The Chain Rule for Multivariable Vector-Valued Functions .... ....

[Math Amateur]

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:
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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

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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:
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Hope that helps ...

Peter

 

[Opalg]

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

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

The answer to all three of those questions is that the equations in (17) are meant to be the definition of $E(y)$. In other words, if you define \[E(y) = \begin{cases}0&\text{if }y=0,\\ f'(b) [ E_a(y) ] + \frac{ \| v \| }{ \| y \| } E_b (v)&\text{if }y\ne0,\end{cases}\] then the second line of (16) follows immediately from the first line.

 

[Math Amateur]

The answer to all three of those questions is that the equations in (17) are meant to be the definition of $E(y)$. In other words, if you define \[E(y) = \begin{cases}0&\text{if }y=0,\\ f'(b) [ E_a(y) ] + \frac{ \| v \| }{ \| y \| } E_b (v)&\text{if }y\ne0,\end{cases}\] then the second line of (16) follows immediately from the first line.

Thanks Opalg ...

Indeed ... you are right, of course!

Should have seen that ...

Peter