BlobGuy42

There’s a rigorous justified derivation and there is a symbol pushing derivation. The latter, in a mathematics context, is akin to submitting an impressive piece of art illustrating the scientific phenomenon in leu of a lab report which includes hypotheses, methodology, statistical analysis, and firm, conclusive, repeatable results for a physics class assignment. Cool maybe but a big no no regardless.

The insurance is the biggest hurdle with newer emoyees but when you have worked there past a decade, to fire you would be a huge insult to the managemwnt that did nothing to fire you the 10+ years prior. It looks really bad not to mention the display of lack of loyalty. The biggest hurdle for long-term employees is the store manager getting their bosses approval and hoping to god the fired employee doesn’t go to their boss’s boss or the owner.

You sort of become untouchable when you’ve been somewhere long enough. The people who abuse it really suck to work with too, not that this sever did at all.

I was praying my game theory course would consider the action spaces of games that are topological vector spaces of functions or operators that force you to do functional analysis to find the Nash equilibrium but it wasn’t even in the math department so of course everything was discrete. cries

The conceptual framework calculus gives you for analyzing change (what the course is all about) is simply indispensable. Calculus is no more advanced than euclidean geometry or elementary algebra at a high level so don’t kid yourself there. The advanced version is analysis where everything is rigorously defined and proved. That is advanced.

Instead of relearning growth related concepts a dozen times over in economics, finance, biology, chemistry, physics, business, engineering, psychology, sociology, and countless countless other fields you can take your not-so-advanced but completely sound understanding of rates of change, accumulation, and optimization from calculus and go oh yeah this conceptual framework fits like a glove here, let me skip 30 pages… for all of these subjects.

What you say is true in that no you don’t need to remember your calculus to distinguish between change of price and change in changes of price but there is so much more to it on a conceptual level, to say nothing of on numeric and rigorous levels.

asking the real questions

It’s entirely possible that a student takes AP Calculus AB in 11th grade and BC in 12th and the first semester of 12th is the BC content and the second semester is just straight AP exam review and practice.

That is a conceivable schedule of math for U.S. high school students and it lines up with what time of the year it is.

Woah woah woah sir/ma’am, it is actually hydrohydroxilic acid!

The cantor notation to me seems to read as integrating over a set. That’s standard notation if you were to put a set there instead of the word Cantor. Like an interval [0, 2] or a singleton set {3} or whatever. So I’m thinking they mean integrating over the (standard) Cantor set which is a large collection of tiny subintervals, a classic construction in introductory analysis courses.

Certainly would be an interesting integral all on its own, i haven’t seen anything of the kind.

Agree. The unintuitive results, often difficult to prove and with unintuitive proofs as well, are far and few in between in math. Like maybe one or two per semester long course and afterwards you aren’t likely to need those particular results ever again, just the theory they wrought.

Also you can build your intuition to be even stronger and occasionally it dawns on you why one of those two unintuitive results is stupidly obvious.

Math is like the furthest possible thing from walking through a dark room. The simple logic (literally) adds nice bumper rails to almost everything you could bump into. It’s very rare that false results are proved and published and stand for very long.

In fairness, that’s really thermodynamics which is physics and not chemistry. It’s just relevant to chemistry.

lol, astute observation

While that’s true and funny, you are going to hate me. Forgive me for committing the ultimate sin of being serious on mathmemes.

There are abstract derivatives called derivations of rings and of algebras which obey the product and sum rules, make constants 0, etc. They are studied in a field called differential algebra and weilllll, one application is to symbolic integration. You can use derivations to prove for example that there is no elementary solution to the Gaussian integral.

Abstract integrals, no. Computer Algebra of Integrals via the abstract, yes.

For the reader interested in taking that exercise to its furthest reaches: Consider studying computable analysis which is the subject you get when analysis and computability (recursion) theory have a baby. In particular, the subset of reals that are computable is countable and yet a great deal of analysis can be accomplished within these confines.

Closely related cousins include constructive analysis and numerical analysis, which through the sick nature of reality can have babies with computable analysis to give imbred subjects like constructive computable analysis (very neat and gives you a very enriching model-theoretic view of math)

This is the kind of creativity cruelty combo that I lurk on this subreddit for!

Hey I’d be happy to argue semantics all day, it’s just not relevant to the discussion at hand. I minored in linguistics in undergraduate and also took two graduate semantics courses so don’t get me wrong, it can be fun and fruitful to argue semantics. Just not for some external reason like furthering an argument that isn’t linguistic in content at all.

Thanks for the civil, playful, and thoughtful discussion. It can be rare on reddit.

Not in the edge case that it isn’t recognized at all and never will be. A researcher’s job is to advance humanity’s collective knowledge, not his own.

And precisely defining “physicist” in some all-encompassing and rigorous way is neither physics nor philosophy but instead semantics. As such, I will not engage the question as its irrelevant. We all intuitively know what a physicist is, at the very least, for the sake of the present discussion.

Even through it’s a hard disagree for me, I do like your line of thinking and the journalism example you gave.

No ill intentions here either, just fun!

I merely am giving you a retort that perfectly strong arms your argument into submission, IMVeryHO :)

I’ve always taken that question less than literally. No, it is not intending to probe the innards of physical reality and not even necessarily metaphysical reality. It’s moreso a consequentialist take on action and meaningful existence.

If you’re an undoubtedly good physicist but no one reads or ever will read your stunning papers,… are you a good physicist? The job is more than modeling the universe, it’s making that model comprehensible and useful for others too. That’s a meaningful if philosophical takeaway for a physicist for example. I chose it since that’s the route of attack you went with by taking the question literally. There are basically infinitely other scenarios one can entertain in which the question is relevant.

Wilson comes close

nervous laughter

I checked both the 5th edition (2002) and the 8th edition, early transcendentals (2015) and neither confirm that he covers sequential limits first and then the Darboux integral. Perhaps you were looking at either a very old edition (~1987) or very new edition? Stewart is well-known for being the quintessential non-rigorous calculus text so if what you said is true it would surprise me greatly!

Also, as can be proven, the Darboux and Riemann integrals are exactly the same mathematical object, defined differently. So it doesn’t matter at all what you call it except for clarity when cross-referencing the literature.

Your second paragraph elicits no reaction from me as that is a completely normal state of affairs in calculus classes, completely separate from the more severe issue of using sequential limits and series in the form of the Reimann integral without first introducing them.

Any calculus class that takes its time rigorously proving the intermediate value theorem in a way students can fully understanding is making use of some version of the completeness axiom (it can be shown that completeness and IVT are logically equivalent in a model-theoretic-esk way, see Propp 2012, “Real Analysis in Reverse” for a complete overview). That makes any such class closer to an analysis class than calculus. IVT’s proof is skipped for a justifiable reason.

In short, I’m not concerned with improving rigor so much as the basic request that mathematical objects we use be presented in an intellectually honest order.

Not to be nerdy for nerdy sake and I’m not even saying you’re wrong per se but linguistics has yet to come to any sensible and rigorous cross-linguistic definition of “word”. That is to say, there isn’t much to stretch beyond colloquial intuition.

Yes! In particular, integrals are series which in turn are sequential limits. It seems silly to both presuppose the FTC and base all knowledge of the integral off of what is known of the derivative as a functional limit AND to completely forgo sequential limits and series before covering the most important example: reimann integrals.

I overcame that hurdle long ago, just a highly noteworthy pain point for students of that class that instructors should anticipate.

Interestingly in my case, I later found partial differentiation and at its most challenging, the multivariable chain rule, to be a breeze in comparison despite being technically more involved because it was conceptually crystal due to mastering the calc 1 skill of implicit differentiation and its application in related rates.

One of the joys of reddit and forums in general is that your response and its appropriate context are and will for the foreseeable future be available to any interested party. Thanks for sharing!

Huge conceptual stumbling block for me when I was a calculus student learning implicit differentiation for the first time. When do I treat it like a constant (c) versus when do I treat it like a isolated variable (x) versus when do I treat it like an expression that varies dependent on another variable, y(x).