Differential notations

Before I lose track of my own thoughts, I'll put down the general question here: What is the intuition behind the notation "dx"?

Okay, a few things I've picked up on my own:

1/ Differentiating a function is analogical to analyzing the function and outputting its general tendency of change. Since there are many components in a single identity (e.g: 3 spatial dimensions for any physical objects), there are different "perspectives" along which the thing in question can change. That's why we need to note which perspective of change we are studying by appending "dx" and the like, a.k.a "with respect to x (the x direction)".
Thus: d/dx f(x, y, z, ...)

2/ In calculating the area under the curve representation of a function, we treat dx as a very small change in x (infinitely small rectangle width) so that the height of the rectangle can take on a spontaneous value given by the function (the variable can't be assigned a value otherwise, as it changes with respect to x's changes) and thus making it possible to find the area of the rectangle (which is also infinitely small).
And then, we sum up an infinite number of those tiny rectangles to get the area under the curve, naturally, they lie along the "x direction".

Is there any, say, rigorous definition for dx? As we can see, in case 1/, dx simply is a notation to literally say "hey, we're finding the rate of change of this fucker in this direction", but in case 2/, dx actually takes on a value, a physical entity.

dx is not really a single entity. It is usually treated like one, but d is really just an operation performed on x. d means "difference". This definition holds in both (1) and (2).

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First, let's consider four different kinds of "d" 's:

1. d - This means "infinitesimal change in". This is used whenever the function being operated on is completely in the dimension of the operation. For example, f(x) is a function entirely of x, so d/dx f(x) and ∫ f(x) dx are acceptable notations.

2. ∂ - This also means "infinitesimal change in". However, the difference from the regular d is that ∂ is used only partially. In other words, it's used whenever the function isn't entirely in the same variable. For example, when f(x,y) is operated on, it's custom to use ∂/∂x f(x,y) to specify that the function isn't entirely in the x dimension. y is treated as a constant when this computation is done.

3. δ - This is the lowercase Greek letter, "delta", and it also means "infinitesimal change in." This applies more to chemistry related thermodynamic equations, and it arises from the product rule:

W = -PV
d(W) = -PdV -VdP

When only one of the two components are taken, δ is used to denote that the other component is "missing". For example, it's customary to write δW = -PdV.


For 1., 2., and 3., the meaning of "infinitesimal change in" can be seen from this formula for computing the change in f(x) per unit x:

df(x)/dx = lim (f(x+h)-f(x))/h as h approaches 0

This is a rudimentary method of finding a derivative. The limit produces a function that measures the "slope" of the (x,f(x)) plane. In other words, it measures the greatest rate of change at any point of x. To obtain a better picture of the meaning of d, df is just the above limit multiplied by dx, where:

dx = lim (x+h)-x as h approaches 0

Really this is just 0, and multiplying that limit by 0 just returns a 0/0 is an undeterminate form, and l'Hopital's rule needs to be applied among other considerations. This seems somewhat circular (and I'm not even sure that it works), but this is just an illustration of the meaning of d.

4. Δ - This is the uppercase Greek letter, "delta". This means the "total change in". For example, Δx = x1-x0. To illustrate what I mean, here's an example of its usage:

∫ f '(x) dx [x0,x1] = Δf(x) = f(x1)-f(x0)

This "total difference" is what distinguishes Δ from d or ∂.

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In your brainstorming, section (1), you're questioning the notation of ∂/∂x f(x,y,z,...)

[∂/∂x](f) is the same thing as ∂f/∂x. The top computes the infinitesimal difference in f, and the bottom computes the infinitesimal difference in x. The division shows that this partial derivative is some kind of slope. In this case, it produces the x component of a gradient vector ∇f, where ∂f/∂x is the line in slope of the greatest rate of change along x at a point. There's no violation of this "difference in" definition.

 More on ∇

∇, the Greek letter "nabla", stands for "del", which is another operation performed similarly to ∂, but instead, it takes the scalar function f and produces a vector field indicating the greatest rate of change. The formula for this is:

∇f(x,y,z) = (∂f/∂x, ∂f/∂y, ∂f/∂z)

What's meant by the "greatest" rate of change is that on a surface, it considers the direction in which all the x,y, and z components are changing, since altering one of these would produce a lesser rate of change of the function. In this case, f is also called the scalar potential function. It's called a scalar because it returns a single value as the solution, and it's called a potential because it gives rise to a new function that have supposedly influenced the individual components x, y, z. In other words, ∇f(x,y,z) alters x, y, and z in some way, and f summarizes ∇f(x,y,z).

∇f indicates the direction of the greatest rate of change. Like in functions of two dimensions, differentiation produces a tangent. For a surface, it produces a tangent plane. Note, however, that all the vectors produced from ∇f are normal to the surface, or perpendicular. This is because it's customary to write planes based on their normal vectors, since only one vector and its scalar multiples are unique to a plane. For any vector that's not normal, there can be infinitely many solutions found by rotating the vector about its tail.

In your second condition, you're questioning ∫ f '(x) dx. I hope you've seen something like this before:

∫ f(x) dx = ∑ f(x) dx

In other words, the integral just means "sum up all of f along x". For the single variable function f(x), this just means the area under the curve. However, it doesn't have to be just this. It's also possible to take many other integrals.

 Here is a list of many types of integration

Double integral for taking volumes:

∫ f(x,y) dA = ∫∫ f(x,y) ∂y ∂x

Triple integrals for summing up values associated with every point in a three-dimensional figure. (For example, the temperature at every point in a room.)

∫ f(x,y,z) dV = ∫∫∫ f(x,y,z) ∂z ∂y ∂x

Scalar line integrals (summing up a density function along a line)

∫ ρ(x,y,z) ds

ds is not even a linear function. It's the length computed with the Pythagorean theroem over a fictitious quantity dt. (For example, if f is a function of x, y, and z, then ds is sqrt(x(t)^2+y(t)^2+z(t)^2) dt, and ρ(x,y,z) will have to be reparametrized to this fictitious quantity.)

Vector line integrals (summing up the vectors along a line)

∫ F(x,y,z) • T ds

Scalar surface integrals (summing up surface area along a surface)

∫∫ f(x,y) ds

Vector surface integrals

∫∫ f(x,y) • e_n ds

And whatever else you want.

It is clear from all above types of integration that the "d"s are detached from whatever follows it, such as the x in dx, and that they always means "infinitesimal difference in". The only one to be careful of is Δ, which is different from first three types of "d"s.

Well, I was aware of the meaning of d, which was why I put dx in quotation marks. I was also aware of the shift from:
a/ slope = Δy/Δx; to
b/ f'(x) = df(x)/dx
Which are essentially the same thing: finding the rate of change one way or another: a/ does it over a range, and b/ does it at any spontaneous spot, an infinitesimal small range, which can only be managed when we put the whole thing in a limit and let the distance approach zero.

What I pondered about was just the use of dx in the intuition behind computing the area under the curve, and eventually it gets messy when I get to the volume under a surface and so on so forth.
I see what confused me though: I forgot that f(x) on a graph does not refer to a point y units away from the x-axis, it actually refers to the distance from the x-axis to that point, a.k.a a line. This way, my two different intuitions behind this seems to merge into one.

- Given x=a, evaluating f(a) gives a line connecting x=a and y=f(a). Summing up an infinite number of those lines along the x direction gives the desired area, but we can't sum things with zero area, so we have to divide the x distance into infinitely many dx's to give those lines some width.
- The same is true for finding the slope of a curve. Since Δy can't be computed along a curve, it has to be reduced to the point where the curve becomes a line, then divided by the corresponding amount of x changes, which consequently is also infinitesimally small.