A place for majestic STEMLORD peacocking, as well as memes about the realities of working in a lab.
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This is a science community. We use the Dawkins definition of meme.
But df/dx is a fraction, is a ratio between differential of f and standard differential of x. They both live in the tangent space TR, which is isomorphic to R.
What's not fraction is \partial f / \partial x, but likely you already know that. This is akin to how you cannot divide two vectors.
The world has finite precision. dx isn't a limit towards zero, it is a limit towards the smallest numerical non-zero. For physics, that's Planck, for engineers it's the least significant bit/figure. All of calculus can be generalized to arbitrary precision, and it's called discrete math. So not even mathematicians agree on this topic.
Software engineer: 🫦
1/2 <-- not a number. Two numbers and an operator. But also a number.
In Comp-Sci, operators mean stuff like >>, *, /, + and so on. But in math, an operator is a (possibly symbollic) function, such as a derivative or matrix.
Youre not wrong, distinctively, but even in mathematics "/" is considered an operator.
https://en.m.wikipedia.org/wiki/Operation_(mathematics)
oh huh, neat. Always though of those as "operations."
The thing is that it's legit a fraction and d/dx actually explains what's going on under the hood. People interact with it as an operator because it's mostly looking up common derivatives and using the properties.
Take for example ∫f(x) dx to mean "the sum (∫) of supersmall sections of x (dx) multiplied by the value of x at that point ( f(x) ). This is why there's dx at the end of all integrals.
The same way you can say that the slope at x is tiny f(x) divided by tiny x or d*f(x) / dx or more traditionally (d/dx) * f(x).
The other thing is that it's legit not a fraction.
it's legit a fraction, just the numerator and denominator aren't numbers.
No 👍
try this on -- Yes 👎
It's a fraction of two infinitesimals. Infinitesimals aren't numbers, however, they have their own algebra and can be manipulated algebraically. It so happens that a fraction of two infinitesimals behaves as a derivative.
Ok, but no. Infinitesimal-based foundations for calculus aren't standard and if you try to make this work with differential forms you'll get a convoluted mess that is far less elegant than the actual definitions. It's just not founded on actual math. It's hard for me to argue this with you because it comes down to simply not knowing the definition of a basic concept or having the necessary context to understand why that definition is used instead of others...
Why would you assume I don't have the context? I have a degree in math. I could be wrong about this, I'm open-minded. By all means, please explain how infinitesimals don't have a consistent algebra.
I also have a masters in math and completed all coursework for a PhD. Infinitesimals never came up because they're not part of standard foundations for analysis. I'd be shocked if they were addressed in any formal capacity in your curriculum, because why would they be? It can be useful to think in terms of infinitesimals for intuition but you should know the difference between intuition and formalism.
I didn't say "infinitesimals don't have a consistent algebra." I'm familiar with NSA and other systems admitting infinitesimal-like objects. I said they're not standard. They aren't.
If you want to use differential forms to define 1D calculus, rather than a NSA/infinitesimal approach, you'll eventually realize some of your definitions are circular, since differential forms themselves are defined with an implicit understanding of basic calculus. You can get around this circular dependence but only by introducing new definitions that are ultimately less elegant than the standard limit-based ones.
We teach kids the derive operator being ' or ·. Then we switch to that writing which makes sense when you can use it properly enough it behaves like a fraction
If not fraction, why fraction shaped?
Having studied physics myself I'm sure physicists know what a derivative looks like.
I still don't know how I made it through those math curses at uni.
Calling them 'curses' is apt
clearly, d/dx simplifies to 1/x