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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 with an example based only on Proposition 8.12 ... and the material in Section 8.2 preliminary to Proposition 8.12 (see scanned text at end of post) ... namely Definitions 8.9 and 8.10 and Proposition 8.11 ...

Proposition 8.12 reads as follows:

View attachment 9436

For Definitions 8.9 and 8.10 and Proposition 8.11 ... see scanned text below ...

Now consider the following example ...

\(\displaystyle f(x,y) = (xy, x/y)\) where we have \(\displaystyle f_1(x, y) = xy\) and \(\displaystyle f_2(x, y)= x/y

\)

I wish to determine \(\displaystyle L = \text{df}_p\) where \(\displaystyle p = (a, b)\)

I also wish to determine the Jacobian matrix of \(\displaystyle f\) at \(\displaystyle p\), namely, \(\displaystyle f'(p)\) again

I am unable to carry out the above without using partial derivatives ... which Browder has not introduced yet... indeed, it may not be possible ...

Hope someone can help ...

Peter

=======================================================================================

Browder Section 8.2 (preliminary to Proposition 8.12) reads as follows:

View attachment 9437

View attachment 9438

Hope that helps ...

Peter

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

I need some help with an example based only on Proposition 8.12 ... and the material in Section 8.2 preliminary to Proposition 8.12 (see scanned text at end of post) ... namely Definitions 8.9 and 8.10 and Proposition 8.11 ...

Proposition 8.12 reads as follows:

View attachment 9436

For Definitions 8.9 and 8.10 and Proposition 8.11 ... see scanned text below ...

Now consider the following example ...

\(\displaystyle f(x,y) = (xy, x/y)\) where we have \(\displaystyle f_1(x, y) = xy\) and \(\displaystyle f_2(x, y)= x/y

\)

I wish to determine \(\displaystyle L = \text{df}_p\) where \(\displaystyle p = (a, b)\)

*Proposition 8.12 and the material in Section 8.2 preliminary to Proposition 8.12 (see scanned text at end of post) ... namely Definitions 8.9 and 8.10 and Proposition 8.11 ...***using only**I also wish to determine the Jacobian matrix of \(\displaystyle f\) at \(\displaystyle p\), namely, \(\displaystyle f'(p)\) again

*Proposition 8.12 and the material in Section 8.2 preliminary to Proposition 8.12 (see scanned text at end of post) ... namely Definitions 8.9 and 8.10 and Proposition 8.11 ...***using only**I am unable to carry out the above without using partial derivatives ... which Browder has not introduced yet... indeed, it may not be possible ...

Hope someone can help ...

Peter

=======================================================================================

Browder Section 8.2 (preliminary to Proposition 8.12) reads as follows:

View attachment 9437

View attachment 9438

Hope that helps ...

Peter