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The Chain Rule for Multivariable Vector-Valued Functions ....
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[QUOTE="Math Amateur, post: 6078721, member: 203675"] Hi fresh_42 ... Thanks again for the help ... Just a clarification ... You write ... " ... ... The general formula for differentiation in direction ##v## goes: ##f\,'(a) \cdot v = f(a+v)-f(a)+r(v)## ... ..." Changing ##a## to ##c## to conform with Apostol's notation we have ##f\,'(a) \cdot v = f(a+v)-f(a)+r(v)## ... ... ... (1) Apostol's definition of differentiation (Equation (5) Section 12.4, page 346) reads as follows: ##f(c+v) = f(c) + f\, '(c)(v) + \| v \| E_c(v)## ... ... ... (2) Now (2) ##\Longrightarrow f\, '(c)(v) = f(c+v) - f(c) - \| v \| E_c(v)## ... ... ... (3) So comparing (1) and (3) we find they will be equivalent if ##f\,'(a) \cdot v = f\, '(c)(v)## ... ... ... (4) and ##r(v) = - \| v \| E_c(v)## ... ... ... (5)Can you please explain why (4) and (5) hold true ... ?... and just another minor issue ... You define r in the following sentence ... " ... ... The general formula for differentiation in direction ##v## goes: ##f\,'(a) \cdot v = f(a+v)-f(a)+r(v)## with a remainder function ##r()## that has to obey ##\lim_{v \to 0} \dfrac{r(v)}{||v||} = 0##... " and then later you assert "Now if ##v=0##, we have ##r(v)=0##." ... ... Can you explain why ##r(0) = 0## ... Thanks again for your help ... Peter [/QUOTE]
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The Chain Rule for Multivariable Vector-Valued Functions ....
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