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

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,881
Hobart, Tasmania
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:



Tapp - Equations 3.4 and 3.5 .png


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