Reconciling Differences in Vector-Valued Function Derivatives

[Math Amateur]

I am trying to reconcile apparent differences between the definitions of the derivative of a vector-valued function \(\displaystyle f: U \rightarrow \mathbb{R}^n\) (where \(\displaystyle U \subset \mathbb{R}^m\) ) of a vector variable from two textbooks ...

The textbooks are as follows:

Andrew Browder: "Mathematical Analysis: An Introduction"

and

Susan Jane Colley: "Vector Calculus" (Second Edition)Browder's definition of the derivative of f is explained in the following notes:View attachment 7464
Colley's definition is as follows:View attachment 7465My question is regarding how to reconcile the differences between these two definitions ... Colley has "length" or norm signs ( \(\displaystyle \lvert \lvert \cdot \rvert \rvert\) ) around the numerator and denominator terms while Browder only has modulus signs (that is "length" or norm in his case) around h in the denominator ...

Can someone please explain or reconcile this difference ...

Peter========================================================================================*** NOTE 1 ***

Colley's Definition of the total derivative mentions df(a) ...

Colley defines df(x) as the matrix of partial derivatives of f ... as follows:View attachment 7466
Peter

 

[Pereskia]

Hi,

For a vector function you have $$\lim_{x->0} f(x) = 0$$ if and only if $$\lim_{x->0} \Vert f(x)\Vert = 0$$
The norm is needed in the denominator since you can't divide a vector (or scalar) with a vector.

Hope this helps
David

 

[Ackbach]

My question is regarding how to reconcile the differences between these two definitions ... Colley has "length" or norm signs ( \(\displaystyle \lvert \lvert \cdot \rvert \rvert\) ) around the numerator and denominator terms while Browder only has modulus signs (that is "length" or norm in his case) around h in the denominator ...

Different authors use different notations, but it's been my experience that those two notations mean the same thing on vectors. One caution: the double lines $\|\cdot\|$ can have different meanings if it has a subscript. So, for example,
$$
\|\mathbf{x}\|_{p}:=
\begin{cases}
\displaystyle\left(\sum_{i=1}^{n}|x_i|^p\right)^{\!\! 1/p}, &\quad 1\le p<\infty \\
\displaystyle\max_{i}|x_i|, &\quad p=\infty.
\end{cases}
$$
The default is $p=2$, in which case you get the Euclidean norm.