Proof of Proposition 2 in Ch.7 Spivak Vol.2: n-Dimensional Distribution?

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SUMMARY

The discussion centers on the proof of Proposition 2 in Chapter 7 of Spivak's Volume 2, specifically regarding the n-dimensional distribution defined as \(\Delta_{p}=\bigcap^{n}_{i,j=1}ker\Lambda^{i}_{j}(p)\) in \(\mathbb{R}^{(n+n^2)}\). The participant initially seeks clarification on the integrability and dimensionality of this distribution but later resolves their query independently. The key takeaway is the affirmation of \(\Delta_{p}\) as an n-dimensional distribution, highlighting its significance in the context of differential geometry.

PREREQUISITES
  • Understanding of differential geometry concepts
  • Familiarity with Spivak's "Calculus on Manifolds" Volume 2
  • Knowledge of kernel and image in linear algebra
  • Basic comprehension of n-dimensional spaces
NEXT STEPS
  • Study the properties of n-dimensional distributions in differential geometry
  • Review the concept of kernels in linear transformations
  • Explore integrability conditions for distributions
  • Examine examples of n-dimensional distributions in \(\mathbb{R}^{(n+n^2)}\)
USEFUL FOR

Mathematicians, students of differential geometry, and anyone studying advanced calculus concepts will benefit from this discussion.

joe2317
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Hi.

I have a question on proof of proposition 2 in chater 7 Spivak volume2.
In the proof, he says that the n-dimensional distribution
\Delta_{p}=\bigcap^{n}_{i,j=1}ker\Lambda^{i}_{j}(p)
in R^(n+n^2) is integrable.
Could anyone explain why it is an n dimensional distribution?
Thanks.
 
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