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What I feel about calculus, esp school calculus, is that it seems to comprise largely of specific tricks of symbol manipulation, rather than general approaches

What I feel about calculus, esp school calculus, is that it seems to comprise largely of specific tricks of symbol manipulation, rather than general approaches that work in all cases. Any ways, analytic integration/differentiation seems to be full of such tricks, of which how the core intuition was first obtained remains a mystery most of the time(rather than it being a result of scientific/mathematical method or process).

adamnemecek 8y ago
Look into automatic differentiation. It's magic.
throwawaymath 8y ago
Putting aside the common issues with pedagogy, the reason you learn a "bunch of tricks" for calculus is because that's more or less all you have in many cases. If you take integration as an example, there is no single approach to solving every integral. More importantly, it's extremely common to encounter integrals for which there exists no closed form antiderivative. In fact it's technically exceptional to find an integral which can be neatly solved in the space of all possible integrals. As a direct result, solving an integral becomes an (often frustrating) exercise in transforming it into something equivalent integrals up to a negligible constant. Nonlinear optimization problems and differential equations are similar in this regard. There is something to be said for the depth of analysis, which does provide a deeper meaning and rigor to the "bag of tricks" in calculus. Outside the US it's somewhat common to skip calculus entirely and begin straight away with analysis, and I think there's merit to that. But the profundity and power of analysis doesn't provide you with any fundamentally more complete methods of solving calculus problems except insofar as they become more advanced and rigorous. Ultimately analytic mathematical work (as opposed to algebraic) is characterized by this kind of pattern-matching; this frequently results in seemingly inspired, bizarre looking proofs compared to how neat everything is in algebra.
- billfruit 8y ago
  Perhaps for some of the students,a course that skips indefinite integrals, and focusing on definite integrals and numerical integration may be sufficient.
commandlinefan 8y ago
When I was learning it, I always felt like calculus (and honestly algebra and trigonometry too) struggled to “justify” itself: whenever they presented real-world applications, they always seemed so artificial that I ended up feeling even more confident that higher math was just complex for the sake of being complex. Although I understand that there are professions that “directly” apply calculus (i.e. solving integrals or derivatives) I went into computer programming where direct applications of calculus are few and far between, and ever the places where it’s applied like gradient descent and elliptic curve cryptography can be used without any understanding or appreciation of the base theory anyway. Still, I’ve come to appreciate that the real-world applications were deliberately simplified because the real real-world applications are mind-meltingly complex and that the presentation of calculus teaches you more to solve problems methodically than to apply specific processes and procedures.
dhfromkorea 8y ago
I thought so.... until I took a real analysis course. Most calculus tricks build upon on a "small" set of big ideas. The big ideas (such as compactness, convergence, continuity, diff/integration..) are limited in numbers to convince oneself on, yet are generalizable tools to think about A LOT of complicated mathematical phenomena in a concise, clear way. Should they fall short to evaluate a certain math phenomenon, they should do so unambiguously rather than opaquely. Real analysis is a start to developing such tools.

adamnemecek 8y ago

Look into automatic differentiation. It's magic.

throwawaymath 8y ago

Putting aside the common issues with pedagogy, the reason you learn a "bunch of tricks" for calculus is because that's more or less all you have in many cases. If you take integration as an example, there is no single approach to solving every integral. More importantly, it's extremely common to encounter integrals for which there exists no closed form antiderivative. In fact it's technically exceptional to find an integral which can be neatly solved in the space of all possible integrals. As a direct result, solving an integral becomes an (often frustrating) exercise in transforming it into something equivalent integrals up to a negligible constant. Nonlinear optimization problems and differential equations are similar in this regard. There is something to be said for the depth of analysis, which does provide a deeper meaning and rigor to the "bag of tricks" in calculus. Outside the US it's somewhat common to skip calculus entirely and begin straight away with analysis, and I think there's merit to that. But the profundity and power of analysis doesn't provide you with any fundamentally more complete methods of solving calculus problems except insofar as they become more advanced and rigorous. Ultimately analytic mathematical work (as opposed to algebraic) is characterized by this kind of pattern-matching; this frequently results in seemingly inspired, bizarre looking proofs compared to how neat everything is in algebra.

billfruit 8y ago
Perhaps for some of the students,a course that skips indefinite integrals, and focusing on definite integrals and numerical integration may be sufficient.

commandlinefan 8y ago

When I was learning it, I always felt like calculus (and honestly algebra and trigonometry too) struggled to “justify” itself: whenever they presented real-world applications, they always seemed so artificial that I ended up feeling even more confident that higher math was just complex for the sake of being complex. Although I understand that there are professions that “directly” apply calculus (i.e. solving integrals or derivatives) I went into computer programming where direct applications of calculus are few and far between, and ever the places where it’s applied like gradient descent and elliptic curve cryptography can be used without any understanding or appreciation of the base theory anyway. Still, I’ve come to appreciate that the real-world applications were deliberately simplified because the real real-world applications are mind-meltingly complex and that the presentation of calculus teaches you more to solve problems methodically than to apply specific processes and procedures.

dhfromkorea 8y ago

I thought so.... until I took a real analysis course. Most calculus tricks build upon on a "small" set of big ideas. The big ideas (such as compactness, convergence, continuity, diff/integration..) are limited in numbers to convince oneself on, yet are generalizable tools to think about A LOT of complicated mathematical phenomena in a concise, clear way. Should they fall short to evaluate a certain math phenomenon, they should do so unambiguously rather than opaquely. Real analysis is a start to developing such tools.