# Tagged Questions

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

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### matrix valued integrating factor of one forms (reference request)

I have $N$ 1-forms $\omega_1(x), \ldots, \omega_N(x)$. I want to know if there exists an invertible linear combination of these forms which yields $N$ closed forms. In other words: does an invertible ...
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### Boundary of a $k-$Cell Definition

In the book I am reading, the Boundary of a $(k+1)-$cell $\varphi$ is defined to be the $k-$chain $$\partial\varphi=\sum_{j=1}^{k+1}(-1)^{j+1}(\varphi\circ \iota^{j,1}-\varphi\circ \iota^{j,0})$$ ...
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### Closed Form and Pullback compatibility

given: $U,V \subset \mathbb{R}^N, f\in C^1(V,U)$ a diffeomorphism Let $\omega$ be a k-Form on U and $f^*\omega$ a closed Form. Then with $0 = df^*(\omega) = d \omega(df)$ we have ,that $\omega$ ...
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### Constructing the Hodge Laplacian from the Laplace-Beltrami one

I know, in principle, how to construct the Hodge Laplacian with the aid of the Hodge star. Is it possible, though, to construct it only from the Laplace-Beltrami operator? I have already found a way, ...
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### Total Derivative and Composition

We are given that $C$ is a function of $Y_D$ and $Y_D=Y-Y\tau$. What would be the total differential of $Y=C(Y_D)$? So far I have the following: $$dY=C_{Y_D}(1-\tau)dY+C_{Y_D}(-Y)d\tau$$ However I ...
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### Differential equation

Hello if I have differential equation which is a function of x = differential equation which is function of t Can I say that the differential equation which is function of x = C= the ...
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### Checking that a two-form transforms correctly under Lorentz transformations

This is exercise $7.22$ in Supergravity by Freedman and Van Proeyen, but I did not understand it and would appreciate if you clear it out. Given the below, I still don't get how, if we define the ...
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### Can we see this integral as the line integral of a 1-form

In Stein and Shakarchi's complex analysis, the following definition is given on pg. 21 Let $z:[a, b]\to \mathbf C$ be a parameterization of smooth curve $\gamma$ in $\mathbf C$ and $f$ be a ...
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### Is the curl of a vectorfield expressable in terms of a Lie derivative?

I am loving differential forms and the general version of Stokes theorem. However, I am also fond of the intuitive pictures of my older physics days of $grad$, $div$ and $curl$. I recently stumbled ...
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### Formula for the curvature $2$-form.

I'm currently reading a textbook to do with curvature and $k$-forms. It says that the curvature $2$-form given connection $1$-form, $A$, is $$F =d^A A = dA+A \wedge A$$ It then goes on to say that ...
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### $k$-form on $\mathbb{P}^n(\mathbb{R})$

Let $\pi$ be the canonical projection from $\mathbb{R}^{n+1}/\{0\}$ to $\mathbb{P}^n(\mathbb{R})$. Given a $k$-form $\alpha$ on $\mathbb{R}^{n+1}/\{0\}$ find necessary and sufficient conditions such ...
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### When are differential forms related by a base space automorphism?

Let $w$ and $u$ be nowhere-vanishing smooth differential forms fields of degree $n$ on a smooth manifold $M$ (aka smooth sections of $\Omega^n(M)$). When does there exist an automorphism $f: M \to M$ ...
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### Show that $\alpha \wedge \mathrm d \alpha =0$ when $\alpha \in \Omega ^1(M)$ and $d(f\alpha) = 0$ for some nowhere zero function $f$

Let $\alpha \in \Omega ^1(M)=\text{Tens}_1(M)$ be a differential form of degree $1$ on the smooth manifold $M$. Suppose that there is $f\in \mathcal C^\infty (M)$ s.t. $f(x)\neq 0$ for all $x\in M$ ...
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### Finding a one-form $\lambda$ such that $d\lambda = \omega$

Let $\omega = 2xz dy\wedge dz + dz\wedge dx -(z^2 + e^x)dx\wedge dy$. We have just started out with differential forms and need to find a one-form $\lambda$ so that $d\lambda = \omega$. \begin{gather}...
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### Integrating a density over a Mobius strip

According to this link one can integrate over a Mobius strip by using "densities". That has me very excited but I can't seem to find a reference on this. Can someone provide a book/ online source ...
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### How does the wedge of dual vectors act on a wedge of vectors?

Suppose $dx_i$ is the dual basis in $R^n$ so that $$dx_i (e_j) = \delta_{ij}.$$ It makes sense to me how 1-forms work: a 1-form evaluated at a point gives some linear functional, which takes tangent ...
Given a smooth real manifold $M$, and a real vector space $V$, when we talk about a $V$-valued $k$-form on $M$, do we mean an element of $\Gamma(\wedge^{*k}M)\otimes V$, where $\Gamma$ denotes ...