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

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2
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1answer
259 views

Concept of integration to differential form

How to integrate differential form actually. As far as I know, a differential form is a multilinear function mapping from a vector space to a real number. Let's take $\int_c fdx+gdy$ as an example. It ...
2
votes
1answer
92 views

Is $\zeta=\frac{x dy \wedge dz+y dz \wedge dx+z dx \wedge dy}{r^3}$ exact in the complement of every line through the origin?

$r=\sqrt{x^2+y^2+z^2}$ of course. If the line is the $z$ axis, it is given in the book (Rudin) that $\zeta=d \left( -\dfrac{z}{r} \dfrac{xdy-ydx}{x^2+y^2} \right)$ I've managed to figure out 2 ...
2
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1answer
137 views

Integrals of Differential Forms

I am working out of Munkres Analysis on Manifolds and I see that he claims for $\eta = f dx_1 \wedge \ldots \wedge dx_k$. $$ \int_A\eta = \int_{x \in A} \eta(x)\big((x;a_1) ,\ldots, (x;a_k) \big)$$ ...
2
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1answer
677 views

non-vanishing k-form on a k-manifold in $\mathbb{R}^n$ implies orientability

I want to know how to prove the theorem: If M is a k-manifold in $\mathbb{R}^n$, then it is orientable if and only if there is a volume form defined globally on M. I'm currently stuck at this step: ...
2
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1answer
473 views

Leibniz rule for exterior derivative of a contraction

If I have a contraction of a vector field with a 1-form valued 2-form, what would be the appropiate product rule? $$d_{\left[a\right.} \left(P_{[bc]i} v^i \right)_{\left. \right]} = \, ?$$ This ...
2
votes
1answer
486 views

property to be exact a 1-form on $\mathbb R^2 -\{(0,0)\}$

(a) Let $\omega$ a $1-$form defined on the open set $ U \subset \mathbb R ^n$ and $ c:[a,b] \to U$ a $ C^1 -$differentiable curve such that $ |\omega (c(t))| \leq M \quad \forall t \in [a,b]$ Prove ...
2
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1answer
227 views

Differential forms on a $S^1$-manifold

I am reading about differential forms on manifolds with group actions and there is an 'obvious' formula which I don't quite understand. Let $X$ be a manifold with a smooth circle action, that is a ...
2
votes
1answer
30 views

Pullback Solid Angle, Stereographic projection

I have an issue with a differential geometry task. Given is the solid angle form: $$\omega = \frac{\epsilon_{ijk} x^i dx^j \wedge dx^k}{2 [ (x^1)^2 + (x^2)^2 +(x^3)^2]^{3/2}}$$ The aim of the task ...
2
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1answer
69 views

An interpretation of $ \frac{\partial^{2}}{\partial x^{2}} $.

$ \left( \dfrac{\partial}{\partial x} \right)_{p} $ is both an element of the tangent space $ {T_{p}}(M) $ and a linear functional on $ {C^{1}}(M) $, while $ (\mathrm{d}{x})_{p} $ is an element of the ...
2
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1answer
61 views

Tangent space and differential forms of quotient groups

I have difficulty understanding the following argument because it seems that many details are swept under the rug and I am looking for a detailed rigorous exposition on this (I'll try to make clear ...
2
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1answer
59 views

Relationship between differential forms and coordinates

For the purpose of this question let us restrict our considerations to smooth $3$-manifolds. So the manifold $M$ we consider here is endowed with smooth coordinate charts $(x,y,z)$. What I have ...
2
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1answer
74 views

Constructing functions such that integral along any closed curve is non-zero

Consider smooth maps $f: \mathbb R^2 \setminus \{(0,0)\} \to \mathbb R$. How can I construct such an $f$ with the property that $$ \oint_C f \neq 0$$ for any closed curve $C$ around the origin? ...
2
votes
2answers
70 views

When are the exterior derivative and contraction of forms inverses?

I am trying to get a better feel for both the exterior derivative of a form and the contraction of a form by a vector field $X$. Basically, when are these inverses? If I have a one-form $\omega$ and ...
2
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2answers
101 views

Differential forms on $S^1$

I'm reading this old question and there are some things I don't understand. For example, why in the case of $S^1$ can every $1$-form be written in the form $f(\theta)d\theta=c d\theta+dg(\theta)$ ...
2
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1answer
52 views

exterior product of forms is exact.

I don't know what to do to prove the following statement: Let $U \subset \mathbb R^n$ be an open set and let $\alpha$ be a $k$-form on $U$ and $\beta$ be an $l$-form on $U$. Suppose both $\alpha, ...
2
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1answer
84 views

Identities for differential forms and vectorfields (reference request)

Recently I found the slides of a talk of J. E. Marsden, (Differential Forms and Stokes' Theorem). These slides introduce the required objects and summarize the basics of the corresponding theory. In ...
2
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1answer
80 views

Pullback of a form under the retraction $r\colon \mathbb{R}^n\setminus\{0\}\to S^{n-1}$.

The following is from Spivak's DG Lemma 7 in Chapter 8, but I'm muddled in a computation. Define two $(n-1)$-forms on $\mathbb{R}^n\setminus\{0\}$ by $$ ...
2
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1answer
45 views

differential form identity and permutations

If $t^1,...,t^k$ are the coordinates of a k-cube. Then apparently $$dt^{\sigma(1)} \wedge \ldots \wedge dt^{\sigma(k)}= (\operatorname{sgn} (\sigma)) dt^1 \wedge dt^k $$ I cannot see how this ...
2
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1answer
63 views

Differential Form Over $S^2$

I was checking problems on differential forms and I found the following one. Consider the sphere $S^2 \subseteq R^3$ and the map $\omega_p : T_pS^2 \times T_pS^2 \rightarrow \mathbb{R}$ given by ...
2
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1answer
64 views

Conceptual Question on different representations of Hyperplanes, Higher Standpoint, Coordinate-free

In a vector space $V$ over some field $F$ a hyperplane is the kernel of some linear transformation $T : V \to F$, i.e. the kernel of an element of the dual space (this could be taken as the definition ...
2
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1answer
72 views

How to make an ideal generated by differential forms into a differential ideal?

Let $M=\mathbb{R}^4$ with standard coordinates $x_1,x_2,x_3,x_4$. Let $\alpha=x_2dx_1+x_3dx_3+dx_4$ and $\beta=2dx_2+x_1^2dx_3+x_1dx_4$ How to find a 1-form $\gamma$ such that the ideal generated ...
2
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1answer
92 views

How to define integration over the boundary of a curve?

When learning about Stokes' theorem ($\int_{\partial \Omega} \omega=\int_{\Omega} \mathrm d \omega$), we are told that it is just a generalization of the 2nd Fundamental Theorem of Calculus $(\int_a^b ...
2
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1answer
25 views

Definition of an integrand

General Question: Say we have integral $$ \int f(z)\ dz $$ Is the integrand in this context (i) $f(z)$ or (ii) $f(z)\ dz$? In any case, is $f(z)\ dz$ a formally defined mathematical object in its ...
2
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1answer
84 views

Why generalize vector calculus with $k$-forms instead of $k$-vectors?

The motivation usually given to differential forms is that they generalize vector calculus nicely. That's true, but there are also $k$-vectors, i.e., objects from $\Lambda^k(V)$ instead of ...
2
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2answers
237 views

Integration of a 2-form

$\textit{What is}$ $\int_C{\omega}$ $\textit{where}$ $\omega=\frac{dx \wedge dy}{x^2+y^2}$ $\textit{and}$ $C(t_1,t_2)=(t_1+1)(\cos(2\pi t_2),\sin(2\pi t_2)) : I_2 \rightarrow \mathbb{R}^2 - ...
2
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1answer
145 views

About Stokes' theorem

I noticed that in the proof of Stokes' theorem on manifolds, the condition that the form $\omega$ is compactly supported ensures that the sum is finite so that we can change the order of sum and ...
2
votes
1answer
472 views

Closed but not exact one-form on $S^2$

I would like to know whether there is any nice prescription to give an example of a closed but not exact one-form on $S^2$ (not the $3$-ball). I assume to take some points out of this surface, e.g. 3. ...
2
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2answers
106 views

Show that $a \wedge * b = g(a,b) \operatorname{vol}$

$\newcommand{\vol}{\operatorname{vol}}$ Let $\omega$ be a $p$-form on a Riemannian manifold $M^n$ with metric $g$ and let $\vol_{i_1,\ldots,i_n}=\sqrt{\lvert g\rvert} \epsilon_{i_1,\ldots,i_n}$ be a ...
2
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1answer
50 views

Proof and counter-example that a chain $c_{R, n} \ne \partial c$. Where is the error?

If $R > 0$ and $n \in \mathbb{Z}$ we can define the singular 1-cube $c_{R, n}\colon [0, 1] \rightarrow \mathbb{R}^2$ by $$c_{R, n}(t) = (R\cos(2\pi n t), R\sin(2\pi n t))$$ We know that $c_{R, n} ...
2
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1answer
138 views

Computing $n$-th external power of standard simplectic form

I need some help: Define a 2-form on $R^n$ by $\omega=dx_1\wedge dx_2+dx_3\wedge dx_4+...+dx_{2n-1}\wedge dx_{2n}$. How to compute $\omega^n:=\omega\wedge\omega\wedge\ldots\wedge\omega$?
2
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1answer
391 views

Differential Forms and Area

I think I'm just misunderstanding something here, but in $\mathbb{R}^2,$ there exists a $1$-form (in fact infinitely many) $\omega$ such that for any region $S,$ we have $\int_{\partial S} \omega = ...
2
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1answer
80 views

Orienting curves with differential forms

Consider the circle given by the equation $x^2+y^2=1$. We can orient this curve by choosing the tangent vector field $(-y,x)^T$, which defines a direction. Supposedly we can do this with by taking an ...
2
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2answers
172 views

How is differential form different from ordinary calculus objects?

I am going to learn differential form soon, but after reading some introductory parts of my texts, I couldn't get why differential form is needed and how it is different from ordinary mathematics ...
2
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1answer
121 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 ...
2
votes
2answers
390 views

Hodge dual on orthonormal basis: two inconsistent answers

I'm trying to learn differential geometry using Göckeler & Schücker's book and I have some problems with the hodge star. As an example, say we have two orthonormal bases $e^i$ and ...
2
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1answer
49 views

Problem with integration of $1$-form on surface

I have some problem with integration of differential forms on algebraic surfaces (I'm reading Cartan's book on analytic functions). Let $X \subseteq \mathbb{C}^2$ be an algebraic curve given by ...
2
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1answer
239 views

Relevance of Differential Forms

I recently started reading about differential forms, and I am trying to figure out their purpose. Lets say $\omega=y\,dx+x\,dy$, and we want to evaluate $\int_C \omega$ over the curve parametrized by ...
2
votes
1answer
244 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 ...
2
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1answer
102 views

Equality of integrals of differential forms

I have two $(n-1)$-forms $\omega_{1}$ and $\omega_{2}$ on $\mathbb{R}^n$ and a smooth function $g(x) \colon \mathbb{R}^n \to \mathbb{R}$ ($dg$ doesn't vanish anywhere) such that $dg \wedge \omega_1 = ...
2
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1answer
237 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 ...
2
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1answer
308 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
386 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 ...
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0answers
33 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 ...
2
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0answers
37 views

Volume form on $(n-1)$-sphere $S^{n-1}$

Let $\omega$ the (n-1) form on $\mathbb{R}^n$ $$\omega=\sum_{j=1}^{n}(-1)^{j-1}x_{j}dx_{1}\wedge\cdots\wedge \hat{dx_{j}}\wedge\cdots dx_{n}$$ Show that the restriction of $\omega$ to $S^{n-1}$ in ...
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0answers
32 views

Does the wedge product of bundle-valued forms induce a universal object?

Given a smooth manifold $M$ and a vector bundle $E$ over $M$, the $C^\infty(M)$-module of $E$-valued $p$-forms on $M$ is defined to be $$\Omega^p(M; E) := \Gamma_M\left( \bigwedge^p T^*M \otimes E ...
2
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1answer
31 views

Express a quadratic form as a sum of squares using Schur complements

So I was able to figure out the first part of this problem, but I have no concept of how it relates to Schur complements, so I'm not sure (no pun intended) how to proceed. The question is as follows: ...
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0answers
60 views

How to define the boundary operator using the exterior derivative?

I am looking for a way to define the boundary operator $\partial : M^n \to N^{n-1}$ from an $n$-dimensional manifold $M$ to its boundary $N$ using the the expression \begin{equation*} \int_M d \alpha ...
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0answers
40 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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0answers
50 views

Clarification of vector valued forms (sections and tensor products)

I am seeking a bit of clarification on vector valued forms. Intuitively, a vector valued differential form is a differential form on $M$ whose values lie in some vector space. More accurately, Let ...
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0answers
24 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 $c^*d\omega$ ...