Functions and Infinitesimals? My textbook says the following:


  $ads = v dv$


where $a$, $s$, $v$ are functions.

I was introduced to integral calculus a semester ago, and Riemann Sums (and integrals) are the only notion of multiplying functions and infinitesimals I'm familiar with.

In the above excerpt, $a$ is not necessarily a function of $s$ yet, you are multiplying it with an infinitesimal of $s$. Could someone give me an intuitive explanation of what it means when you multiply a function with with an infinitesimal of a variable that the function is not of?
IMO, thinking of differentials as infinitesimals is the wrong way to go: if you think of $s$ as a variable quantity, then you should be thinking of $ds$ as a gadget that represents how $s$ varies in different directions. As such, it makes perfect sense to multiply it by other variables, or to add two differential forms.

(don't try to multiply two differential forms: the meaning of that is tricky business, having to do with how things vary in a two-dimensional direction, and has odd properties like $dx dy = -dy dx$. It's not so bad once you're used to it, though; you'll probably be shown the basic idea when you encounter surface integrals)

"In what directions?" you might ask. "Any direction!" is the answer! The really neat thing about doing algebra with differential forms is that they don't really care about how many independent variables you have, or the shape of your parameter space or anything.

If you know your parameter space and have a direction to plug in -- e.g. in $3$-space with coordinates $x,y,z$, if you decide to look at motion in the direction of the vector $(1,2,0)$, you would plug in $dx = 1$, $dy=2$, and $dz=0$ to obtain an actual value for the "rate" at which various quantities are varying. Or if you decide you want to work on the unit sphere rather than all of $3$ space, you add the equation $x^2 + y^2 + z^2 = 1$ to relate your variables... and also the equation $2x dx + 2y dy + 2z dz = 0$ to relate your differentials.

Quick differential geometry aside! If we want to see how things vary in the direction $(1,2,0)$ and we stick on the unit sphere, we would get $2x \cdot 1 + 2y \cdot 2 + 2z \cdot 0 = 0$, or $2x + 4y = 0$. This tells us that the only points on the sphere where we can move in the direction $(1,2,0)$ -- i.e. the points to which $(1,2,0)$ is a tangent vector to the sphere, are the ones where $2x + 4y = 0$.

It is often simpler to use differential forms to relate rates of different quantities. e.g. if $a,v,s$ are all functions of some other variable $t$ -- e.g. $a = \alpha(t)$, $v = \nu(t)$, and $s = \sigma(t)$ -- then $da$, $dv$, and $ds$ are all multiples of $dt$. Specifically,

$$ da = \alpha'(t) dt \qquad dv = \nu'(t) dt \qquad ds = \sigma'(t) dt $$

and so we see how each of the variables $a,v,s$ vary as compared to how $t$ varies. We can even plug these into the original equation to get

$$ a \sigma'(t) dt = v \nu'(t) dt $$

and so everywhere in parameter space, we have either

$$ a \sigma'(t) = v \nu'(t) \qquad \text{or} \qquad dt = 0 $$

The first of these options can be written in the more suggestive form

$$ a \frac{ds}{dt} = v \frac{dv}{dt} $$

In fact, my first understanding of differential forms (long before I ever heard the term) was that things like $ds$ meant "the derivative with respect to some variable, but I haven't decided which, and I'll fill the same one in everywhere once I've decided". 

Be careful that you don't try to divide differential forms; doing such a thing only has a chance to be meaningful if you already know one differential form is a multiple of another, which is usually the case when you only have one independent variable. That usually doesn't happen when you have multiple independent variables; e.g. $dx/dy$ is total nonsense in the $xy$-plane.

(the partial derivative $\partial x / \partial y$ makes sense, but I really don't like doing multivariable calculus that way: I usually prefer to stick to differentials, and if I want to see how things vary while holding $x$ constant, I'll simply add in the equation $dx=0$)



Take notice that I said things like "variable" and "variable quantity" rather than "function". IMO, when using this notation style, it's a bad idea to think of your variables as secretly being functions -- doing so gets in the way of being able to use this style of notation fluently, and doubly so when you need to switch back and forth between this style and the style of writing everything as a function.

Even if $a$ is functionally dependent on $s$, it's not a good idea to think of $a$ as being literally a function. Instead, the dependence is expressed by saying that there is a function $f$ such that $a = f(s)$ is true. (and correspondingly, $da = f'(s) ds$)
So good that you asked.  It shows you are thinking, which has been known to improve one's mathematics.

There is not actually such a thing as multiplying by infinitesmals.  When it looks that way, it is always a shortcut for the chain rule.  David H. demonstrates it for this case very nicely in his answer.

You can see that you have to look for the common independent variable, and then for the relationships between the derivatives of the various functions with respect to that variable.

Notational shortcuts such as this one can be very helpful in simplifying computations, or in checking your work.You'll see this shortcut a lot when you take differential equations.

Here's another example: in a physics computation you may have an equation like

$a ft^2 * b (sec)/ ft^2$ = ab (sec). (I wrote "second" here as "sec" rather than the usual s, just for clarity). We know we are not "multiplying" these units of measure, but the notation is very handy.  It allows us to keep the units straight; also if you wind up in units of seconds and you were trying to compute  work, you know you made a mistake.

The main thing is to understand what the shortcut is, so you understand what you are doing.  Notation is to serve you, so you can write things any way that makes them clear to you, but of course you don't want to misuse it because you misunderstand what it means.