Reconciling Differences in Vector-Valued Function Derivatives

In summary, there are apparent differences between the definitions of the derivative of a vector-valued function f: U \rightarrow \mathbb{R}^n from the textbooks of Andrew Browder and Susan Jane Colley. Browder only uses modulus signs around h in the denominator while Colley uses norm signs for both the numerator and denominator terms. However, both notations ultimately mean the same thing for vectors. It is important to note that the meaning of the double lines $\|\cdot\|$ can change if it has a subscript, as shown in the example provided.
  • #1
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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
 
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  • #2
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
 
  • #3
Peter said:
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.
 

Related to Reconciling Differences in Vector-Valued Function Derivatives

1. What are vector-valued function derivatives?

Vector-valued function derivatives are derivatives of functions that have multiple outputs, or vectors, rather than just a single output. They are used to describe how the outputs of a function change with respect to the inputs.

2. Why is it important to reconcile differences in vector-valued function derivatives?

Reconciling differences in vector-valued function derivatives is important because it ensures that the derivatives are consistent and accurate. This is crucial for making accurate predictions and calculations in fields such as physics, engineering, and economics.

3. What are some common differences that need to be reconciled in vector-valued function derivatives?

Some common differences that need to be reconciled in vector-valued function derivatives include differences in notation, differences in the order of the derivatives, and differences in the way the derivatives are calculated.

4. How can differences in vector-valued function derivatives be reconciled?

Differences in vector-valued function derivatives can be reconciled by using the chain rule, product rule, and quotient rule, which are all methods for finding derivatives of composite functions. Additionally, using standard notation and double-checking calculations can help ensure consistency.

5. What are some potential challenges in reconciling differences in vector-valued function derivatives?

Some potential challenges in reconciling differences in vector-valued function derivatives include complex functions with multiple inputs and outputs, inconsistent notation conventions, and human error in calculations. It is important to carefully check and double-check all calculations to ensure accuracy.

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