Calculus is completely unrigorous ? (1 Viewer)

[Zen2613]

Hey all,
So I've just finished the hsc this year and now that it's all over, I'm thinking about all the things in maths that made absolutely no sense to me. One of these is good old integration. Like when first learning calculus, we were told dy/dx is NOT a fraction. But then when we get to integration and substitutions, we treat it as one, so how is this justified ? Also the worst bit is how we write an integrand. The integral dx of something magically works perfectly with dy/dx so how can we be sure that the dx should go there like that, if all it means is taken with respect to the x-axis. Specifically it only doesn't make sense with integration by substitution, we're just told this is what to do, and don't question it, because it is beyond you. I hope when I get to uni, they approach calculus more rigorously so that it won't be just being told what to do, and more so actually understanding it...

 

[glittergal96]

Yep, these are good concerns to have.

In HS they sweep a bunch of details under the carpet and you just have to trust them (not specifically your HS teachers, many of whom don't know better themselves lol, more the people who write the syllabus and books). If they tried to do things more properly in high school they would:

a) Lose the interest of many many students.
b) Be able to cover way less material, which kind of obscures just how useful calculus is, even for people who don't understand its inner workings.

In uni you do things from scratch and properly, and it is good to forget lots of how you thought about things in high school maths. (Although the practice at computing things quickly is occasionally useful, and these skills should not be forgotten.)

dy/dx is just notation for an abstract operation (differentiation of a function labelled by y with respect to the variable x). We can choose whatever notation we like, y'(x), y', etc etc. The major advantage of "dy/dx" is precisely the fact that it does behave in a fraction-like way when it comes to the chain rule and the fundamental theorem of calculus.

Also dx is not just an indicator of what variable we are integrating with respect to. It is an abstract object called a differential form. Their usefulness is hard to explain when one has only learned single-variable calculus, but they allow us to generalise integration to SIGNIFICANTLY more abstract settings.

It is also a pleasant fact that differential forms transform with respect to change of variables in the "fraction-like" way one would expect from the "dy/dx" notation. This won't be surprising by the time you encounter them, as they are actually in a sense the "dual" object to things like the operation of differentiation with respect to a variable.

Don't worry, real maths is far more satisfying and rigorous.

 

[Flop21]

Yep, these are good concerns to have.

In HS they sweep a bunch of details under the carpet and you just have to trust them (not specifically your HS teachers, many of whom don't know better themselves lol, more the people who write the syllabus and books). If they tried to do things more properly in high school they would:

a) Lose the interest of many many students.
b) Be able to cover way less material, which kind of obscures just how useful calculus is, even for people who don't understand its inner workings.

In uni you do things from scratch and properly, and it is good to forget lots of how you thought about things in high school maths. (Although the practice at computing things quickly is occasionally useful, and these skills should not be forgotten.)

dy/dx is just notation for an abstract operation (differentiation of a function labelled by y with respect to the variable x). We can choose whatever notation we like, y'(x), y', etc etc. The major advantage of "dy/dx" is precisely the fact that it does behave in a fraction-like way when it comes to the chain rule and the fundamental theorem of calculus.

Also dx is not just an indicator of what variable we are integrating with respect to. It is an abstract object called a differential form. Their usefulness is hard to explain when one has only learned single-variable calculus, but they allow us to generalise integration to SIGNIFICANTLY more abstract settings.

It is also a pleasant fact that differential forms transform with respect to change of variables in the "fraction-like" way one would expect from the "dy/dx" notation. This won't be surprising by the time you encounter them, as they are actually in a sense the "dual" object to things like the operation of differentiation with respect to a variable.

Don't worry, real maths is far more satisfying and rigorous.

When you say they teach you from scratch and properly, is that like 4u level math in uni you're talking about?

 

[InteGrand]

When you say they teach you from scratch and properly, is that like 4u level math in uni you're talking about?

Lol no, 4U isn't even rigorous. It means you learn everything from the foundations rigorously (so including 2U level things).

 

[Flop21]

Lol no, 4U isn't even rigorous. It means you learn everything from the foundations rigorously (so including 2U level things).

Really?? So say some courses/subjects have 3u assumed, I'd still be okay going in with 2u (with some study during holidays)?

 

[glittergal96]

From scratch as in from the definitions of a function, continuity, differentiability, etc etc. Basically "jumping in" right after the real numbers are constructed and their elementary properties understood.

In later years if you take further maths courses, you can then look at the construction of the real numbers themselves. How are they defined and why does this make them have the properties that they do?

Pretty much anyone who wants to work in pure mathematics has to understand at least this much. Going any deeper into the foundations of mathematics you get into things like formal logic and philosophy which are a little less "mathematical" in flavour, and not really needed to work in mathematics. (And there are many unsatisfying features of the current foundations of mathematics thanks to crazy results like Godel's incompleteness theorems.)

 

[glittergal96]

Really?? So say some courses/subjects have 3u assumed, I'd still be okay going in with 2u (with some study during holidays)?

The "assumed knowledge" doesn't really refer to stuff you need to know otherwise you won't be able to do the course. You will have books/course notes and stuff that are pretty self-contained.

It's just that the average student who has only taken 2U might struggle in some first year uni courses as they will have seen fewer things in the course before and will generally be less mathematically "cultured" for lack of a better term.

 

[Silly Sausage]

Calculus in high school such as Differentiation and Integration is extremely mechanical in that you just "do it" and not really appreciate what is happening"behind the scenes".

 

[Zen2613]

Ok cool, I was just making sure because a lot of my teachers as you said could not explain it and it made me wonder if we even learn it at uni at all, unless you get to a very high level of math (graduate/post-graduate). Can't wait til uni !

 

[glittergal96]

It's roughly 2nd year of advanced maths courses when you learn enough about analysis (a branch of maths concerning things like limits) that I would consider your knowledge of HSC level calculus to be "rigorous".

First year is usually only as rigorous as is possible without going into any of the deeper analysis of the real numbers themselves. (Which is still quite a bit more rigorous than how it is taught in HS for the topics in common.)

 

[Zen2613]

Awesome, well I'll have to just wait and see