 #1
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I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ...I am focused on Chapter 12: Multivariable Differential Calculus ... and in particular on Section 12.4: The Total Derivative ... ...I need help in order to fully understand Theorem 12.3, Section 12.4 ...Theorem 12.3 (including its proof) reads as follows:
View attachment 8505
Regarding the above Theorem, I am finding it difficult to understand how, when \(\displaystyle T_c(u) = f'(c;u)\), that a function can have a finite directional derivative \(\displaystyle f'(c;u)\) for every \(\displaystyle u\) but may fail to be continuous at \(\displaystyle c\) ... whereas! ... for a total derivative ... when a function has a total derivative it is continuous ... YET! ... \(\displaystyle T_c(u) = f'(c;u)\) ...Can someone please explain what is going on ...
Help will be appreciated ...
Peter
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It may help MHB readers of the above post to have access to Section 12.4 on the total derivative ... so I am providing access to the same ... as follows...
View attachment 8506
View attachment 8507
It may also help MHB readers of the above post to have access to Section 12.2 on the directional derivative ... so I am providing access to the same ... as follows...
View attachment 8508
View attachment 8509
Hope that helps ...
Peter
View attachment 8505
Regarding the above Theorem, I am finding it difficult to understand how, when \(\displaystyle T_c(u) = f'(c;u)\), that a function can have a finite directional derivative \(\displaystyle f'(c;u)\) for every \(\displaystyle u\) but may fail to be continuous at \(\displaystyle c\) ... whereas! ... for a total derivative ... when a function has a total derivative it is continuous ... YET! ... \(\displaystyle T_c(u) = f'(c;u)\) ...Can someone please explain what is going on ...
Help will be appreciated ...
Peter
=====================================================================================
It may help MHB readers of the above post to have access to Section 12.4 on the total derivative ... so I am providing access to the same ... as follows...
View attachment 8506
View attachment 8507
It may also help MHB readers of the above post to have access to Section 12.2 on the directional derivative ... so I am providing access to the same ... as follows...
View attachment 8508
View attachment 8509
Hope that helps ...
Peter
Attachments

Apostol  Theorem 12.3 .png13.5 KB · Views: 78

Apostol  1  Section 12.4 ... PART 1 .png32.7 KB · Views: 92

Apostol  2  Section 12.4 ... PART 2 ... .png41.1 KB · Views: 84

Apostol  1  Section 12.2 ... PART 1 ... .png22.5 KB · Views: 81

Apostol  2  Section 12.2 ... PART 2 .png19.5 KB · Views: 76