Why is Mu Differentiable in McInerney Example 3.1.5?

[Math Amateur]

I am reading Andrew McInerney's book: First Steps in Differential Geometry: Riemannian, Contact, Symplectic ... and I am focused on Chapter 3: Advanced Calculus ... and in particular on Section 3.1: The Derivative and Linear Approximation ...

In Section 3.1 McInerney defines what is meant by a function $f: \mathbb{R}^n \to \mathbb{R}^m$ being differentiable and also defines the derivative of f at a point $a \in \mathbb{R}^n$ ...

... see the scanned text below for McInerney's definitions and notation ...

... McInerney then gives several examples ... I need help with several aspects of Example 3.1.5 which reads as follows:

I have two questions with respect to Example 3.1.5 ...Question 1

In the above text from McInerney we read the following:

"... ... Then $\mu$ is differentiable for all $a = (a_1, a_2) \in \mathbb{R}^2$ and $D \mu (a_1, a_2) (h_1, h_2) = a_1 h_2 + a_2 h_1$ ... ..Can someone explain exactly how/why ...

(a) $\mu$ is differentiable for all $a = (a_1, a_2) \in \mathbb{R}^2$ ... ...

and ...

(b) $D \mu (a_1, a_2) (h_1, h_2) = a_1 h_2 + a_2 h_1$ ... .. that is how/why is this true ...

(especially given that McInerney has only just defined differentiable and the derivative!)Question 2

In the above text from McInerney we read the following:

"... ... $\displaystyle \lim_{ \mid \mid h \mid \mid \to 0} \frac{ \mid h_1 h_2 \mid }{ \sqrt{ h_1^2 + h_2^2 } } = 0$... ...Can someone please show and explain exactly how/why it is that $\displaystyle \lim_{ \mid \mid h \mid \mid \to 0} \frac{ \mid h_1 h_2 \mid }{ \sqrt{ h_1^2 + h_2^2 } } = 0$ ... ...
Help will be much appreciated ... ...

Peter============================================================================================================================

It may help members reading the above post to have access to the text at the start of Section 3.1 of McInerney ... if only to give access to McInerney's terminology and notation ... so I am providing access to the text at the start of Section 3.1 ... as follows:

Hope that helps ...

Peter

 

[andrewkirk]

For question 1, just interpret the colon ":" as saying "because". The answer to Question 1 is what comes after the colon. The function $T_\mathbf a$ maps $\mathbf h=(h_1,h_2)$ to $a_1h_2+a_2h_1$. What comes after that shows that the required limit condition in Definition 3.1.1 holds.

To answer Question 2, write $\mathbf h$ in polar coordinates as $(r\cos\theta,r\sin\theta)$. Then the denominator of the ratio is $\|\mathbf h\|=r$. What is the numerator? What can we conclude about what the ratio does as $r\to 0$?

 

[Ray Vickson]

I am reading Andrew McInerney's book: First Steps in Differential Geometry: Riemannian, Contact, Symplectic ... and I am focused on Chapter 3: Advanced Calculus ... and in particular on Section 3.1: The Derivative and Linear Approximation ...

In Section 3.1 McInerney defines what is meant by a function $f: \mathbb{R}^n \to \mathbb{R}^m$ being differentiable and also defines the derivative of f at a point $a \in \mathbb{R}^n$ ...

... see the scanned text below for McInerney's definitions and notation ...

... McInerney then gives several examples ... I need help with several aspects of Example 3.1.5 which reads as follows:
View attachment 240802
I have two questions with respect to Example 3.1.5 ...Question 1

In the above text from McInerney we read the following:

"... ... Then $\mu$ is differentiable for all $a = (a_1, a_2) \in \mathbb{R}^2$ and $D \mu (a_1, a_2) (h_1, h_2) = a_1 h_2 + a_2 h_1$ ... ..Can someone explain exactly how/why ...

(a) $\mu$ is differentiable for all $a = (a_1, a_2) \in \mathbb{R}^2$ ... ...

and ...

(b) $D \mu (a_1, a_2) (h_1, h_2) = a_1 h_2 + a_2 h_1$ ... .. that is how/why is this true ...

(especially given that McInerney has only just defined differentiable and the derivative!)

You say you cannot see why $D \mu (a_1, a_2) (h_1, h_2) = a_1 h_2 + a_2 h_1.$ Have you looked at $(a_1+h_1)(a_2+h_2) - a_1 a_2?$ Have you looked at what you get when you keep terms that are of first-order only in $h_1, h_2?$

 

[fresh_42]

Maybe it helps to consider the following. Let $f \, : \,\mathbb{R}^n \longrightarrow \mathbb{R}^m$ be a differentiable function. Then the derivative $D_af$ at $x=a$ in direction $v$ which is a linear function in $v$ satisfies the following equation:
$$
f(a+v) = f(a) + (D_af)(v) + r(v)
$$
where $r(v)$ is an error function with the property $\lim_{v \rightarrow 0}\frac{r(v)}{||v||}=0$, i.e. it converges faster to zero than linear.

Try to write your function in this form.

 

[Math Amateur]

Thanks to all who responded ...

Most helpful ...

Now ... just a followup question ... how would you determine/calculate the derivative of $\mu$ ... in an efficient way ... indeed, what formula would you use ... ?

Peter

 

[Math Amateur]

After reflecting on my last post ... I thought I would try to answer my own question ...

So ... we have $\mu(x, y) = xy$

and

$D \mu ( a_1, a_2) = ( D_1 \mu ( a_1, a_2), D_2 \mu ( a_1, a_2) )$

$= \left( \frac{ \partial \mu}{ \partial x } ( a_1, a_2) , \frac{ \partial \mu}{ \partial y } ( a_1, a_2) \right)$

$= ( y \mid_{ ( a_1, a_2) }, x \mid_{ ( a_1, a_2) } )$

$= (a_2, a_1 )$

and so we also have

$D \mu ( a_1, a_2) \left( \begin{array}{cc} h_1 \\ h_2 \end{array} \right)$

$= a_2 h_1 + a_1 h_2$

Is that correct?

Peter

 

[andrewkirk]

After reflecting on my last post ... I thought I would try to answer my own question ...

So ... we have $\mu(x, y) = xy$

and

$D \mu ( a_1, a_2) = ( D_1 \mu ( a_1, a_2), D_2 \mu ( a_1, a_2) )$

$= \left( \frac{ \partial \mu}{ \partial x } ( a_1, a_2) , \frac{ \partial \mu}{ \partial y } ( a_1, a_2) \right)$

$= ( y \mid_{ ( a_1, a_2) }, x \mid_{ ( a_1, a_2) } )$

$= (a_2, a_1 )$

and so we also have

$D \mu ( a_1, a_2) \left( \begin{array}{cc} h_1 \\ h_2 \end{array} \right)$

$= a_2 h_1 + a_1 h_2$

Is that correct?

Peter

Yes that is correct.

It helps to nail down some terminology. I think 'differential' is used for the linear map $T_\mathbf a:\mathbb R^n\to\mathbb R^m$, which we can call the 'differential' of $\mu$ at $\mathbf a$.
'Derivative' is used for the map from $\mathbb R^n$ to $S$, where $S$ is the set of linear maps from $\mathbb R^n$ to $\mathbb R^m$, which maps each point $\mathbf a\in\mathbb R^n$ to the differential of $\mu$ at that point, ie to $T_\mathbf a$.

The method for calculating the differential at $\mathbf a$ is to calculate the Jacobian matrix of the function $\mu$ at $\mathbf a$. The differential is the linear map represented by that matrix. that is what you have done in your latest post above. The wiki link shows you how to calculate the Jacobian matrix in the general case. Note the para that starts with "The Jacobian matrix is important", which explains how the linear map associated with the matrix is the $T_\mathbf a$ that your text writes about.

No doubt your text will get on to discussing Jacobians a little further along.

 

[mathwonk]

Perhaps "differential" is more common today, (or at least it is familiar to me, since I learned the subject from Lynn Loomis), but the usage has evolved over several decades. In the book by Dieudonne' from 1960, one of the first where this abstract approach was introduced in the US the term was "derivative", and elsewhere (especially for Banach spaces), it was known as the "Frechet derivative", but in the slightly later book of Loomis and Sternberg in 1968 it was already called the "differential". Still in Spivak's Calculus on Manifolds in 1965 it was called derivative, p.16. Also in Lang's Analysis I, in 1968, it is called also the derivative, p. 303, as also in Guillemin and Pollack, 1974, p. 8 of Differential Topology. in John Lee's book Intro to Smooth manifolds, in the context of a manifold, on p. 46 he calls this linear map a "push forward", (acting on derivations). I have also heard the lin ear map calle the "total derivative", as contrasted with "partial derivatives". In coordinates, the total derivative is the linear map whose matrix has the partial derivatives as entries.

I have also heard dF called the differential of F, and dFa called the diffrential of F at a. Calling one of these the differential and the other the derivative is new to me.

In Spivak's Introduction to Differential Geometry, in chapter 7 he uses the term "differential" rather for the concept of "exterior derivative", taking a k form to a (k+1) form. Here the word differential is also apparently used for the process of taking a differential, i.e. the word seems to apply just to d, and not only to a particular dF.

So terminology is not entirely universal, since it depends on where you learned it first.

 

[fresh_42]

I think differential is a consequence of physicists' language: something with $dx$ at the end.

We use a separate word (Ableitung) which is literally derivative or derivation. The words derivative and derivation themselves are reserved to a certain chemical process, resp. a map $(fg) \longmapsto D(f)g+fD(g)$.

The distinction between $p \longmapsto D_p(f)$ and $v \longmapsto D_p(f)(v)$ is rarely made. I think it also depends in English as well much more on the author as there is a clear distinction of the two. To name the first a derivative and the second a differential seems to me more of a holy wish than an actual fact.

 

[mathwonk]

Apparently it was in Courant's Differential and Integral Calculus vol 2, p. 66, that I encountered the term "total differential". Since this book was apparently translated from the German, around 1936 perhaps, that term would presumably be in existence there. Since also Courant and Hilbert wrote Methods of Mathematical Physics (also in German), they were also steeped in physics.

 

[fresh_42]

I only have Courant - Hilbert: Methods of Mathematical Physics Vol.1 which doesn't have either term in the index. Within the book they are right in the middle of functional analysis after some linear algebra. So they are directly dealing with integration, and as a measurement, not to reverse differentiation. The word differential occurs probably the first time as part of differential equation. I couldn't find neither differential nor derivative nor Ableitung, and it seems they handle everything with integrals. The closest term I saw was $G(x,y,z,x',y',z')=0$ referred to as differential expression. This leads to the assumption that in German it is simply Ableitung and differential the according adjective. Derivativ or Derivat is widely used in all kind of areas to name the result of a derivation, e.g. in finance.

Dieudonné's book about mathematical history (German ed.) isn't of much help either, as it uses Ableitung and quotes Cauchy as reference (1823, Résumés ... sur le calcul infinitésimal). The equation $f(p+v)=f(p) + D_p(f)(v) + r(v)$ was apparently introduced by Weierstraß (1864) and thus again Ableitung.

All other constructions as differential equations, diffeomorphism, or differential forms are differential here as well. I still think that the differential is primarily used by physicists who need basis vectors for their cotangent vector space and want to avoid or abbreviate the term differential forms. Derivative sounds natural in a language which calls integrals antiderivatives. In any case, I couldn't find a formal distinction between derivative ($D.f$) and differential ($Df.$) anywhere.

The German Wikipedia offers Differentialquotient as alternative, and the French writes

La dérivée d'une fonction $f$ est une fonction qui, à tout nombre pour lequel $f$ admet un nombre dérivé, associe ce nombre dérivé. La dérivée en un point d'une fonction de plusieurs variables réelles, ou à valeurs vectorielles, est plus couramment appelée différentielle de la fonction en ce point, et n'est pas traitée ici.

The derivative of a function f is a function which, at any number for which f admits a derived number, associates this derived number. The one-point derivative of a function of several real variables, or vector-valued variables, is more commonly referred to as the differential of the function at this point, and is not dealt with here.

What a great hint - a third convention: Differential in the one dimensional case and derivative in general!

 

[mathwonk]

Interesting. I looked also in Goursat, Course in Mathematical Analysis, from 1902, translated (presumably from French) by Hedrick in 1904, and found the term differential for the object assopciates to f and represented as ∂f/∂x dx + ∂f/∂y dy, but rigorously defined. Thus it is displayed as a linear combination of the variations dx and dy, hence visibly a linear function, but without using that language. i.e. it is defined there as the "principal part" of the function f, which is a certain linear approximation to it, differing from f by an "infinitesimal", which is defined much as Loomis does.