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

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De Rham Differential for Vector Valued Forms?

Let $M$ be a smooth manifold of dimension $n$ and let $V$ be a $\mathbb R$-vector space of finite dimension $\ell$. A $k$-form on $M$ with values on $V$ is a map $\omega$ on $M$ such that: ...
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2answers
35 views

Grammatically confused: $\omega=4dV$ for 3-form $\omega$ and volume in $\Bbb R^4$?

Background: Against the advice I should have been given but wasn't, I'm taking a Lie theory course with no background in differential geometry. We finally made it into the part of the course where we ...
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1answer
39 views

Volume Forms Induced by Embedding

Let $(M, g)$ be a Riemannian Manifold of dimension $d$, $g$ naturally gives rise to an invariant volume form $V_M \in \Omega^d(M)$. Let $\Sigma$ be a smooth embedded submanifold of dimension $d-1$ in ...
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1answer
157 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 $[[P,{}]]$ ...
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41 views

The formula for the differential of a vector-valued function

If we have a vector, $\,U=U\left(x_1,x_2,x_3\right)$, in the coordinate axis $\left(x_1,x_2,x_3\right)$, then why does the following differential relation hold? $$ dU= \left(\frac{\partial ...
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Is there a commutative algebra for which multiderivations are not generated by order 1?

By multiderivations (of order $k$) I mean polylinear skew-commutative operations with values in my algebra which satisfy Leibniz rule in each of the arguments. (That is, the dual module to ...
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52 views

Problem proving Cartan's identity

There is a famous identity stating that, if $X$ is a field and $\omega$ a form, then: $$\mathcal{L}_X\omega=\iota_Xd\omega+d\iota_X\omega.$$ I'm trying to prove it. Thanks to Anthony Carapetis, I ...
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14 views

Find curves that generate a dual basis to a space of one-forms

Let $V$ be the vector space of one-forms on the plane that have quadratic functions as coefficients of $dx$ and $dy$, with basis $\{x^2dx,xy\;dx,y^2dx,x^2dy,xy\;dy,y^2dy\}$. For any curve $\Gamma$ ...
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1answer
64 views

Interior product and exterior product

I have seen around the internet that this should hold: $$\iota_X(\alpha\wedge\beta)=\iota_X\alpha\wedge\beta+(-1)^k\alpha\wedge\iota_X\beta,$$ where $X$ is a vector field, $\alpha$ a $k$-form, ...
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1answer
26 views

primitive of closed forms along a continuous curve

Here is the theorem: Suppose $\omega$ is a closed differential form in open domain $\Omega$. There is $\gamma:[a,b]\to\Omega$, that defines a continuos curve. Then $\omega$ has a primitive along ...
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2answers
118 views

The inegral $\int_1^2 2x \sqrt{x^2 + 1}\; dx$ using differential forms

I am trying to learn about differential forms. I think I understand what they are and the basic idea of integrals. But I wanted to make sure that I understand the process of integration by ...
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1answer
46 views

How do you integrate a $0$-form?

I am trying to learn about differential forms. I know that A $0$-form is just a scalar function $f$. My question is: How is the integral of a $0$-form defined? In particular, if $f$ is a function of ...
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2answers
51 views

Calculating $d(xdx + z^2dy + xydz)$

Inspired by this answer, I am trying to learn about differential forms. I am going through these notes were (on page 3) $d(xdx + z^2dy + xydz)$. I believe that the formula that I should use is $$ df = ...
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1answer
27 views

How to show that the metric $\frac{|dz|^2}{y^2}$ and the 2-form $\frac{dz \wedge d\overline{z}}{(-2i)y^2}$ are $SL_2$-invariant measures?

I am reading the lecture notes. On page 19, let $G=SL_2$. It is said that the metric $\frac{|dz|^2}{y^2}$ and the 2-form $\frac{dz \wedge d\overline{z}}{(-2i)y^2}$ are $SL_2$-invariant measures on the ...
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1answer
33 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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26 views

Evaluating an integral without a parametrization

From what I understand one of the main benefits of differential forms over Riemann integrals is that you're supposed to be able to integrate differential forms without parametrizing your curve (or ...
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2answers
128 views

How do I derive this formula from gauge theory?

This is Exercise 3.4.14 in R. W. Sharpe's Differential Geometry. Suppose $G$ is a Lie group with Lie algebra $\mathfrak{g}$ and $H$ is a Lie subgroup of $G$. Let $\theta$ be a ...
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1answer
35 views

Show that gravity is described by this 1-form

From Harold Edwards' Advanced Calculus: A Differential Forms Approach, section 2.1, exercise 1: The central force field. Newton's law of gravitational attraction states that the force exerted by ...
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1answer
28 views

How to prove this equation? Is $dx_i=e^*_i$?

Suppose U$\subseteq R^m$, F:U$\to R^m$ is $C^{\infty}$, f$\in C^{\infty}(R^m)$. And $x_1,x_2...x_m$:U$\to$R are coordinates on U.{$e^*_1,e^*_2...e^*_m$} is the basis of $(R^m)^*$ dual to the basis ...
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1answer
29 views

Pullback on differentials of the 2-Sphere

I'm considering the diffeomorphism of the 2-Sphere given by the antipodal map and the pullback given by this map on the differentials $d\theta$ and $d\phi$. Let $\psi$ be such a map. My intuition ...
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1answer
67 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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32 views

Understanding the definition of the support of a differential form on a manifold

I am an undergraduate student currently learning differential forms to be used in the context of multivariable calculus, namely to prove the generalized Stokes' Theorem. I'm studying by the book ...
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63 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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1answer
65 views

integrate over a cube given some differential form

What is process of integrating a differential form given some cube (hyperdimensional obejcts)? I read a lot qualitative problems on this, but seem to find rare examples on how to compute such ...
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2answers
51 views

if $x = 2y$ what does it mean to say that $\frac{d}{dx} = \frac{1}{2}\frac{d}{dy}$?

If $x = 2y$ what does it mean to say that $\frac{d}{dx} = \frac{1}{2}\frac{d}{dy}$? The question is the following: let $f$ be differentiable (smooth if necessary). I believe that we could try to ...
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1answer
59 views

De Rham cohomology group of the Klein Bottle

I need to compute all the cohomology rings of the Klein Bottle. I want to apply the Mayer-Vietoris sequence. Here I'm using the same good open cover suggested by the Wikipedia page :-) It's ...
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2answers
55 views

Confusion with notation on Indices, differential forms

I think I have a little confusion with index notation (concerning p-form). For example, do the coordinates on $\mathbb{R}^n$ $\{x^1,x^2, \dots , x^k\}$ and $\{x^{i_1},x^{i_2}, \dots , x^{i_k}\}$ mean ...
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0answers
34 views

de Rham Cohomology of the complex projective spaces

I want to compute the de Rham cohomology of the complex projective spaces $P^n_{\mathbb{C}}$. I know what the result is, and I've seen many posts in the forum asking the same thing. The problem is ...
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1answer
69 views

Complex differential forms on $CP^n$

Why Complex projective spaces don't admit some differential forms? To be more specific, I know that the space of complex forms is decomposed as direct sum of holomorphic and anti-holomorphic part; ...
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1answer
17 views

Relating existence of a “potential” with exactness of a certain form

Let $\Omega \subseteq \Bbb R^2$ be an open set, and let $\omega = \omega_1\,{\rm d}x_1 +\omega_2 \,{\rm d}x_2$ be a $1$-form in $\Omega$. Consider the field: $$L = \omega_2 \frac{\partial}{\partial ...
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2answers
31 views

1 Form Integral along the curve

Let $\alpha$ be the $1$-form on $D=\mathbb{R}^2-\{(0,0)\}$ defined by, $$ \alpha=\frac{xdx+ydy}{x^2+y^2}, $$ where $(x,y)$ are cartesian coordinates on $D$. Evaluate the integral of the $1$-form ...
2
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1answer
29 views

Chain Rule for Differentials

I'm trying this problem from Lee's Smooth Manifolds, but I'm not sure where I'm making a mistake: Problem: Let $M$ be a smooth manifold, and let $f,g\in C^{\infty}(M)$. If $J\subset\mathbb{R}$ ...
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1answer
45 views

Why functions can't be integrated on manifolds

I'm trying to teach myself about differential forms, and my book says that functions can't be integrated on manifolds because the integral isn't coordinate independent, but if the manifold has ...
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0answers
52 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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62 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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1answer
47 views

Pullback of differential form is zero [closed]

Let $ f:\Bbb R^m \to \Bbb R^n $ be differentiable map. Assume $ m<n$ and let $ w $ be a differential $k$-form in $\Bbb R^n $ , with $ k>m $. Show $ f^*w $ =0 Here $ f^* $ is the pullback of the ...
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1answer
43 views

Correspondence between Differential Forms and Vector Fields

I did not understand the highlighted text. Could anyone please explain it to me. There is a related post here- Differential Forms and Vector Fields correspondence. The first paragraph of the first ...
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1answer
28 views

Explanation of theorem on differential forms

This is text from Do Carmo's Differential forms and applications Page-10. Could anyone explain the highlighted step?If f* is applied to each of the term then how do we get the RHS of the highlighted ...
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2answers
42 views

Even dimensional real projective spaces cannot be combed

I have to prove that the even-dimensional real projective space cannot be combed, i.e. there isn't any non-vanishing smooth vector field. (I can't use Hopf theorem since those manifolds are not ...
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1answer
42 views

Euler number zero for odd dimensional compact manifolds

I need to prove that every compact manifold of odd dimension has Euler number zero. The Euler number of $M$ compact and oriented is $$ e(M):=\int_Ms_0^*\phi(TM) $$ where $s_0$ is the zero section of ...
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Basis of a linear map

A map, also known as function, is the set of ordered pairs $\left \{(a\in A;b \in B)| \forall a \exists!b(a\in A,b\in B) \right \}$ Manfredo and Do Carmo, in "Differental Forms and Applications" ...
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27 views

How the canonical symplectic form acts

I've read that the canonical symplectic form $\omega$ on $\mathbb R^{2n}$ is given by $$\omega=\sum_{i=1}^n dp_i\wedge dq_i,$$ where $(p_1,\dots,p_n,q_1,\dots,q_n)$ are the coordinates on $\mathbb ...
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1answer
20 views

Exterior Derivative in overlapping charts

Given a smooth manifold $M$, say of dimension $n$ and two charts $ (U,x)$ & $ (V,y)$, I want to prove that if $U \cap V \neq \emptyset $, then the exterior derivatives $d_x$ and $d_y$ coincide in ...
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1answer
52 views

What is an ordinary 2-form?

What is the difference between an ordinary 2-form and just a 2-form in general? Cant seem to find the definition for ordinary 2-form anywhere. Thanks in advance! Edit: If it helps, it was used in ...
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Composition of Lie Derivatives

I was wondering if anyone has any insight into what the composition of the lie derivative would look like. For example say you take the lie derivative of some p-form $ \omega $ with $X$ and then ...
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24 views

Geometrical interpretation of closed form

I'm studying Differential forms and my teacher generally gives geometrical interpretation of most things. So, I was wondering, what is the geometrical interpretation of the closed form and exact ...
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1answer
23 views

how (dx)^2 affect a formula

I am given α = xdx − ydy, β = zdxdy + xdydz, and required to compute αβ. After distributions and reorderings of differentials I get: $$xz(dx^2dy)+x^2(dxdydz)+yz(dy^2dx)-yx(dy^2dz)$$ Since I am not ...
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3answers
38 views

Where are the covectors in a line integral?

I'm trying to connect the idea of covectors to integration. From what I understand, given a basis $\{e_1, \dots, e_n\}$ of a vector space $V$, there exists a dual basis of covectors $\{f^1, \dots, ...
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1answer
38 views

2-forms as wedge product of 1-forms

Why can all $2$-forms on $\operatorname{T}_p (\mathbb{R}^3)$ be written as product of $1$-forms? What is a counter-example to show this isn't true in a space other that $\operatorname{T}_p ...
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1answer
32 views

Integrals of non-compactly supported forms

Let's recall the definition of integral of a compactly supported smooth $m$-form $\omega$ over an orientable $m$-differential manifold $M$: if $\{(U_\alpha,\phi_\alpha )\}$ is a differentiable atlas, ...