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I am reading the book "Several Real Variables" by Shmuel Kantorovitz ... ...

I am currently focused on Chapter 2: Derivation ... ...

I need help with an element of the proof of Kantorovitz's Proposition on pages 61-62 ...

Kantorovitz's Proposition on pages 61-62 reads as follows:https://www.physicsforums.com/attachments/7803

View attachment 7804

I am trying to understand the above proof in terms of the definitions of directional and partial derivatives (in terms of limits) ... but I am having trouble understanding equation (2.3) above ...

I will explain my difficulties in terms of Kantorovitz's definitions as he develops them on page 60 ... as follows:View attachment 7805Now ... I am trying to understand how the definition of

so then for \(\displaystyle F_j\) in the proof (see the expression that is above the expression (2.3)) ... we have

\(\displaystyle F_j (t) = f ( x + h^{j-1} + te^j )\) and we appear to be dealing (for some reason?) with \(\displaystyle ( x + h^{j-1} )\) instead of \(\displaystyle x\) ...

... which is OK ... just put \(\displaystyle x = x + h^{j-1}\) ...

What is the definition of \(\displaystyle F_j'(t)\) ... and working strictly and rigorously from the definition how do we obtain

\(\displaystyle F'_j (t) = \frac { \partial f }{ \partial x_j } f ( x + h^{j-1} + te^j )\)

Hope someone can help ...

Peter***NOTE***

I have to say I find it somewhat confusing in trying to work from the definition of partial derivative, that Kantorovitz gives the definition for partial and directional derivative in terms of expressions where \(\displaystyle t\) tends to zero ... and then equation (2.3) above is a partial derivative with \(\displaystyle t\) as a variable ... as in \(\displaystyle F'_J(t)\) ... surely \(\displaystyle t \rightarrow 0\) as per the definition ...

I am currently focused on Chapter 2: Derivation ... ...

I need help with an element of the proof of Kantorovitz's Proposition on pages 61-62 ...

Kantorovitz's Proposition on pages 61-62 reads as follows:https://www.physicsforums.com/attachments/7803

View attachment 7804

I am trying to understand the above proof in terms of the definitions of directional and partial derivatives (in terms of limits) ... but I am having trouble understanding equation (2.3) above ...

I will explain my difficulties in terms of Kantorovitz's definitions as he develops them on page 60 ... as follows:View attachment 7805Now ... I am trying to understand how the definition of

*applies to equation (2.3) in the proof of the proposition ... so for equation (2.1) of the definition we put \(\displaystyle u = e^j\) (because we are dealing with partial derivatives) ... ... and so (2.1) becomes:\(\displaystyle F(t) = f( x + t e^j)\)***partial derivative**so then for \(\displaystyle F_j\) in the proof (see the expression that is above the expression (2.3)) ... we have

\(\displaystyle F_j (t) = f ( x + h^{j-1} + te^j )\) and we appear to be dealing (for some reason?) with \(\displaystyle ( x + h^{j-1} )\) instead of \(\displaystyle x\) ...

... which is OK ... just put \(\displaystyle x = x + h^{j-1}\) ...

*In Definition 2.1.1 Kantorovitz defines the partial derivative this way:\(\displaystyle \frac { \partial f }{ \partial x_j } := F'(0) = \lim_{ t \rightarrow 0 } \frac{ F(t) - F(0) }{t}\) \(\displaystyle = \lim_{ t \rightarrow 0 } \frac{ f ( x + h^{j-1} + te^j ) - f(x) }{t} \)... ... is the above correct?Now ... my question is as follows: (pertaining largely to equation (2.3) )***... BUT ...**What is the definition of \(\displaystyle F_j'(t)\) ... and working strictly and rigorously from the definition how do we obtain

\(\displaystyle F'_j (t) = \frac { \partial f }{ \partial x_j } f ( x + h^{j-1} + te^j )\)

Hope someone can help ...

Peter***NOTE***

I have to say I find it somewhat confusing in trying to work from the definition of partial derivative, that Kantorovitz gives the definition for partial and directional derivative in terms of expressions where \(\displaystyle t\) tends to zero ... and then equation (2.3) above is a partial derivative with \(\displaystyle t\) as a variable ... as in \(\displaystyle F'_J(t)\) ... surely \(\displaystyle t \rightarrow 0\) as per the definition ...

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