- #1
CJ2116
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Hi everyone, first of all I have been a lurker here for years and have benefited greatly from many of the discussions in the math and physics sections. Thanks, I have received a lot of helpful information from these forums!
I have been working through Wald's General Relativity book and I am having trouble following the reasoning behind one part of a theorem. From page 15, the theorem and part of the proof is (For those who don't have the book):
Let M be an n-dimensional manifold. Let p \in M and let V_p denote the tangent space at p. Then dim V_p=n
Proof We shall show that dim V_p=n by constructing a basis of V_p, i.e. by finding n linearly independent tangent vectors that span V_p. Let \psi : O \rightarrow U\subset R^n be a chart with p\in O. If f\in \mathfrak{F}, then by definition f\circ \psi^{-1}:U\rightarrow R is C^{\infty}. For \mu=1,...,n define X_{\mu}:\mathfrak{F}\rightarrow R by
$$X_{\mu}(f)=\frac{\partial}{\partial x^{\mu}}(f\circ \psi^{-1})\bigg|_{\psi (p)}$$
$$\vdots$$
I can't seem to figure out how the term \frac{\partial}{\partial x^{\mu}}(f\circ \psi^{-1})\bigg|_{\psi (p)} is a mapping from \mathfrak{F}\rightarrow R. f\circ \psi^{-1} was defined to be a mapping from U\rightarrow R. In other words, I don't see why these last two terms should be equal. I think I am missing something obvious here. Is there maybe some sort of chain rule argument?
Thanks, any pointer in the right direction would be greatly appreciated!
I have been working through Wald's General Relativity book and I am having trouble following the reasoning behind one part of a theorem. From page 15, the theorem and part of the proof is (For those who don't have the book):
Let M be an n-dimensional manifold. Let p \in M and let V_p denote the tangent space at p. Then dim V_p=n
Proof We shall show that dim V_p=n by constructing a basis of V_p, i.e. by finding n linearly independent tangent vectors that span V_p. Let \psi : O \rightarrow U\subset R^n be a chart with p\in O. If f\in \mathfrak{F}, then by definition f\circ \psi^{-1}:U\rightarrow R is C^{\infty}. For \mu=1,...,n define X_{\mu}:\mathfrak{F}\rightarrow R by
$$X_{\mu}(f)=\frac{\partial}{\partial x^{\mu}}(f\circ \psi^{-1})\bigg|_{\psi (p)}$$
$$\vdots$$
I can't seem to figure out how the term \frac{\partial}{\partial x^{\mu}}(f\circ \psi^{-1})\bigg|_{\psi (p)} is a mapping from \mathfrak{F}\rightarrow R. f\circ \psi^{-1} was defined to be a mapping from U\rightarrow R. In other words, I don't see why these last two terms should be equal. I think I am missing something obvious here. Is there maybe some sort of chain rule argument?
Thanks, any pointer in the right direction would be greatly appreciated!
