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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 12.9: The Chain Rule ... ...I need help in order to fully understand Theorem 12.7, Section 12.9 ...Theorem 12.7 (including its proof) reads as follows:

View attachment 8523

View attachment 8524

In the proof of Theorem 12.7 we read the following:

" ... ... Using (14) in (15) we find\(\displaystyle f(b+v) - f(b) = f'(b) [ g'(a) (y) ] + f'(b) [ \| y \| E_a(y) ] + \|v \| E_b(v) \)\(\displaystyle = f'(b) [ g'(a) (y) ] + \| y \| E(y)\) ... ... ... (16)Where \(\displaystyle E(0) = 0\) and \(\displaystyle E(y) = f'(b) [ E_a(y) ] + \frac{ \| v \| }{ \| y \| } E_b (v) \ \ \ \ \text{ if } y\neq 0\) ... ... ... (17)... ... ... "

My questions are as follows:

Can someone show how Equation (16) follows ... that is ...

... how exactly does \(\displaystyle f(b+v) - f(b) = f'(b) [ g'(a) (y) ] + \| y \| E(y)\)

follow from

\(\displaystyle f(b+v) - f(b) = f'(b) [ g'(a) (y) ] + f'(b) [ \| y \| E_a(y) ] + \|v \| E_b(v)\)?

What is \(\displaystyle E(0)\) ... I know what \(\displaystyle E_a\) and \(\displaystyle E_b\) are ... but what is \(\displaystyle E\)?

Similarly ... what is \(\displaystyle E(y)\) in (16) and in (17) ... shouldn't it be \(\displaystyle E_a(y)\) ... ?

Further ... why (formally and rigorously) does \(\displaystyle E(0) = 0\)

Can someone please demonstrate how/why

\(\displaystyle E(y) = f'(b) [ E_a(y) ] + \frac{ \| v \| }{ \| y \| } E_b (v)\)

Help will be appreciated ...

Peter

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

It may help MHB readers of the above post to have access to Apostol's section on the Total Derivative ... so I am providing the same ... as follows:

View attachment 8525

View attachment 8526

Hope that helps ...

Peter

View attachment 8523

View attachment 8524

In the proof of Theorem 12.7 we read the following:

" ... ... Using (14) in (15) we find\(\displaystyle f(b+v) - f(b) = f'(b) [ g'(a) (y) ] + f'(b) [ \| y \| E_a(y) ] + \|v \| E_b(v) \)\(\displaystyle = f'(b) [ g'(a) (y) ] + \| y \| E(y)\) ... ... ... (16)Where \(\displaystyle E(0) = 0\) and \(\displaystyle E(y) = f'(b) [ E_a(y) ] + \frac{ \| v \| }{ \| y \| } E_b (v) \ \ \ \ \text{ if } y\neq 0\) ... ... ... (17)... ... ... "

My questions are as follows:

**Question 1**Can someone show how Equation (16) follows ... that is ...

... how exactly does \(\displaystyle f(b+v) - f(b) = f'(b) [ g'(a) (y) ] + \| y \| E(y)\)

follow from

\(\displaystyle f(b+v) - f(b) = f'(b) [ g'(a) (y) ] + f'(b) [ \| y \| E_a(y) ] + \|v \| E_b(v)\)?

**Question 2**What is \(\displaystyle E(0)\) ... I know what \(\displaystyle E_a\) and \(\displaystyle E_b\) are ... but what is \(\displaystyle E\)?

Similarly ... what is \(\displaystyle E(y)\) in (16) and in (17) ... shouldn't it be \(\displaystyle E_a(y)\) ... ?

Further ... why (formally and rigorously) does \(\displaystyle E(0) = 0\)

**Question 3**Can someone please demonstrate how/why

\(\displaystyle E(y) = f'(b) [ E_a(y) ] + \frac{ \| v \| }{ \| y \| } E_b (v)\)

Help will be appreciated ...

Peter

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

It may help MHB readers of the above post to have access to Apostol's section on the Total Derivative ... so I am providing the same ... as follows:

View attachment 8525

View attachment 8526

Hope that helps ...

Peter

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