What is the interpretation of dx in calculus?

["Don't panic!"]

So are the functions $dx^{i}$ essentially the coordinate functions of the tangent space?
I guess all this approximation business has been confusing me. I've heard before that $df$ is the best linear approximation of $f$ at a particular point, but the "approximation" part is what's troubling me. In elementary calculus we say that $df$ is an infinitesimal change in $f$ near a point (which I know is not strictly correct) and this equals $\frac{df}{dx}dx$ (in one dimension), but is this essentially saying that $dx$ is the coordinate function of the tangent line to $f$, and so when $dx$ is evaluated on a particular tangent vector at that point $df$ quantifies how much $f$ f is changing in that direction (and at what rate it is doing so)?

 

[mathwonk]

"So are the functions dxi essentially the coordinate functions of the tangent space?" yes exactly.

 

["Don't panic!"]

So is the reason why we can qualitatively think of $dx^{i}$ as an infinitesimal change in the coordinate function $x^{i}$ at a particular point because $dx^{i}$ is only defined at that point, but encodes the information about how $x^{i}$ changes along all possible directions (encoded in the tangent vectors) as one "moves" through that point?

(P. S. If you have any good references on this particular discussion it'd be much appreciated).

 

[Fredrik]

So are the functions $dx^{i}$ essentially the coordinate functions of the tangent space?

Right. As you know, there's a theorem that says that if $v\in T_pM$ and $x$ is a coordinate system with p in its domain, we can write
$$v=v(x^i)\bigg(\frac{\partial}{\partial x^i}\bigg)_p.$$ This means that the ith component of v in the coordinate system x is $v(x^i)$. And the definition $(df)_p(v)=v(f)$ implies that
$$(dx^i)_p(v)=v(x^i).$$ So for each p in the domain of x, $(dx^i)_p$ takes each tangent vector at p to its ith component. In particular, we have
$$(dx^i)_p\bigg(\frac{\partial}{\partial x^j}\bigg)_p =\bigg(\frac{\partial}{\partial x^j}\bigg)_p x^i =(x^i\circ x^{-1})_{,j}(x(p))=I^i_{,j}(p)=\delta^i{}_j,$$ where $I$ is the identity map on $\mathbb R^n$. This means that $\big((dx^1)_p,\dots,(dx^n)_p\big)$ is the ordered basis dual to $\big(\big(\frac{\partial}{\partial x^1}\big)_p,\dots,\big(\frac{\partial}{\partial x^n}\big)_p\big)$.

I guess all this approximation business has been confusing me. I've heard before that $df$ is the best linear approximation of $f$ at a particular point, but the "approximation" part is what's troubling me. In elementary calculus we say that $df$ is an infinitesimal change in $f$ near a point (which I know is not strictly correct) and this equals $\frac{df}{dx}dx$ (in one dimension), but is this essentially saying that $dx$ is the coordinate function of the tangent line to $f$, and so when $dx$ is evaluated on a particular tangent vector at that point $df$ quantifies how much $f$ f is changing in that direction (and at what rate it is doing so)?

The term "infinitesimal" doesn't belong in a calculus book. In calculus, given a function $f:\mathbb R^n\to\mathbb R$, you can define a function $df:\mathbb R^{2n}\to\mathbb R$ by
$$df(x,h)=f_{,i}(x)h^i$$ for all $x,h\in\mathbb R^n$. This makes $df(x,h)$ the first-order approximation of the difference $f(x+h)-f(x)$. In differential geometry, we have the definition $(df)_p(v)=v(f)$ for all $v\in T_pM$. To see the connection between these two, let's use the differential geometry definition with the identity map as our coordinate system, on a function $f:\mathbb R^n\to\mathbb R$. We get
$$(df)_p(v)=v(f)=v^i \bigg(\frac{\partial}{\partial I^i}\bigg)_p f =v^i (f\circ I^{-1})_{,i}(I(p)) =v^i f_{,i}(p)=df(p,v),$$ where the df on the right is the one from calculus, and the v on the right is the n-tuple of coordinates (in the coordinate system $I$) of the v on the left.

 

["Don't panic!"]

Thanks for your descriptive explanations Fredrik.
My confusion initially arose from reading Spivak's book on differential geometry in which he says:

"Classical differential geometers (and classical analysts) did not hesitate to talk about ‘infinitely small’ changes $dx^{i}$ or the coordinates $x^{i}$, just as Leibniz had. No one wanted to admit that this was nonsense, because true results were obtained when these infinitely small quantities were divided into each other (provided one did it in the right way).
Eventually it was realized that the closest one can come to describing an infinitely small change is to describe a direction in which this change is supposed to occur, i.e., a tangent vector. Since $df$ is supposed to be an infinitesimal change of $f$ under an infinitesimal change of the point, $df$ must be a function of this change, which means that df should be a function on tangent vectors. The $dx^{i}$ themselves then metamorphosed into functions, and it became clear that they must be distinguished from the tangent vectors $\frac {\partial} {\partial x^{i}}$."

Now, I understand that the notion of an infinitesimally small number is nonsense, as if a number is infinitesimally close to zero then it should be equal to zero according to the definition of the limit, but I'm struggling to understand what $dx^{i}$ means intuitively in this new (more rigorous approach), and also what the intuition behind the definition $df(v) =v(f)$?

 

[lavinia]

Now, I understand that the notion of an infinitesimally small number is nonsense, as if a number is infinitesimally close to zero then it should be equal to zero according to the definition of the limit, but I'm struggling to understand what $dx^{i}$ means intuitively in this new (more rigorous approach), and also what the intuition behind the definition $df(v) =v(f)$?

$dx^{i}$ is the differential of the coordinate function $x^{i}$. It is no different than the differential of any function.

The differential of a function,df, is a linear function defined on vectors. The value of df on a vector v is df(v). One can also think of vectors as operators on functions. The value of v on a function is called v(f) and is evaluated as df(v). Operators on functions can be defined abstractly. One can show that if an operator satisfies certain conditions it is in fact a tangent vector.

 

["Don't panic!"]

dxi dx^{i} is the differential of the coordinate function xi x^{i} . It is no different than the differential of any function.

I'm confused about one should interpret a differential now though (especially after reading that paragraph given in Spivak's book). I'm confused as to whether one can interpret it as an infinitesimal change in a function anymore, or whether now $df$ denotes a new function which encodes how $f$ changes locally. As $df_{p}$ is defined at a given point then we can interpret it as describing an infinitesimal change in $f$ along all possible directions passing through that point. As such infinitesimal changes are induced by tangent vectors at that point, $df$ must be a function of tangent vectors?!

Should $df(v) =v(f)$ be interpreted as a differential change in $f$ along $v$ at a point, which is equal to evaluating the directional derivative of $f$ along $v$ (although how is it possible to equate a differential to a derivative)?

 

[lavinia]

I'm confused about one should interpret a differential now though (especially after reading that paragraph given in Spivak's book). I'm confused as to whether one can interpret it as an infinitesimal change in a function anymore, or whether now $df$ denotes a new function which encodes how $f$ changes locally. As $df_{p}$ is defined at a given point then we can interpret it as describing an infinitesimal change in $f$ along all possible directions passing through that point. As such infinitesimal changes are induced by tangent vectors at that point, $df$ must be a function of tangent vectors?!

As has been explained, today in mathematics, df is considered to be a function on tangent vectors.

Also as as been explained small increments in f along a curve with velocity vector,v, satisfy the equation

Δf = Δtdf(v) + small error term that goes to zero faster than the square of Δt.

If Δt is chosen small enough so that the error term is negligible -e.g. too small to be measured -then Δf is often written as df and is called an infinitesimal increment and Δt is written as dt. In this case one could write df = df(v)dt. Notice that the notation is being used in two ways. I think you are confusing the two. The notation df and dt is common in Physics books.

I don't know the history but I think that df was called an infinitesimal change in f because it represented an increment that could only be perfectly measured in the limit of no increment at all. I guess that Leibniz and others thought of this as an infinitesimal increment. Maybe they thought of df(v) as something real. Today we reformulate this idea by saying that v is a tangent vector and that df is a cotangent vector. Just as the infinitesimal increment of Leibniz could only be inferred from measurement and never directly observed, so tangent and cotangent vectors can not be directly observed since they exist in a separate space, the tangent space of the manifold.

I suppose one could use language that says that df is the infinitesimal tendency of f to change because it exists on its own ( as a cotangent vector in modern terms) but is not realized as a finite increment until f is evaluated along a curve.

 

["Don't panic!"]

If Δt is chosen small enough so that the error term is negligible -e.g. too small to be measured -then Δf is often written as df and is called an infinitesimal increment and Δt is written as dt. In this case one writes df = df(v)dt. Notice that the notation is being used in two ways. I think you are confusing the two

But isn't this appealing to the notion of an infinitesimal quantity again (by letting $\Delta t$ become really small)? Is the point that $df$ represents the first order change in a function near a point, and therefore, as when we are near that point this can exactly describe the change in the function, we can consider the linear change in $f$ as the infinitesimal change in $f$ at that point?!

Is Spivak basically saying that we can no longer consider infinitesimal changes in functions directly, but can do so indirectly by considering them as functions of infinitesimal changes (i. e. tangent vectors) along all possible directions at a particular point. In doing so the differentials of the coordinate functions $dx^{i}$ themselves become functions of tangent vectors, describing how the coordinate maps $x^{i}$ change along all possible directions at that point?!

 

[lavinia]

Is Spivak basically saying that we can no longer consider infinitesimal changes in functions directly, but can do so indirectly by considering them as functions of infinitesimal changes (i. e. tangent vectors) along all possible directions at a particular point. In doing so the differentials of the coordinate functions $dx^{i}$ themselves become functions of tangent vectors, describing how the coordinate maps $x^{i}$ change along all possible directions at that point?!

yes.

Along any curve one can ask what the coordinates are at any time. Thus one can ask what the change in coordinates is over a small time interval. There is no difference between this and asking what the change in any measurement is over a small time interval - e.g. the change in gravitational potential or the change in altitude - whatever.

 

[lavinia]

But isn't this appealing to the notion of an infinitesimal quantity again (by letting $\Delta t$ become really small)? Is the point that $df$ represents the first order change in a function near a point, and therefore, as when we are near that point this can exactly describe the change in the function, we can consider the linear change in $f$ as the infinitesimal change in $f$ at that point?!

In Physics books I have seen this. In this way of looking at things,df and dx are small increments- so small that the error term above the first order approximation can be ignored. I am not sure that Leibniz thought of it this way. I have not read his works but I do know that he was a philosopher and may have thought of df as an infinitesimal increment in a metaphysical way. No idea. I have read that non-standard analysis models his idea of infinitesimal but I know nothing about it.

We often talk of an object as having a speed. We say " The car was going 50 miles per hour when it crashed into the truck." This contains the idea that at any point in time the car actually has something called a speed. But what is this speed really? How can something have a speed in a instant of time? Don't we need to watch it move over a small time interval and calculate its average speed? In Mathematics, the answer is the the speed is a tangent vector - not directly observable because it doesn't exist in the manifold but exists in a separate space,in the tangent space.

 

["Don't panic!"]

Thanks for your help (and patience). I think I'm starting to understand it a bit better.

 

[lavinia]

Thanks for your help (and patience). I think I'm starting to understand it a bit better.

No Problem. It is good to struggle with these difficult ideas.

 

["Don't panic!"]

So, is the reason $df$ is called the differential of $f$ because it describes the linear change in $f$ near a point that has been obtained by differentiating $f$ at that point along a given vector, and thus measuring the local change in $f$ along that vector at that point. We can use this to approximate $f$ near this point as $f(x)\simeq f(p)+d_{p}f(v)$. As we get "nearer" to the point $p$ this approximation becomes more and more exact?!
Would it be correct to say that $dx^{i}_{p}(v)$ represents a finite quantity, which is the component of the tangent vector $v$ along the direction $\frac{\partial}{\partial x^{i}}$ (i.e. the amount of change that occurs in the coordinate map $x^{i}$ along $v$ at the point $p$)?!
Is the reason why such quantities can be considered differential changes in the corresponding function because that describe how it is changing at a specific point and not over some region (or interval)?Restricting to one-dimension for simplicity, is the function $df$ the change in the function describing the tangent line to the function $f$ at a given point; thus the differential is a function describing the change along the tangent line to $f$ at a given point (as we can no longer consider infinitesimals, the closest we can come to this notion is to describe the change along a particular direction at a given point, i.e. along the tangent line at that point)? Thus, locally to this point we can obtain a good linear approximation of $f$ by using this change along the tangent line to $f$.

 

[lavinia]

In one dimension df(∂/∂x) = df/dx. It is the ordinary derivative. It is the rate of change of f with respect to x.

When you evaluate df on a vector it is a rate of change - just like in one variable calculus.

An increment in f as a function along a curve, is close to Δtdf(v) for small Δt and Δf/Δt is close to df(v) so df(v) is just the limit of Δf/Δt as Δt goes to zero.
That is the whole story.

But all of this has already been explained.

 

["Don't panic!"]

But all of this has already been explained.

Yes, sorry I was trying to summarize what I'd gleamed from the discussion and to see if I'd understood it correctly. I think the block I'm struggling to get around in my brain, is that (in the past) I've always associated $df$ as describing an actual change in the function, but in differential geometry it seems to be that the derivative (rate of change of $f$) are merged into the same concept?!

Restricting to one-dimension for simplicity, is the function dfdf the change in the function describing the tangent line to the function ff at a given point; thus the differential is a function describing the change along the tangent line to ff at a given point (as we can no longer consider infinitesimals, the closest we can come to this notion is to describe the change along a particular direction at a given point, i.e. along the tangent line at that point)? Thus, locally to this point we can obtain a good linear approximation of ff by using this change along the tangent line to ff.

In this paragraph I was trying to justify in my mind (at least intuitively) why we still refer to $df$ as the differential of $f$ (which I've, incorrectly, always associated with an infinitesimal quantity) despite it being a finite quantity, and how it can still capture the notion of a change in $f$?!

Referring to Spivak's comments on the subject of infinitesimal changes in functions, is the point that we can no longer consider infinitesimals, however we can utilize the notion of rates of change in given directions (tangent vectors) to capture the notion of an infinitesimal change in a function at a point. We do this by noting that such a change should depend on which direction we consider and rate of change in that direction, i.e. it should depend on the tangent vectors at that point. Thus, $df$ maps tangent vectors at a point to real numbers which describe how $f$ is changing in a given direction at that point. Would this be a correct intuition at all?

 

[WWGD]

Yet another take : When f is differentiable at x df(x) is the change of the function f along the linear approximation to f (tangent line, plane, etc.), which is as good as you want ( in a rigorous $\delta - \epsilon$ sense when f is differentiable). So, e.g., for $f(x)=x^2$ , 2xdx gives you a(n) local approximation to the actual change of x^2 at $x= x_0$ (change along the tangent line) between two values of x close to each other. For any choice of $\epsilon$ you can find an interval $(x -\delta, x+\delta)$ where $| (x^2-x_0^2)-(2x(x-x_0))| < \epsilon$ for $|x-x_0| < \delta ; \delta >0$ .

 

[&quot;Don&#039;t panic!&quot;]

So is the point that $dx$ is a finite change along the tangent line to a function $f$ (at a given point $x_{0}$) and this $f(x_{0})+f'(x_{0})dx$ is the equation of the tangent line to the function $f$ at $x=x_{0}$. We can then use an $\epsilon -\delta$ limit approach to make this approximation approach the exact change in $f$ at $x=x_{0}$, such that $\lim_{x\rightarrow x_{0}}\left[(f(x)-f(x_{0}))-f'(x_{0})(x-x_{0})\right]=0$ (where $dx=x-x_{0}$). In other words if we construct a linearised function at a point that approximates $f$ locally, then $df$ is a change in this linear function. As such a linearised function is obtained through computing the derivative of $f$ at a given point, we thus call it the differential of $f$?!

Also, is any of what I put in my previous post correct?

(Sorry to bug everyone with this; I feel like I'm going round in circles at the moment :-( )

 

[WWGD]

Yes,df is finite I finite since you are dealing with continuous functions ( differentiable implies continuous) and it is in a sense the best linear approximation to the change of f locally, and the linearization is done through the derivative/gradient. Same applies to higher dimensions, e.g., you can approximate the change of the function locally in the same sense by studying the change of f at (x,y) along the tangent plane at (x,y). .