On p. 22 of "Applied Differential Geometry" by William Burke he says that Df(u) must be the unique linear operator satisfying

Why not just say:

\(\displaystyle

Df(u) \equiv \lim_{h\rightarrow0}||\frac{f(u+h) - f(u)}{h}||

\)

??

Then in the following example he says that Df(u) is a map.

\(\displaystyle \lim_{h\rightarrow0}\frac{||f(u+h) - f(u) - Df(u)||}{||h||} = 0\)

Why not just say:

\(\displaystyle

Df(u) \equiv \lim_{h\rightarrow0}||\frac{f(u+h) - f(u)}{h}||

\)

??

Then in the following example he says that Df(u) is a map.

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