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I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...

I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... ...

I need some help in formulating a proof of Proposition 8.12 ...

Proposition 8.12 reads as follows:

View attachment 9428

Can someone please help me to demonstrate a formal and rigorous proof of Proposition 8.12 using on the definitions and propositions preceding the above proposition ...I am most interested in how/why we know that \(\displaystyle \text{df} (h) = \text{df}_1 (h), \ ... \ ... \ ... \ \text{df}_m (h) )\)... and also that ...\(\displaystyle f' (p) = \begin{bmatrix} f'_1 (p) \\ f'_2 (p) \\ . \\ . \\ . \\ f'_n (p) \end{bmatrix} \)... ... ... The definitions and propositions pertaining to the differential preceding the above proposition read as follows:

View attachment 9429

View attachment 9430

Hope that someone can help ...

Help will be much appreciated ...

Peter

I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... ...

I need some help in formulating a proof of Proposition 8.12 ...

Proposition 8.12 reads as follows:

View attachment 9428

Can someone please help me to demonstrate a formal and rigorous proof of Proposition 8.12 using on the definitions and propositions preceding the above proposition ...I am most interested in how/why we know that \(\displaystyle \text{df} (h) = \text{df}_1 (h), \ ... \ ... \ ... \ \text{df}_m (h) )\)... and also that ...\(\displaystyle f' (p) = \begin{bmatrix} f'_1 (p) \\ f'_2 (p) \\ . \\ . \\ . \\ f'_n (p) \end{bmatrix} \)... ... ... The definitions and propositions pertaining to the differential preceding the above proposition read as follows:

View attachment 9429

View attachment 9430

Hope that someone can help ...

Help will be much appreciated ...

Peter

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