8.12 in Browder's "Mathematical Analysis: An Introduction"

[Math Amateur]

I am reader Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...

I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ...

I need some help in fully understanding Proposition 8.12 and the notes leading up to it ... ...

Proposition 8.12 and the definitions, remarks and propositions leading up to it read as follows:View attachment 7457
View attachment 7458
Now in the above notes from Browder:

\(\displaystyle \text{df}_p\) is the differential (derivative) of \(\displaystyle f\) at \(\displaystyle p\) and denotes (is) the linear map \(\displaystyle L\)

and

f ' (p) is the matrix of the linear map with respect to the standard basis My question relates to the notation (and more importantly the meaning) of \(\displaystyle \text{df} (h)\) in Proposition 8.12 ...... what exactly is \(\displaystyle \text{df} (h)\) ... ? What is its meaning in geometric terms? Has Browder left off the subscript p ... ?
It appears that \(\displaystyle \text{df} (h)\) is \(\displaystyle h\) mapped under the matrix \(\displaystyle df\) ... but evaluated at \(\displaystyle p\) ... ? How do we interpret this ...

and

... how is \(\displaystyle \text{df} (h)\) related to the differential (total derivative) \(\displaystyle \text{df} (h)_p\) ...?
Hope someone can clarify the above ...

Peter

 

[GJA]

Hi Peter,

... what exactly is \(\displaystyle \text{df} (h)\) ... ?

At each point $p$, $df_{p}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ is a linear operator/mapping. There is no difference between $df_{p}(h)$ here and $Tv$ ($df_{p}=T$ and $h=v$ <-- Thank you to Krylov for noting that this should be $v$ not $p$) that we were discussing in your other post today, other than before $T$ was a fixed linear mapping and now the linear mapping is allowed to change from point to point, as indicated by the subscript $p$. This is just like single variable calculus where $f(x)\Longrightarrow df_{x}(dx)=f'(x)dx.$ Here $dx$ is the independent variable and corresponds to $h$ above. $f'(x)$ corresponds to the matrix representation of the linear operator $df_{x}$ that we multiply $dx$ by to compute the output of $df_{x}$ applied to $dx$. Finally, the subscript $x$ indicates that the linear operator (and therefore its corresponding matrix representation, $f'(x)$) is allowed to change/vary from point to point.

What is its meaning in geometric terms?

Geometrically $df_{p}$ maps vectors in $\mathbb{R}^{n}$ based at $p$ (i.e., that emanate from $p$ = whose tails are based at $p$) to vectors in $\mathbb{R}^{m}$ based at $f(p)$, and does so in such a way that it is the best linear approximation to the function $f$ at the point $p$. In other words, if you had to pick a linear function that most closely mimics the behavior of $f$ at the point $p$, you would pick $df_{p}$.

Has Browder left off the subscript p ... ?

Yes, this is common and is not considered an abuse of notation. This is similar to writing $f'$ instead of $f'(x)$.

It appears that \(\displaystyle \text{df} (h)\) is \(\displaystyle h\) mapped under the matrix \(\displaystyle df\) ... but evaluated at \(\displaystyle p\) ... ? How do we interpret this ...

It is true that there are two variables here, but thinking of them in the correct order should help clear things up. You start by choosing/fixing a value for $p$. This then fixes the linear operator $df_{p}$. The domain of this linear operator is now all vectors $h$ based at $p$ in $\mathbb{R}^{n}$.

... how is \(\displaystyle \text{df} (h)\) related to the differential (total derivative) \(\displaystyle \text{df} (h)_p\) ...?

I did not see where the author wrote $df(h)_{p}$ anywhere. At any rate, $df(h)$ and $df_{p}(h)$ are the same thing. One has the $p$ dependence suppressed and the other does not.

 

[S.G. Janssens]

There is no difference between $df_{p}(h)$ here and $Tv$ ($df_{p}=T$ and $h=p$) that we were discussing in your other post today, other than before $T$ was a fixed linear mapping and now the linear mapping is allowed to change from point to point, as indicated by the subscript $p$.

I think that in the above sentence you meant that $h$ corresponds to $v$, not to $p$?

In fact, I agree with most of what GJA wrote, but I also have some remarks.

To me, the formulation of Prop. 8.12 is not very good. If I were to learn this material for the first time, I would be confused. "To be evaluated at $\mathbf{p}$" indeed wrongly suggests somehing like $d\mathbf{f}(\mathbf{h})_{\mathbf{p}}$. I do find it abuse of notation to write $d\mathbf{f}(\mathbf{h})$ when $d\mathbf{f}_p(\mathbf{h})$ (or: $d\mathbf{f}_p\mathbf{h}$, since the derivative at a point is a linear map) is meant, just as I find it abuse of notation to write $f'$ instead of $f'(x)$.

This abuse of notation is indeed fairly common, but it can easily lead to misunderstandings, particularly when differentiating mappings between function spaces.

 

[GJA]

I think that in the above sentence you meant that $h$ corresponds to $v$, not to $p$?

You are most certainly right, Krylov! Thanks for spotting that typo.

 

[Math Amateur]

You are most certainly right, Krylov! Thanks for spotting that typo.

Thanks to GJA and Krylov for their explanations and thoughts ...

... still reflecting on what you have written ...

Peter