Focusing on Differentials (Refactoring Calculus: Part 1)

[@rrr00bb1]

to really "think like a computer",we should treat "a^b" like a binary operator, just define d for all binary operators, and not necessarily hold either operand constant: d[a^b] = a^b [(b/a) * d[a] + log[e,a] * d[b]]. And even log[b,v] is a binary operator, where neither b or v are necessarily held constant until the end where if b is "constant", then d[b]=0. One thing I found when using your way of doing things: I can get a lot farther in making this work for discrete analogues of differential calculus, except I had trouble where d[x]=1 for discrete cases where log is involved.

[@XetXetable]

The entire thing about infinitesimals is vestigial. If you view d as a linear operator which performs the product rule, then it's exactly a thing that will take any expression made up of variables, numbers, sums, and multiplication and turns it into a linear expression tangent to a point, up to translation. The expression in question is linear with respect to dx, dy, etc. where dx and dy are simply newly introduced variables. This is exceptionally obvious in the 2D case. Example; if z = 2 x^2 - x y then dz = (4 x - y) dx - x dy fixing a particular x and y, dz becomes a formula for a plane with variables dx and dy; the linear approximation for any 2D surface. This is, ultimately, much simpler and has less philosophical baggage than the infinitesimal explanation. You don't actually get anything out of either derivatives or integrals by treating differentials as infinitesimals in anything but the most formal sense (e.g. squaring to 0 is a secondary, not primary property); they really should be thought of as variables in a linear approximation.

[@APaleDot]

15:18 I want to add that if you take 'x' and 'z' to be independent inputs into the function, then there is no relation between them and dz/dx would be zero, leaving you with the standard definition of a partial derivative.

[@BPLearningTV]

Not really. If they are independent, then the ratio can be *any* value, not zero. dz/dx can only be zero *if* Z is not changing. What the equation actually means is that the slope dy/dx will depend on *which* slope of dz/dx you choose. Take a basketball and a ruler. Get the ruler tangent to the basketball. Note that you can spin the ruler around and still have a tangent. However, as you spin it, your dy/dx is coordinating with your dz/dx. There *may* be a slope of dy/dx that occurs when dz/dx is zero, but it is not the only valid slope.

[@BigDmitry]

Great video, thanks! I myself am a programmer, and when I look at calculus it very much resembles a novice level spaghetti code mess in desperate need of refactoring. And its mostly due to the thing you're pointing out -- the missed abstraction of differentials and infinitesimals. So the video resonates strongly with me. I also wanted to point out, that you probably want to preserve the second and higher order differential terms ((dx)^2, dudv, ...). They contain additional information, that might be useful in case you want to make some further operations on the results and not just take the real part of them immediately.

[@BPLearningTV]

Thanks for that reply! I've been playing with preserving those terms, but I have yet to find anything worthwhile to do with them. I originally thought that maybe it could help solve higher-degree differential equations, but I've attempted several times and have failed each one. I haven't totally given it up, but I haven't found a path forward yet, either.

[@BigDmitry]

@@BPLearningTV Well, it seems logical that extra terms will make solving equations harder, not easier. It's an added complexity after all... Unless they cancel out with some other terms for some reason. But they might help make more accurate models for quantum physics or some macroscopic chaotic systems, where the smallest terms blow out exponentially. So non-linear differential equations seems like the right way to go.

[@noshiko5398]

Definitely agree. Standard notation is also great at making types incredibly unclear