# Directional Derivative ... Tapp,Equations (3.4) and (3.5) ... ...

#### Peter

##### Well-known member
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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 [FONT=MathJax_AMS]R[/FONT][FONT=MathJax_Math]m[/FONT] to [FONT=MathJax_AMS]R[/FONT][FONT=MathJax_Math]n[/FONT] ... ...

I need help in order to fully understand exactly how/why equations (3.4) and (3.5) follow ... ...

The relevant portion of Tapp's text reads as follows: My questions are as follows:

Question 1

In the above text from Tapp we read the following:

" ... ... If the directional derivative $$\displaystyle df_p(v) = \displaystyle \lim_{ t \to 0 } \frac{ f(p + tv) - f(p) }{t}$$ exists, then the limit can be reexpressed as the claim that

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

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

$$\displaystyle df_p(v) = \displaystyle \lim_{ t \to 0 } \frac{ f(p + tv) - f(p) }{t}$$

implies that

$$\displaystyle \displaystyle \lim_{ t \to 0 } \frac{ \mid f(p + tv) - f(p) - t df_p(v) \mid }{t} = 0$$ ... ... ?

Question 2

In the above text from Tapp we read the following:

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

which really just says that

(3.5) $$\displaystyle \ \ \ \ \ \ \ \ f(p + tv) - f(p) \approx t df_p(v)$$ ... ...

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

$$\displaystyle \displaystyle \lim_{ t \to 0 } \frac{ \mid f(p + tv) - f(p) - t df_p(v) \mid }{t} = 0$$

means the same as

$$\displaystyle f(p + tv) - f(p) \approx t df_p(v)$$ ... ...?

Hope that someone can help ...

Peter