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I am reading the book "An Introduction to Differential Manifolds" (Springer) by Jacques Lafontaine ...

I am currently focused on Chapter 1: Differential Calculus ...

I need help in order to fully understand a remark by Lafontaine following his definition of differentials ...

Lafontaine's definition of differentials followed by the remark in question read as follows:View attachment 8514

View attachment 8515

At the start of the above remark, Lafontaine writes the following:"We can rewrite the definition in the form \(\displaystyle \overrightarrow{ f(a) f(x) } = L \cdot \vec{ax} + o( \vec{ax} )\) ... ... "

Can someone please explain (simply and in detail) how Lafontaine's definition can be rewritten in the form \(\displaystyle \overrightarrow{ f(a) f(x) } = L \cdot \vec{ax} + o( \vec{ax} )\)

Hope someone can help ...

Peter

I am currently focused on Chapter 1: Differential Calculus ...

I need help in order to fully understand a remark by Lafontaine following his definition of differentials ...

Lafontaine's definition of differentials followed by the remark in question read as follows:View attachment 8514

View attachment 8515

At the start of the above remark, Lafontaine writes the following:"We can rewrite the definition in the form \(\displaystyle \overrightarrow{ f(a) f(x) } = L \cdot \vec{ax} + o( \vec{ax} )\) ... ... "

Can someone please explain (simply and in detail) how Lafontaine's definition can be rewritten in the form \(\displaystyle \overrightarrow{ f(a) f(x) } = L \cdot \vec{ax} + o( \vec{ax} )\)

Hope someone can help ...

Peter

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