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I am reading Loring W.Tu's book: "An Introduction to Manifolds" (Second Edition) ...
I need help in order to fully understand Tu's section on tangent vectors in $$\mathbb{R}^n$$ as derivations... In his section on tangent vectors in \mathbb{R}^n as derivations, Tu writes the following:View attachment 8637
View attachment 8638In the above text from Tu we read the following:
" ... ... If $$f$$ is $$C^{ \infty }$$ in a neighborhood of $$p$$ in $$\mathbb{R}^n$$ and $$v$$ is a tangent vector at $$p$$, the directional derivative of $$f$$ in the direction of $$p$$ ... ... "
My questions are as follows:
Question 1
What are these functions $$f$$ that Tu is introducing ... and further, what is the point of them ... ?
Question 2
The domain of $$f$$ is clearly $$\mathbb{R}^n$$ ... BUT ... what is the range of $$f$$ ... I am guessing it is $$\mathbb{R}$$ ... is that correct ... but why is $$f$$ real-valued?
Hope that someone can clarify these issues ...
Peter
I need help in order to fully understand Tu's section on tangent vectors in $$\mathbb{R}^n$$ as derivations... In his section on tangent vectors in \mathbb{R}^n as derivations, Tu writes the following:View attachment 8637
View attachment 8638In the above text from Tu we read the following:
" ... ... If $$f$$ is $$C^{ \infty }$$ in a neighborhood of $$p$$ in $$\mathbb{R}^n$$ and $$v$$ is a tangent vector at $$p$$, the directional derivative of $$f$$ in the direction of $$p$$ ... ... "
My questions are as follows:
Question 1
What are these functions $$f$$ that Tu is introducing ... and further, what is the point of them ... ?
Question 2
The domain of $$f$$ is clearly $$\mathbb{R}^n$$ ... BUT ... what is the range of $$f$$ ... I am guessing it is $$\mathbb{R}$$ ... is that correct ... but why is $$f$$ real-valued?
Hope that someone can clarify these issues ...
Peter