Differentiation of Vector Valued Functions - Browder, Proposition 8.12 ....

[Math Amateur]

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 help in proving Proposition 8.12 ... ...

Proposition 8.12 and the definitions, remarks and propositions leading up to it read as follows:
View attachment 7467
https://www.physicsforums.com/attachments/7468Although Browder states that Proposition 8.12 is easy to prove I am unable to make an effective start on the proof ...

Can someone please demonstrate how Proposition 8.12 is proved ...

[Note that I am unsure about the definition ... and the nature ... of the $$f'_i (p)$$ ... ... ]

Help will be much appreciated ...

Peter

 

[HallsofIvy]

When you are "not sure" about definitions, look at simple examples. Start with a two dimensional case, say [math]\vec{f}(x, y)= \begin{pmatrix} f_1(x, y) \\ f_2(x, y)\end{pmatrix}= \begin{pmatrix}x^2y \\ e^x sin(y)\end{pmatrix}[/math].

The "rows" referred to are [math]\begin{pmatrix}\frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y}\end{pmatrix}= \begin{pmatrix}2xy & x^2\end{pmatrix}[/math] and [math]\begin{pmatrix}\frac{\partial f_2}{\partial x} & \frac{\partial f_2}{\partial y}\end{pmatrix}= \begin{pmatrix} e^x sin(y) & e^x cos(y)\end{pmatrix}[/math] so that the 2 by 2 "derivative matrix" is
[math]\begin{pmatrix}\frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y} \\ \frac{\partial f_2}{\partial x} & \frac{\partial f_2}{\partial y}\end{pmatrix}= \begin{pmatrix} 2xy & x^2 \\ e^x sin(y) & e^x cos(y)\end{pmatrix}[/math]