Directional and Partial Derivatives .... working from the definitions ....

[Math Amateur]

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 partial derivative 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:\(\displaystyle F(t) = f( x + t e^j)\)

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 \(\displaystyle x = x + h^{j-1}\) ...... BUT ...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) )

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 ...

 

[jvicens]

but how does that help us to understand the equation?
Dear Peter,

Thank you for reaching out for assistance with understanding Kantorovitz's Proposition on pages 61-62. I can empathize with your struggles to understand complex mathematical concepts and proofs. I will do my best to explain the concepts and equations in a clear and concise manner.

Firstly, let's review the definitions of directional and partial derivatives as given by Kantorovitz in Chapter 2. For a function f(x), the directional derivative in the direction of a unit vector u is defined as:

f_u'(x) = \lim_{h \rightarrow 0} \frac{f(x + hu) - f(x)}{h}

This definition can be rewritten using the definition of a partial derivative as:

f_u'(x) = \frac{\partial f}{\partial u}(x)

where \frac{\partial f}{\partial u}(x) is the partial derivative of f with respect to u.

Now, in the proof of Kantorovitz's Proposition, we are dealing with the partial derivative \frac{\partial f}{\partial x_j}(x). To better understand this, we can rewrite it using the definition of a partial derivative as:

\frac{\partial f}{\partial x_j}(x) = \lim_{t \rightarrow 0} \frac{f(x + te^j) - f(x)}{t}

Note that e^j is a unit vector in the jth direction. This is why we use it in the definition of the partial derivative.

Now, let's look at equation (2.3) in the proof. This equation is using the chain rule to calculate the partial derivative \frac{\partial f}{\partial x_j}(x). The chain rule states that for a composite function F(t) = f(g(t)), the derivative of F with respect to t is given by:

F'(t) = f'(g(t))g'(t)

In this case, our composite function is F(t) = f(x + h^{j-1} + te^j). The first term, x + h^{j-1}, is a constant with respect to t, so its derivative is 0. The second term, te^j, is a function of t, so its derivative is e^j. Therefore, the derivative of F(t) is given by:

F'(t)