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

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1answer
297 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 $Q\in U$ ...
2
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1answer
371 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 ...
2
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0answers
23 views

Verify stokes theorem example [duplicate]

Let $c:I^2\rightarrow\mathbb{R}^3$ be the singular $2$-cube given by $$c(s,t)=(\frac{1}{2}s^2,st,\frac{1}{2}t^2)$$ Let $$\omega=xy^2dz$$ Questions: i) Compute $c^*\omega$ ii) Compute $c^*d\omega$ ...
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0answers
36 views

Laplacian on sphere with differential forms [closed]

I want to express the Laplacian on the 2-sphere in terms of differential forms. Does anybody know how this can be done? I am not so familiar with submanifolds, thus I would appreciate help very much. ...
2
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0answers
31 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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0answers
49 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 ...
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0answers
53 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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1answer
94 views

Moving frame in a semi-Riemannian manifold

Can someone point me some reference for the moving frame theory in semi-Riemannian manifolds, using differential forms? In special, I'm looking for a version of Cartan's structural equations. I've ...
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0answers
52 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 ...
2
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2answers
51 views

Arc Length and Differential Forms

Suppose $\gamma$ is circle in $\mathbb{R}^3$ defined by coordinates $\begin{pmatrix}r\cos\theta\\r\sin\theta\\0\end{pmatrix}$, and function $F: \gamma \rightarrow \mathbb{R}^3$ is defined by ...
2
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0answers
62 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)\, ...
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0answers
64 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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0answers
40 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
69 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
69 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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1answer
204 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 ...
2
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0answers
158 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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0answers
54 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
34 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
52 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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1answer
49 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 ...
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172 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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2answers
71 views

Why is $\theta \not \in C^{\infty}(S^1)$?

Why is $\theta \not \in C^{\infty}(S^1)$? I know that since $\int_{S^1} d\theta = 2\pi$ then $d\theta$ is not exact. Thus since $d(\theta)=d\theta$, $\theta$ must not be $C^{\infty}$ but it seems ...
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0answers
480 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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0answers
64 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
153 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 ...
2
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1answer
350 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 ...
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3answers
89 views

A question on differential topology

Let $\mathbb{C}P(1)$ denote the complex projective line. I am attempting to show that there does not exist a nonzero holomorphic differential $1$-form on $\mathbb{C}P(1)$. My intuition is as ...
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2answers
51 views

Stokes or homotopy?

The problem states Show that if $X$ is a simply connected manifold, then $\oint_{\gamma}\omega=0$ for all closed 1-forms $\omega$ on X and all closed curves $\gamma$ in $X$. However I have ...
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2answers
72 views

Exactness of $dx,dy$

Let $dx,dy$ denote differential $1$-forms. It is easy to verify that they are closed. My question is: Does there exist a space such that either $dx$ or $dy$ or both are exact? (A ...
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2answers
78 views

Relation between volume form and cross product

Euclidean three-dimensional space (it's simpler). Defining $\eta={e^*}^1 \wedge {e^*}^2 \wedge {e^*}^3$, with $\{{e^*}^1,{e^*}^2,{e^*}^3\}$ dual of the orthonormal basis, and indicating the classic ...
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2answers
49 views

Compute the contraction of a 1-form with a vector field

Question: Let $\alpha$ be the $1$-form on $\mathbb{R}^3$ given by $\alpha=zdy-ydz$ and let $\mathbb{X}$ be the vector field on $\mathbb{R}^3$ given by $\mathbb{X}=(0,y,-z)$. Compute ...
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1answer
48 views

De Rham cohomology group

We know $m$-th de Rham cohomology group on $U$ is defined to be, $H^{m}_{dR}(U)=ker(d^m)/im(d^{m-1})$ where $d^m:\Omega^m(U)\to \Omega^{m+1}(U)$'s are usual exterior derivative maps. Now its saying ...
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1answer
82 views

diffeomorphism preserve a volume form

Let $\omega_1$, $\omega_2$ two volume form on a compact manifold $M$, we know that there exists a never-vanishing function $f$, s.t. $\omega_1=f\omega_2$. If $h$ is a diffeomorphism $M \to M$ ...
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1answer
185 views

Computing a Differential Form

Apologies in advance, I don't know TeX, so this might look a bit gross... I'm given a 1-form $A=f_1dx_1+...+f_ndx_n$, infinitely differentiable and closed on $R^n$. I want to show that $dg=A$ for ...
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1answer
85 views

1-form is exact

Let $\omega = f_1 dx_1 +f_2dx_2 + \cdots + f_ndx_n$ be a closed $ C^{\infty}$ $1-$form on $ \mathbb R ^n$. Define a function $g$ by $\displaystyle{ g(x_1, x_2,\cdots, x_n) = \int_{0}^{x_1} f_1(t,x_2 ...
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2answers
173 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 + ...
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1answer
53 views

$df\wedge \alpha=0$ implies $\alpha=g\,df$ for some $g$

Let $A \subset \mathbb{C}$ be an open set and $f\in {C}^{\infty}(A)$ with $df\neq0$. I consider $\alpha$ a $1$-form such that $df\wedge \alpha=0$ and I want to prove that exists a function $g$ such ...
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1answer
51 views

Reference to finite coverings causing injections on deRham cohomology

So, I've heard that if you have a finite degree covering of a compact connected manifold by another compact connected manifold of dimension $n$ (So $\pi :M \rightarrow N$) gives an injection on the ...
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1answer
79 views

Identity concerning Lie derivative of $k$-form $\omega$

Let $X$ and $Y$ be vector fields on $\mathbb{R}^n$. Show that for $\omega$, a $k$-form on $\mathbb{R}^n$, $(L_XL_Y-L_YL_X)\omega=L_{[X,Y]}\omega $. I try using Cartan's magic formula and get that ...
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1answer
47 views

$d(\beta \wedge d\beta)=0$ if $k$ is even.

Let $\beta$ be a $k$-form. Show that $d(\beta \wedge d\beta)=0$ if $k$ is even. I get that $d(\beta \wedge d\beta)=d\beta \wedge d \beta + (-1)^k\beta \wedge d^2\beta=d\beta \wedge d \beta$. Why ...
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1answer
36 views

Is there a Poincare lemma for codifferential?

Is every co-closed form also locally co-exact? That is for each $k$-form $\omega$ such that $\delta \omega = 0$ there exists $(k-1)$-form $\eta$ for which locally $\omega = \delta \eta$. My current ...
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2answers
139 views

Question about integrating differential forms

Maybe it's stupid question, by why: $$\int_S Fdx\wedge dy=\int_S Fdxdy$$ And is calculating a surface integral $$\int_S Fdx\wedge dy+Gdy \wedge dz+H dz\wedge dx=\int_S Fdxdy+\int_SGdydz + ...
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2answers
86 views

Flat Surfaces in $\mathbb{R}^3$ Can Be Bent Only Along Straight Lines

This is a problem out of Elementary Differential Geometry by Barrett O'Neill (Chapter 6 Section 3 Number 2). Let $M$ be a flat surface in $\mathbb{R}^3$ with principal curvatures $k_1$ and $k_2$, ...
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1answer
50 views

If differential 1-forms agree on chains with integer coefficients, are they equal?

Let $M$ be a real, smooth manifold. Let $\omega_1$ and $\omega_2$ be differential 1-forms on $M$, and let $C_1(\mathbb Z,M)$ denote the set of 1-chains with integer coefficients. If \begin{align} ...
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3answers
128 views

Prove $d(f\alpha)=d(f \wedge \alpha)$

I am reading the article http://en.wikipedia.org/wiki/Exterior_derivative and a definition of an exterior derivative from Axioms for the exterior derivative. How could I show that if $f$ is a function ...
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2answers
72 views

Equality involving exterior product..

suppose you have a differential form $\omega$ writting in local coordinates as $$\omega=\sum_{i=1}^ndx_i\wedge dy_i.$$ Can anyone help me showing the following equality: $$\omega^n=n!(dx_1\wedge ...
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1answer
101 views

Use the Fundamental Theorem to deduce the formula for the area of an ellipse.

Use the Fundamental Theorem (Green's Theorem) to deduce the formula for the area of an ellipse. Hint: find a 1-form whose exterior derivative is $ dxdy $.
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1answer
158 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 ...
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1answer
55 views

What does $d\zeta_1\wedge\cdots\wedge d\zeta_n$ mean in the context of Cauchy formula (on polydiscs)?

A Polydisc of center $z^o=(z_1^o,\dots,z_n^o)\in\Bbb C^n$ and multiradius $r=(r_1,\dots,r_n)\in(\Bbb R^+)^n$ is defined as $$ P_{z^o,r}:=\prod_{j=1}^n\Delta_{z_j^o,r_j} $$ where ...