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

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A closed $1$-form on a convex open set is exact

Baby Rudin Exercise 10.24: Let $\omega = \sum a_i(\mathbf x) \, dx_i$ be a $1$-form of class $\mathscr{C}''$ in a convex open set $E \subset \mathbb{R}^n$. Assume $d \omega = 0$ and show that ...
2
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1answer
39 views

$M \times N$ orientable iff both $M$ and $N$ are orientable proof in terms of volume forms

I'm studying differential forms, and in my homework I'm asked to show that the product of two manifolds $M \times N$ is orientable if and only if both $M$ and $N$ are orientable. I want to show this ...
5
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1answer
53 views

Induced bilinear form on exterior powers - Towards a global Hodge Star Operator

In all constructions of the hodge star operator I've seen so far there was a part where an inner product on the exterior power of the tangent space was defined by the ungodly local formula: ...
2
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1answer
59 views

Computing the integral of a differential form in $\mathbb{R}^{2}$.

Let $D$ be the disk \begin{equation} D=\{(x,y)\in\mathbb{R}^{2}\:|\:x^{2}+y^{2}\leq 1\}, \end{equation} which is easily verified to be a compact $2$-differentiable manifold with boundary. Let ...
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2answers
42 views

Proving df is the sum of partial derivatives

Wikipedia states: Since any vector $v$ is a linear combination $\sum v_je_j$ of its components, $df$ is uniquely determined by $df_p(e_j)$ for each $j$ and each $p \in U$, which are just the ...
0
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1answer
84 views

Multivariable calc “second course” that does differential forms

I've worked through a computation-heavy, "standard" but quite nonrigorous treatment of multivariable calculus in the past. What book would do well as a rigorous (but not overly) "second course"? In ...
3
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1answer
34 views

Definition of integration of differential forms

I am trying to understand precisely the following paragraph: Question Why would he define the support $K$ of a form $\omega$ defined on an open set $U$ as a subset $K\subseteq M$ instead of a ...
3
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2answers
84 views

Computing Lie derivative

Can anyone help me with computing Lie derivative ${L}_{X}Y$ using its definition for these two vector fields: $X=y\frac{\partial}{\partial x}-x\frac{\partial}{\partial y}, ...
3
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1answer
131 views

What is the intuition behind differential forms?

I am comfortable with the way physicists use differentials as elements of area/volume. I know the (algebraic) formal definition of differential forms, but it makes no intuitive sense, especially since ...
0
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1answer
60 views

How to write differential forms on manifolds?

In the book "Differential Forms and applications" by Manfredo do Carmo, he says that a differential $k$-form on a $n$-dimensional smooth manifold $M$ is determined by a choice, for each ...
0
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1answer
22 views

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
39 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 ...
2
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1answer
49 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 ...
3
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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 $[[P,{}]]$ ...
0
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1answer
42 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 ...
2
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0answers
27 views

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 ...
0
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1answer
57 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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0answers
15 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 $\Gamma$ ...
1
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1answer
118 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, ...
2
votes
1answer
32 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
127 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 ...
2
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1answer
48 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
54 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 = ...
0
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1answer
28 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 ...
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 ...
0
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1answer
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 ...
5
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2answers
132 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 ...
2
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1answer
37 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 ...
0
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1answer
33 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 ...
3
votes
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 : ...
0
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1answer
92 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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0answers
66 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 ...
1
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1answer
69 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 ...
2
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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 ...
1
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1answer
83 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 ...
1
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2answers
58 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 ...
0
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0answers
36 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 ...
2
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1answer
82 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; ...
0
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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
32 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
30 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
47 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 ...
3
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0answers
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 ...
3
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0answers
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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1answer
59 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
50 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 ...
2
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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
44 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 ...
1
vote
1answer
50 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 ...