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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

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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