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An Example on Differentials .... Browder Proposition 8.12 ....
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[QUOTE="Math Amateur, post: 6777899, member: 203675"] 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: [ATTACH=CONFIG]9436._xfImport[/ATTACH] For Definitions 8.9 and 8.10 and Proposition 8.11 ... see scanned text below ... Now consider the following example ... [MATH]f(x,y) = (xy, x/y)[/MATH] where we have [MATH]f_1(x, y) = xy[/MATH] and [MATH]f_2(x, y)= x/y [/MATH]I wish to determine [MATH]L = \text{df}_p[/MATH] where [MATH]p = (a, b)[/MATH] [I][B]using only[/B][/I] 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 ...I also wish to determine the Jacobian matrix of [MATH]f[/MATH] at [MATH]p[/MATH], namely, [MATH]f'(p)[/MATH] again [I][B]using only[/B][/I] 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 ...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: [ATTACH=CONFIG]9437._xfImport[/ATTACH] [ATTACH=CONFIG]9438._xfImport[/ATTACH] Hope that helps ... Peter [/QUOTE]
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An Example on Differentials .... Browder Proposition 8.12 ....
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