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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:
In the proof of Theorem 12.7 we read the following:
" ... ... Using (14) in (15) we find
##f(b+v)  f(b) = f'(b) [ g'(a) (y) ] + f'(b) [ \ y \ E_a(y) ] + \v \ E_b(v)##
##= f'(b) [ g'(a) (y) ] + \ y \ E(y)## ... ... ... (16)
Where ##E(0) = 0## and
##E(y) = f'(b) [ E_a(y) ] + \frac{ \ v \ }{ \ y \ } E_b (v) \ \ \ \ \text{ if } y\neq 0## ... ... ... (17)
... ... ... "
*** EDIT ***
It now occurs to me that in fact Apostol is defining E(y) in equations (16) and (17)
I should have seen this earlier ...
Peter
================================================================
My questions are as follows:
Question 1
Can someone show how Equation (16) follows ... that is ...
... how exactly does ##f(b+v)  f(b) = f'(b) [ g'(a) (y) ] + \ y \ E(y)##
follow from
##f(b+v)  f(b) = f'(b) [ g'(a) (y) ] + f'(b) [ \ y \ E_a(y) ] + \v \ E_b(v)##?
Question 2
What is ##E## ... I know what ##E_a## and ##E_b## are ... but what is ##E##?
Similarly ... what is ##E(y)## in (16) and in (17) ... shouldn't it be ##E_a(y)## ... ?
Further ... why (formally and rigorously) does ##E(0) = 0##
Question 3
Can someone please demonstrate how/why
##E(y) = f'(b) [ E_a(y) ] + \frac{ \ v \ }{ \ y \ } E_b (v)##
Help will be appreciated ...
Peter
=========================================================================================
It may help Physics Forum readers of the above post to have access to Apostol's section on the Total Derivative ... so I am providing the same ... as follows:
Hope that helps ...
Peter
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:
In the proof of Theorem 12.7 we read the following:
" ... ... Using (14) in (15) we find
##f(b+v)  f(b) = f'(b) [ g'(a) (y) ] + f'(b) [ \ y \ E_a(y) ] + \v \ E_b(v)##
##= f'(b) [ g'(a) (y) ] + \ y \ E(y)## ... ... ... (16)
Where ##E(0) = 0## and
##E(y) = f'(b) [ E_a(y) ] + \frac{ \ v \ }{ \ y \ } E_b (v) \ \ \ \ \text{ if } y\neq 0## ... ... ... (17)
... ... ... "
*** EDIT ***
It now occurs to me that in fact Apostol is defining E(y) in equations (16) and (17)
I should have seen this earlier ...
Peter
================================================================
My questions are as follows:
Question 1
Can someone show how Equation (16) follows ... that is ...
... how exactly does ##f(b+v)  f(b) = f'(b) [ g'(a) (y) ] + \ y \ E(y)##
follow from
##f(b+v)  f(b) = f'(b) [ g'(a) (y) ] + f'(b) [ \ y \ E_a(y) ] + \v \ E_b(v)##?
Question 2
What is ##E## ... I know what ##E_a## and ##E_b## are ... but what is ##E##?
Similarly ... what is ##E(y)## in (16) and in (17) ... shouldn't it be ##E_a(y)## ... ?
Further ... why (formally and rigorously) does ##E(0) = 0##
Question 3
Can someone please demonstrate how/why
##E(y) = f'(b) [ E_a(y) ] + \frac{ \ v \ }{ \ y \ } E_b (v)##
Help will be appreciated ...
Peter
=========================================================================================
It may help Physics Forum readers of the above post to have access to Apostol's section on the Total Derivative ... so I am providing the same ... as follows:
Hope that helps ...
Peter
Attachments

Apostol  1  Theorem 12.7  Chain Rule  PART 1 ... .png75.2 KB · Views: 515

Apostol  2  Theorem 12.7  Chain Rule  PART 2 ... ... .png54.7 KB · Views: 693

Apostol  1  Section 12.4  PART 1 ... .png67 KB · Views: 285

Apostol  2  Section 12.4  PART 2 ... .png68.8 KB · Views: 289

?temp_hash=94e326edac58a0ed69338d46334d19ae.png75.2 KB · Views: 282

?temp_hash=94e326edac58a0ed69338d46334d19ae.png54.7 KB · Views: 287

?temp_hash=94e326edac58a0ed69338d46334d19ae.png67 KB · Views: 199

?temp_hash=94e326edac58a0ed69338d46334d19ae.png68.8 KB · Views: 197
Last edited: