# Tagged Questions

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

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### Demonstration of a basic formula involving differential forms

I'm writing some notes on Lie Groups and I'm not sure if I should demonstrate this formula or not. Assume $\omega$ is a differencial form and $X,Y$ fields con a Manifold M, is there a simple way to ...
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### Let $\theta$ a $1-$form. Why $d\theta(X,Y)=X\theta(y)-Y\theta(X)-\theta([X,Y])$?

Let $\theta$ a $1-$form. Why $$\mathrm d\theta(X,Y)=X\theta(Y)-Y\theta(X)-\theta([X,Y])\ \ ?$$ I know that $\theta=\sum_{i=1}^n a\mathrm d x^i$ where $a\in \mathcal C^\infty (U)$ and $U$ an open. ...
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### Orientation form on manifold cut out by $m$ functions

Let $f: \mathbb{R}^n \to \mathbb{R}^m$ be a smooth function. If $a$ is such that $f$ has surjective derivative at all points in $f^{-1}(a)$ then this is an $n-m$ dimensional manifold $X$. I'm trying ...
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Hi have to determinate a primitive of this differential form: $$\omega = \frac{xy}{\sqrt{(x^2+y^2)}}dx + \frac{x^2 + 2y^2}{\sqrt{(x^2+y^2)}}dy$$ As far as I know this should be a radial form which I ...
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### Condition for set of de Rham cohomology classes is linearly independent

If we have a set of 1-forms $w_1, ... w_n$ on a smooth manifold $X$ I can show that $w_i$ are linearly dependent if and only if $w_1 \wedge ... \wedge w_n = 0$. I wondered if this is also true in the ...
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### Area form of $M^2 \subseteq \Bbb R^4$: does $(-1)^{i+j-1}\det_{i'j'}(n,\nu) = dx^i \wedge dx^j$?

This is a follow up question from this one. I'm asking separately since one way or another, that one is sort of answered. Now, we know that if $M^{n-1} \subseteq \Bbb R^n$, then the volume element $dM$...
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### Is $\sigma_u \times \sigma_v \neq \vec{0}$ essential for $\int_{\alpha} \vec{F} \cdot d\vec{r} = \iint_\sigma \mathrm{curl}\,\vec{F} \cdot d\vec{S}$?

I think I have proved the following version of Stokes' theorem: Teorem 1: Let $\beta: [0,4] \to \mathbb{R}^2$ be the curve given by \beta(t) = \begin{cases} (t,0) & \mbox{if }...
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### Show that a k-form can be expressed as wedge product

I'm trying to show that given a $1$-form $\omega$ and a $k$-form $\alpha$ such that $\alpha \wedge \omega = 0$ then there exists a $(k-1)$-form $\beta$ such that $\alpha = \omega \wedge \beta$. I'm ...
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### Intersection of kernels of linearly independent smooth 1-forms on $\mathbb R^n$

I'm trying to solve the following problem: Let $\omega^1,\dots,\omega^k$ be smooth $1$-forms on $\mathbb R^n$ that are linearly independent at each point of $\mathbb R^n$. For $p\in\mathbb R^n$, ...
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### Definition of exact form

That's the definition of exact form in $E$. But if we look at the definition 10.18 we see that $\lambda \in C'$ but Rudin skip this condition. Can anyone please explain this moment
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### Stokes' Theorem from PMA Rudin. Confusing moment with simplex

I am reading the proof of Stokes' theorem from PMA Rudin but one moment seems to very weird. Why Rudin considers the case when $\sigma=[0,\mathbf{e}_1,\dots, \mathbf{e}_k]$? After all $\sigma$ may ...
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But paragraph which I marked by red line seems to me confusing. Let $k=3$ then $\sigma=[\mathbf{p}_0,\mathbf{p}_1,\mathbf{p}_2,\mathbf{p}_3]$ and $\partial \sigma=[\mathbf{p}_1,\mathbf{p}_2,\mathbf{p}... 1answer 63 views ### Finding A 1-form on$R^2 - {\{(0,0)}\}$I want to find a 1-form on$R^2 - {\{(0,0)}\}$such that$w(Y) = 0$and$w(X) = 1$. Here, $$X = -y\frac{\partial }{\partial x} + x\frac{\partial}{\partial y}\ \text{and}\ Y = x\frac{\partial }{\... 1answer 61 views ### A necessary and sufficient condition for the admittance of integrating factor Let \omega be a smooth 1-form on a smooth manifold M. A smooth positive function \mu on some open subset U\subset M is called an integrating factor for \omega if \mu\omega is exact on U. ... 2answers 42 views ### Specific example of integrating a 1-form over a curve I was given the following definition in my course but no corresponding examples: Supppose \gamma:[a,b]\rightarrow{M} is a smooth curve and \omega a 1-form on M (so \omega:M\rightarrow{T^*M}). ... 1answer 60 views ### Affine chains from PMA Rudin. Confusing examples I understood the definition of affine k-chain and that he defines \int \limits_{\Gamma} \omega as (82). But I can't understand the last two above examples. What does they mean? Can anyone ... 0answers 26 views ### Equivalent conditions for \mathfrak{F} to be a differential ideal Heres the question: Let \mathfrak{F} be an ideal of forms on a manifold M locally generated by r independent 1-forms. Say \mathfrak{F} is generated by \omega_1,\ldots, \omega_r on U. ... 0answers 29 views ### Let f : \mathbb{R}^n \to \mathbb{R}^n, suppose that \mu = dx^1\wedge\ldots\wedge dx^n, then f^{\ast}\mu = \det (df)\mu Let f : \mathbb{R}^n \to \mathbb{R}^n, suppose that \mu = dx^1\wedge\ldots\wedge dx^n, then f^{\ast}\mu = \det (df)\mu. I am trying to prove this. I made several low dimensional cases and ... 1answer 41 views ### Prove the exterior derivative of the following (n-1) form is zero Let \omega(x)=\frac{1}{{\parallel x \parallel}^n}\displaystyle\sum_{i=1}^{n}(-1)^{i-1}x_{i} dx_{1} \wedge \dots \wedge \widehat{dx_{i}} \wedge \dots \wedge dx_{n} be a differential (n-1) form on \... 1answer 56 views ### Confusing moment in Theorem 10.27 from PMA Rudin Theorem 10.27 If \sigma is an oriented rectilinear k-simplex in an open set E\subset \mathbb{R}^n then$$\int \limits_{\overline{\sigma}}\omega=\varepsilon\int \limits_{\sigma}\omega \qquad (81)$...
Consider the differential 1-form $\omega = ydx+dy$. I need to show that this is not exact, and find an example of a function $G(x,y)$ such that $G\omega=G(x,y)(ydx+dy)$ is an exact form. I have done ...
We know that $(dy_I)_T=dt_{i_1}\land \dots \land dt_{i_k}$ and using definition 10.18 we get $$d((dy_{I})_{T})=d1\land dt_{i_1}\land \dots \land dt_{i_k}=0$$ since $dc=0$ for any $c\in \mathbb{R}^1$. ...