form. Suppose that there is such a form: $$\omega=f_3(x,y,z)dx ...
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1answer
105 views

How to visualize differential forms geometrically

I've been attempting to teach myself differential geometry and I have heard that one can visualise them geometrically and that this can sometimes be helpful for an intuitive understanding of them. For ...
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1answer
34 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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3answers
108 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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1answer
83 views

Why does a differential form represent a vector field?

I'm trying to learn the Divergence/Stoke's theorem and I can't wrap my head around the meaning of a differential form in this context. What does it mean that a differential form represents a vector ...
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1answer
82 views

Proving that the pullback map commutes with the exterior derivative

I'm trying to prove that the pullback map $\phi^{\ast}$ induced by a map $\phi:M\rightarrow N$ commutes with the exterior derivative. Here is my attempt so far: Let $\omega\;\in\Omega^{r}(N)$ and let ...
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1answer
58 views

Lie derivative for a wedge product $\omega_{1}\wedge\omega_{2}$

I have to prove that $L_X\omega_{1}\wedge\omega_{2}=(L_X\omega_{1})\wedge\omega_{2}+\omega_{1}\wedge(L_X\omega_{2})$ using the definition ...
3
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1answer
136 views

Area form a differential form?

I just read that $\omega_x( \eta, \zeta) := \langle x, \eta \times \zeta \rangle $ is the area form on the sphere, where $x \in \mathbb{S}^2$ and $\eta,\zeta \in T_xM.$ All I see is that this is ...
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149 views

Volume form for a product manifold.

Given two Riemannian manifolds $(M,g^M)$ and $(N,g^N)$, we can construct a product Riemannian manifold $(M\times N,g^{M \times N})$ as described in Product of Riemannian manifolds? . Is there a ...
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1answer
64 views

Well-definedness of a coboundary map between a reduced $L^2$ de Rham cohomology group and a relative cohomology group

I'm working right now with this paper of Carron. And I think I'm stuck at a relatively simple question. On page 11 he is defining a coboundary map $b : H^k_{2, \text{reduced}}(M - K) \to H^{k+1}(K, ...
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1answer
137 views

Homework: calculation about differential form

Here is the question: Let $\omega = A dy\wedge dz + B dz \wedge dx + C dx \wedge dy$ in $\mathbf{R}^3$, and $d\omega = 0$. Denote \begin{eqnarray} \alpha = \int_0^1 tA(tx,ty,tz)dt\cdot(ydz-zdy)\\ ...
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1answer
696 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 ...
3
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1answer
59 views

1-form on $S^n$ with non-degenerate differential.

How it can be solved? Whether there is a $1$-differential form $w$ on $S^n$, such that $dw$ is non-degenerate on $T_aS^n$ for each $a \in S^n$?
3
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1answer
152 views

Independence of coordinate charts in the definition of the order of a pole of a meromorphic 1-form on a Riemann surface

I am currently reading Forster. Let $X$ be a Riemann suface, and $Y$ an open subset of $X$. Assume we have a meromorphic 1-form $\omega \in \Omega(Y \setminus \{a\})$ with a pole at $a \in Y$. One ...
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On the meaning of formal sums of $k$-cubes, i.e. $k$-chains (in integration on manifolds)

A singular $k$-cube in $A \subseteq \mathbb R^n$ is a continuous function $c : [0,1]^k \to A$. A singular $0$-cube in $A$ is then a function $f : \{0\}\to A$, what amounts to the same thing, a point ...
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0answers
23 views

Kodaira decomposition of 1-form on a Real manifold

I recently stumbled across the Hodge decomposition theorem, which states that on any compact orientable manifold, for any form the following holds $$ \omega = \text{d}\alpha + \delta \beta + \gamma $$ ...
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0answers
44 views

Complex Analysis with differential forms

I'm studying a little of Complex Anlysis and I have seen that I can thing the integrals of complex functions as integrals of differential forms in $\mathbb{R}^n$. For example I know that Cauchy ...
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1answer
167 views

Why is Schouten-Nijnhuis bracket trivial on Poisson cohomology?

For a commutative algebra $A$, let a biderivation $P$ be called a Poisson structure if $[[P,P]]=0$ (the bracket is Schouten-Nijenhuis). Then one obtains a complex of multiderivations with $[[P,{}]]$ ...
3
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1answer
70 views

If $\alpha \wedge \omega = 0$ then $\alpha = f \omega$ for some $f$

Question: Let $\alpha$,$\omega$ be $1$-forms of class $C^1$ in $\mathbb R^3$. If $w(x) \neq 0$, for every $x \in \mathbb R^3$ and $\alpha \wedge \omega = 0$. Then $\alpha = f\omega$, where $f : ...
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57 views

Suggestion for reference book for differential forms, differentiable manifolds and other topics

I am currently taking a course on multivariable calculus and our professor is following the book by Do Carmo: Differential forms and applications. I feel the text is too rigorous, which I really ...
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81 views

Find the interior product of a basic p-form $\alpha = dx_1 \wedge dx_2 \wedge \ldots \wedge dx_p$ a and a vector field $X$

I'm reading Differentiable Manifolds by Nigel Hitchin, that is, his class notes for an Oxford course freely available here. In particular, I'm trying to understand the interior product on manifolds, ...
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321 views

push forward of differential form/ integration over fiber

It is elementary that differential forms can be pulled back via a smooth map between manifolds. However, I was reading a paper and came across a construction about push forward of a differential form ...
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1answer
147 views

Vectors, Forms, Multivectors, and Tensors

In researching some of the ways that vectors (and vector fields) generalize I find that there are apparently many different objects that generalize them -- matrices, differential forms/ covectors, ...
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1answer
70 views

An exercise on differential forms

Let $\omega$ be a $1$-form in $U \subset \mathbb{R}^2$. A local integrating factor in $p$ for $\omega$ is a function $g: V \rightarrow \mathbb{R}$ defined in a neighbourhood $V$ of $p$ such that ...
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0answers
33 views

The lie bracket as a limit

Let $M$ be a manifold and $f_t$ be the monoparametric group of local transformations generated by a vector field $X$. Suppose $w$ is a $k$-form and $x_1,\dots,x_k$ are vector fields. How can we see ...
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125 views

Differential Form Pullback Definition

I'm having some difficulty following how Spivak (Calculus on Manifolds) has induced the pullback on page 89. From reading elsewhere online it seems convention is to define the induced map of the ...
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196 views

Product of Two Orientable Manifolds is Orientable

I am trying to show that following: Let $M$ be an oriented smooth manifold of dimension $m$, and $N$ be an oriented smooth manifold of dimension $n$. Then $M\times N$ is orientable. Let ...
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55 views

Can We Write the Differential in Terms of Covectors?

Let $f:\mathbf R^n\to \mathbf R$ be a smooth map. We can write $df:T\mathbf R^n\to \mathbf R$ neatly as $$ df = \sum_{i=1}^n(\partial f/\partial x_i) dx_i $$ For a function $f:M\to \mathbf R$ defined ...
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148 views

Second structural equations in lorentzian space $\Bbb L^3$.

I'm rewriting O'Neill's "Elementary Differential Geometry"'s section on connection forms in Lorentz-Minkowski space $\Bbb L^3$, and I'm having trouble proving the second structural equations $${\rm ...
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154 views

Second fundamental form of a graph of a function using frame fields

I am trying to recover the second fundamental form $II$ for the graph $\Gamma(f)$ of a smooth real function $f$ in the Euclidean space $\mathbb{R}^3$ using first order frame fields. I have worked with ...
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1answer
94 views

Integration on $\mathbb{R}^n$ in terms of differential forms

One defines integration on a smooth manifold as follows: First define $\int_M \omega$ when $\omega$ is supported on a single coordinate chart by pulling back to $\mathbb{R}^n$ an integrating there, ...
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1answer
39 views

Global n-form on Calabi-Yau

I am now reading these lectures by Stefan Vandoren on complex geometry. Everything is fine in general, hiwever I am confused with how he defines 1-form on a Calabi-Yau 1-fold (or 2-form on CY$_2$). ...
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2answers
78 views

Using Stokes's theorem to calculate a value of integral

Use Stokes's theorem to calculate the integral $$I= \int_\Gamma (x^2+2y)dx+(y+z)dy+(z^2+x^2)dz$$ where $\Gamma$ is the boundary of $$\gamma=\left\{ (x,y,z):3x+y+3z=3,x\ge0,y\ge0,z\ge0\right\} $$ ...
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1answer
118 views

de Rham cohomology of $\mathbb{R}P^n$ via action by $SO(m+1)$

In lecture, my teacher proved the theorem that given a smooth $G$-action by a compact, connected Lie group on a manifold $M$, the de Rham cohomology of the $G$-invariant differential forms $H^p_G(M)$ ...
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0answers
123 views

Definition of differential forms

I am reading the book Algebraic Geometric Codes: Basic Notions by Tsfasman, Vladut and Nogin. The authors introduce some definitions about differential forms at the beginning of Section 2.2. These ...
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75 views

Cohomology of volume forms

If g and h are Riemannian metrics on the same manifold, say both of volume 1, then it follows (I guess from Poincaré duality) that their volume forms dvol_g and dvol_h are cohomologous. Question: is ...