Proof of dimension of the tangent space

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SUMMARY

The discussion centers on the proof of the dimension of the tangent space of an n-dimensional manifold as presented in "General Relativity" by Wald. Key equations referenced include 2.2.3 and 2.2.5, which involve the properties of smooth functions and their differentiability. The participants clarify that the equality in equation 2.2.5 is justified through the nature of constant functions and the properties of smooth functions, emphasizing that the tangent space is locally isomorphic to R^n. The proof's validity is further supported by referencing similar proofs in Isham's work.

PREREQUISITES
  • Understanding of smooth functions and differentiability
  • Familiarity with the concept of tangent spaces in differential geometry
  • Knowledge of the gradient theorem and its implications
  • Basic principles of manifolds and their local isomorphism to R^n
NEXT STEPS
  • Study the gradient theorem in detail to understand its application in proofs
  • Explore the concept of differentiable manifolds and their properties
  • Read "General Relativity" by Wald, focusing on pages 79-84 for the proof
  • Investigate similar proofs in "Quantum Gravity" by Isham to compare methodologies
USEFUL FOR

Mathematicians, physicists, and students of differential geometry seeking to deepen their understanding of tangent spaces and manifold theory.

hideelo
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I am attaching a picture of a proof from the book "general relativity" by wald. This is supposed to show that the tangent space of an n dimensional manifold is also n dimensional. I have two questions.

In equation 2.2.3 couldn't the function be anything at a since the (x-a) term is 0?

How is the equality in equation 2.2.5 justified, I'm just not seeing it
 

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I'm sorry, I cannot read the picture attached. Can't you just type it?
 
It's hard to write them out on my phone, so I'll try another picture
 

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Is that better?
 
No, it has to be exacly as written. Otherwise 2.2.2. will not be true.
 
I'm assuming 2.2.2 follows from the gradient theorem, in which case 2.2.3 should really be an integral, otherwise it's not an equality but the linear approximation.
 
A proof very similar to Wald's is included in Isham. Start reading on page 79. The proof is on pages 82-84.

Last time I discussed this proof with someone, I posted my version of it. It's in post #14 here.
 
hideelo said:
In equation 2.2.3 couldn't the function be anything at a since the (x-a) term is 0?
The theorem says that the function is smooth, i.e. differentiable an arbitrary number of times. Differentiable functions are continuous. The value of a continuous function at a specific point in its domain is completely determined by its values in the rest of its domain. For example, if ##f:\mathbb R\to\mathbb R## is continuous then ##f(0)=\lim_{n\to\infty}f\big(\frac 1 n\big)##.

hideelo said:
How is the equality in equation 2.2.5 justified, I'm just not seeing it
Assuming that you meant the last equality, this is how:

First term: f(p) is a constant function (the map that takes an arbitrary q to f(p)), so v(f(p))=0.

Second term: The sum is of the form ##\sum_\mu (A_\mu-A_\mu)B_\mu## so every term is zero.

Third term: For all smooth functions f and all constant functions g, we have v(f-g)=v(f)-v(g)=v(f).
 
an n manifold is by definition locally isomorphic (i.e. diffeomorphic)) to R^n, hence (by chain rule) each tangent space is isomorphic to the tangent space to R^n at the image point. Do you know that R^n is the tangent space to R^n at each point? That would do it.
 

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