Utility of the coordinate free definition of the derivative on manifolds.

Question

Preface: I am not an expert on the topic of smooth manifolds, nor do I have the perspective gained from knowing many theorems proven on smooth manifolds. Please try to look at the problem from the perspective of someone learning this topic for the first time.

For future reference let $f:M \rightarrow N$ be a smooth function where $M$ and $N$ are smooth manifolds. Let the derivative of $f$ at $p\in M$ be denoted as the usual $df_p:T_pM \rightarrow T_{f(p)}N$.

Question:

In what ways is it advantageous to define $df_p$ in a coordinate free way? I cannot see the motivation behind such a definition. To me there are three defining characteristics of the derivative at a point of a function between $\mathbb{R}^n$ and $\mathbb{R}^m$.

1. The derivative of a function offers an approximate relationship between perturbations of the input and corresponding perturbations of the output.

2. This relationship is linear.

3. This approximation is better than any other linear approximation.

Let's check how many of these properties actually survive the abstraction process needed for a coordinate free generalization of the derivative on manifolds.

Property 1: To interpret a tangent vector as a "perturbation" of the point p, we would need some sort of canonical mapping from $T_pM$ to $M$. I do not know of any such mapping which doesn't rely on a local coordinates. Since we can't interpret the inputs and outputs of $df_p$ as perturbations, I assume that the notion of local approximation is abandoned as well. How could $df_p$ "approximate" $f$ when we don't even have a way to relate their respective domains and ranges. Thus I will assume that this property of the derivative has been stripped away in the abstraction process.

Property 2: This is satisfied.

Property 3: Since we have determined that without coordinates $df_p$ does not act as an approximation of $f$, it cannot be the "best" approximation.

Thus only one property survives. $df_p$ seems to be (to my unexperienced eyes) a glorified linear mapping with no substantial importance until local coordinates are introduced. Thus why don't we simply define the derivative in a coordinate dependent manner? This would also make the definition of the tangent space much simpler. The "geometric tangent space", $\mathbb{R}^n_p$, as defined in Lee's "Smooth Manifolds" would suffice as a definition since we are assuming some coordinate representation. Is anything lost by doing this?

In response to nicrot000’s answer:

To summarize, nicrot000 states in his answer that it is the general trend of differential geometry to express concepts in coordinate free ways. This allows said concepts to be independent of arbitrary decisions (i.e. choice of local coordinates). /end of summary\

I guess my question is rooted in the fact that I’m not convinced arbitrary decisions are a bad thing in terms of defining the derivative. From my perspective, $df_p$ can only be interpreted in some meaningful way after local coordinates have been assumed.

An analogy:

Suppose we have a computer which we will refer to as the “idea machine”. The sole purpose of this computer is to run a program which, on command, generates a random idea. After you have clicked “generate idea”, the program will prompt you to choose from a set of languages (all of which can be “smoothly” translated between each other) in which the idea will be displayed on the screen (the choice of a language corresponds to choosing a local coordinate chart). There is however the option to choose no language (corresponding to coordinate free). If no language is chosen, nothing will be displayed on the screen and the idea will forever stay encoded by the ones and zeros in the hardware. We do however know that the idea exists independent of our arbitrary choice of language (hooray!). At the end of the day, if we want to know what the idea was then in some form or another we would have to use a “language” to interpret the ones and zeros in the hardware.

Interpretation of the analogy:

The idea produced by the idea machine represents the information encoded by the coordinate free function $df_p$. The set of languages which one can choose from represents the smooth structure. Choosing a language to display the idea corresponds to choosing a coordinate chart to represent $df_p$. We can choose to use a coordinate free representation however I do not see the purpose of this just as choosing no language is useless. From a developer standpoint, why would we go through the work of adding the no language option? i.e. why would we go through the work of developing a coordinate free definition of $df_p$.

Answer 1

The importance comes in when you replace your spaces by manifolds which are not (open subsets of) $\mathbb{R}^n$. Then, you are by definition of a manifold always able to choose a local coordinate system in order to represent the derivative of $f$ as a matrix, but none of these coordinate systems is distinguished and the descriptions in any two of these are equivalent. So the general agenda in differential geometry is to define any maps/objects in a coordinate free manner so that they depend merely on the space and not on a local coordinate system.

$\bf{Edit~on~your~edit:}$

I can join in with the comments on you "no-language" analogy, but I guess I know what you mean by it.

Note that you can always choose a local chart on any manifold. Indeed, physicists do this all the time. As soon as you open a physics book on relativity you will notice a lot of Greek indices doing precisely what you want to do and any equation for vector fields/tensorial objects is actually an equation for their components. Usually you find something like "Let $x^\mu$ be a vector (-field), then [sth.]". In fact, this approach is very suitable for performing lengthy computations and physicist have a remarkable intuition in dealing with such equation. The downside from the math point of view: You run into some squishy definition like "A vector is something which transforms like a vector", "a tensor is something which transforms like a tensor" and so on.

Mathematicians, on the other hand, prefer being a bit more precise. So if you want to formulate something something like "Let $x^\mu$ be a vector (-field), then [sth.]" as above, I will tell you that $x^\mu$ is not a vector field. Then you will start to specify chart domain and expand the vector field of interest in the local frame of coordinate vector field and then you can finally speak of $x^\mu$ and formulate equations with it. But this will be the very same formulation every time, over and over again, and mathematicians don't like to repeat themselves if it is not necessary. If your objects become more and more complicated, this can get very messy although it contains no relevant information.

Finally, from my own experience as having studied both math and physics, I always had the impression of gaining a way deeper understanding of whats going on from the math point of view than from the physics view. In particular, always being aware what your objects are of course helps a lot. For example, take the formula $$ D_\mu=\partial_\mu-ig~T^aA_\mu^a $$ for the so-called covariant derivative (which I just took from the Wiki of Yang-Mills theory) and try to figure our what kind of map it actually is, which kind of functions it can differentiate. After having a few courses of quantum field theory and theoretical particle physics I wasn't able to tell you. We were just told that you need to subtract this term from each derivative in order to restore gauge invariance, and then, in the usual physicists manner, you get used to it and can do any computation and therefrom somehow "feel like" understanding it. On the other hand, there is a fully worked out mathematical theory on what this thing is, why in any coordinate chart it has to be of that form and why it has to be introduced in order to obey very fundamental physical principles. At least I, and I guess I speak for a few others as well, had the impression to understand a lot more while studying gauge theory (for example) from the mathematical perspective.

So concluding, your suggestion works (if carried out carefully) and is indeed practiced a lot, but for mathematicians, which are more interested in gaining a deep understanding of a theory over a "effective formalism", the coordinate-free way is the way to go (in my opinion).

Answer 2

In addition to nictor000's answer, let me address your quarrels with Property 1.

Indeed, there is no canonical map $T_pM\to M$, but there is something that takes its place. For any $v\in T_pM$ there is a curve $\gamma\colon \mathbb{R}\rightarrow M$ with $\gamma(0)=p$ and $\dot \gamma(0)=v$ and you can view this as perturbation of the point $p$ in direction $v$. Then $g=f\circ \gamma \colon \mathbb{R}\rightarrow \mathbb{R}$ is a smooth map with derivative $g'(0)=df_p(v)$ and so the interpretation of the differential as best linear approximation carries over, if you restrict $f$ to the curve $\gamma$. Of course, given $v$, the curve $\gamma$ is not unique --- but that is not a problem, the preceding holds true for any such curve $\gamma$.

Answer 3

If your properties 1. and 3. did not "survive the abstraction process", and if coordinates somehow magically resolve them, that just means we don't quite understand what's going on conceptually and are sweeping it under the coordinate carpet. I suggest we look at the double "smooth" in the definitions:

...let $f:M \rightarrow N$ be a smooth function where $M$ and $N$ are smooth manifolds. Let the derivative of $f$ at $p\in M$ be denoted as the usual $df_p:T_pM \rightarrow T_{f(p)}N$. ...we have determined that $df_p$ does not act as an approximation (!?) of $f$,

Smooth manifolds can locally at each point's infinitesimal vicinity be approximated by a linear tangent space. That's where we then approximate a smooth $f$ by $df$.

Answer 4

A coordinate free approach is important when dealing with infinite dimensional manifolds. In infinite dimensions, there is typically no natural coordinate system to work with.

You might think this is a niche topic because we live in a finite dimensional world. However, generally a space of functions that maps between two finite dimensional manifolds is an infinite dimensional manifold.

So, even when you start with finite dimensional manifolds, infinite dimensional manifolds arise naturally. This is in keeping with the modern trend in all of mathematics towards studying objects by instead studying spaces of functions that map between those objects.