# Tagged Questions

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

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### Rudin's proof on Poincare Lemma.

Two identical questions are here: A question about differential forms and here: http://mathforum.org/kb/thread.jspa?forumID=13&threadID=2141549&messageID=7208112 In his proof, Rudin says ...
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### Vector-valued differential forms

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 ...
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### Stoke theorem and exterior derivative

$w=x \, dy\wedge dz - 2z f(y) \, dx \wedge dy + y f(y) \, dz \wedge dx$ where $f: \mathbb{R} \to \mathbb{R}$ belong $C^1$ (differentiable and derivative is continuous) with $f(1)=1$. Find $f$ so that ...
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### Differential forms, projections

I have a problem with one exercise from differential geometry. I don't even know how to start. Anyone could help with this problem? Let $M$, $N$ be manifolds, $M$ connected. Let $\pi:M\times N \to N$...
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### Divergence, contraction and lie derivatives

I'm working through this question. I can show the forward direction in (a) but can't show the converse. I have $\delta/\delta t \phi^*_t \mu$ evaluated at t=0 is 0, but I can't see how I conclude from ...
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### $d(\iota_v\rho)=0 \implies d(\phi\iota_v\rho)=d\phi(v)\rho$?

My motivation is physical, but my question is purely mathematical. Everybody knows, that the power of the electric current in a piece of wire is $$P=UI$$ where the wire is regular domain $V$ in a ...
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### Contraction of a vector form

I'm trying to make sense of this definition, but I cannot see why the resulting map is in a space of dimension $k-1$, surely as it is comprised of k vectors this maps a k-form to a (k+1)-form? I ...
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### Hermitian vector space and relation of associated operators.

Here what i want to do is prove proposition 1.1 in chapter 5, on Wells, Differential analysis on complex manifold, The propositions are follows For Hermitian vector space of complex dimension $n$. ...
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### Simplifying the Kahler form

In the link here, p.4, it says that, given a fundamental 2-from $\mathcal{K}$ $$\mathcal{K}=\frac{\sqrt{-1}}{2\pi}g_{i\bar{j}}dz^i\wedge d\bar{z}^{\bar{j}},$$ a manifold is said to be Kahler if this ...
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### 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 ...
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### 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 ...
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 ...
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}, Y=x^2\frac{\partial}{\...