Rather than talking about engineers, I am mostly speaking about the subject which is calculus, aka the study of continuous change, aka infinitesimal change, aka

Rather than talking about engineers, I am mostly speaking about the subject which is calculus, aka the study of continuous change, aka infinitesimal change, aka derivatives and integrals. At first order the engineer need to be able to provide some sort of solution to any problem and get something done. The numerical toolbox is kind of an easy reliable Swiss knife that will get you through anything related to derivatives and integrals. It Ten years down the line when your nuclear core is melting and you need to pull the right lever will you be confident enough to trust your rusty manual computations tricks and not miss some minus sign somewhere in the transcription of your derivative. The derivative and integral game, is kind of an interesting mathematical artefact about the study of functions from before the time we had computers. As far as I know it's not yet a solved game. It's more about heuristics and picking rules than using a specific formula. Rule based Integration https://rulebasedintegration.org/ https://rulebasedintegration.org/ is a kind of collection of all the tips and tricks to play this game, but even then it's not foolproof. Closed formula are nice to have but hard to get by in real problems. The thing which makes the difference between something that is nicely behaved or not, is whether or not the operations are closed ("closure" concept). When it's the case you can build some nice algebra with nice properties. The computations becomes deterministic and formulaic. When it's not you end-up playing a NP-complete game to find the solution. For example in the case of the study of infinitesimal rotations, the natural extension of calculus for physics, we have the Lie Algebra. Even there your numerical toolbox will be more useful than your pages of Feynman Diagrams.