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I am reading Kristopher Tapp's book: Differential Geometry of Curves and Surfaces ... and am currently focused on Chapter 3: Surfaces ... and in particular on Section 1: The Derivative of a Function from \(\displaystyle \mathbb{R}^m\) to \(\displaystyle \mathbb{R}^n\) ... ...

I need help in order to fully understand Definition 3.3 ... ...

Definition 3.3 reads as follows:

In the above text from Tapp we read the following:

\(\displaystyle df_p(v) = \displaystyle \lim_{ t \to 0 } \frac{ f(p + tv) - f(p) }{t} = (f \circ \gamma) (0)\) ... ... "

Can someone please demonstrate (formally and rigorously) exactly why/how

\(\displaystyle \displaystyle \lim_{ t \to 0 } \frac{ f(p + tv) - f(p) }{t} = (f \circ \gamma) (0)\) ... ...?

Hope someone can help ...

Peter

I need help in order to fully understand Definition 3.3 ... ...

Definition 3.3 reads as follows:

In the above text from Tapp we read the following:

\(\displaystyle df_p(v) = \displaystyle \lim_{ t \to 0 } \frac{ f(p + tv) - f(p) }{t} = (f \circ \gamma) (0)\) ... ... "

Can someone please demonstrate (formally and rigorously) exactly why/how

\(\displaystyle \displaystyle \lim_{ t \to 0 } \frac{ f(p + tv) - f(p) }{t} = (f \circ \gamma) (0)\) ... ...?

Hope someone can help ...

Peter

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