All Questions
Tagged with exterior-derivative differential-topology
6 questions
1
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74
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Is $\delta \Delta^{-1} d$ the identity operator?
Let $d, \delta, \Delta = (d\delta + \delta d)$ be the exterior derivative, codifferential and Laplace-de-Rham operator.
Let $\omega$ be a closed $k$-form, one can then say $\Delta \omega = d \delta \...
- 1,384
2
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0
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145
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Closed $n$ form from Harvard Qualifying Exam
In the most recent Harvard Qualifying exam, one is asked to prove that if $M$ is a compact oriented manifold, and there are two nonvanishing orientation forms $\xi$ and $\eta$ who's integrals over $M$ ...
- 4,163
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139
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Exterior derivative of a 2-form defined by composition of metric with an endomorphism
Im interested in calculating the exterior derivative of a 2-form defined by $\omega(x,y) = g(x,Ay)$ where
$A \in \Gamma(End(T)
)$ is skew and $g$ is some metric.
I hope to reach some coordinate-...
- 681
2
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90
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Geometric Interpretation of the Exterior Derivative of a 1-form
I was reading this link where says that the geometric interpretation of the exterior derivative of a 1-form $\varphi$ is “the sum of $\varphi$ on the boundary of the surface defined by its arguments” ...
- 370
3
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1k
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The wedge of an exact form with a closed form is exact.
I'm trying to prove that the wedge of a closed form $\xi$ with an exact form $\omega$ is exact. We already have that half of it is exact. Maybe we can use the equation of $\xi$ being closed to rewrite ...
- 974
1
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1
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319
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Confusion about Tangent vs Cotangent differentials
In Lee's Introduction to Smooth manifolds, let $M$ be a smooth manifolds and given a smooth map $f \in C^\infty(M)$, we have two definitions of the "differential" at $f$:
(1): for each $p \in M$, ...
- 6,814