Prerequesites for Studying Physics and Other Hard Sciences

[Kaffee]

Hullo chaps.
Sorry to bother you with this, but I need some help.
Is it possible to, having taken up to Algebra II (yes I'm underage, yes it's pathetic), quickly review everything you learned during the High School Algebra courses and then use a Calculus text like Apostol I to review what is necessary for understanding, but not learned in HS Alg, while simultaneously taking a Geometry course in school (once it starts up, of course) and be able to truly appreciate and understand Apostol I -> Spivak Calc -> Apostol II before then taking on an undergraduate level Physics textbook?
I would, of course, continue studying other, higher levels of Math while studying Physics, but I can't really use a Physics text without learning Calculus first, and I'm eager to start learning this shít and actually understanding the world around me, without having to resort to oversimplifications of varying degrees of comprehensibility.
Thanks in advance.

 

[chiro]

Hey Kaffee and welcome to the forums.

In terms of understanding calculus for what it is, you can understand that pretty quickly without knowing how to calculate things. The calculation itself and the framework with its results is what you need to do problems, but the idea of calculus is very straightforward.

The idea of calculus is basically to relate changes and sums together: that's basically it.
In physics we typically know how stuff changes (or we have models that are good enough) and then we want to figure out how something responds over time. The characteristic of the change is that we know how it changes 'instantaneously': this is the important thing and the instantaneous change of one variable with respect to another is known as the derivative of one with respect to another.

Differentiation (instanteous changes) and Integration are inverses of each other: integration adds up lots of changes and differentiation tells how something instantaneously changes.

Of course we have conditions on what we can differentiate and integrate with respect to the instantaneous type integral and derivative, but a lot of physics has this property so its OK.

The things that change vary: it can be arc-length, area, volume, projections (used in vector calculus), curl (used in vector calculus) or some other measure. You usually spend a year doing non-vector calculus stuff and the next year you do vector calculus stuff and then differential equations, but the idea doesn't change.

The stuff that you will need to do in university work will be applying all the theorems to problems so that you can actually calculate stuff. For this stuff, you should probably look at Khan Academy for tutorials and practice problems.

 

[Kaffee]

The stuff that you will need to do in university work will be applying all the theorems to problems so that you can actually calculate stuff. For this stuff, you should probably look at Khan Academy for tutorials and practice problems.

This doesn't exactly answer the question I posed, and why would using Khan Academy be preferable to working through a textbook? I'm not about to go into uni either; I'm only 15.

 

[chiro]

This doesn't exactly answer the question I posed, and why would using Khan Academy be preferable to working through a textbook? I'm not about to go into uni either; I'm only 15.

You said you wanted to understand calculus and not have an oversimplification: what I said should aid you to do that.

If you want to just do problems, then you could go to a uni bookshop in your area and grab a copy of a calculus book with plenty of exercises and partial solutions. Order a book of amazon if you have to (and you could get a good 2nd hand one cheaply).

This is all you are going to do in a math-based calculus course: you will get problems where you have to solve specific integrals or derivatives, work out a calculus-based model for some descriptive word problem and solve that, or construct simple proofs.

A book like Stewarts Calculus should cover most of that for Calc II, Calc II and Calc III.

But the understand is more important because you can work on as many problems as you like, but the thing is to know how to derive things. This is why I said what I said because deriving calculus-based models and analyzing them to get results depends on how something actually changes with respect to the other variables: if you don't learn this, then all the textbook problems in the world won't make an ounce of difference.

If you want to do the problems, just do the problems. If you are stuck on something post it in one of the forums and you should get a reply pretty quickly.

 

[Millennial]

Because Khan Academy explains it intuitively. Given your age, it is likely you will find it better than a textbook that gives everything rigorously. As an example, take limits. A textbook is likely to use the epsilon-delta definition of Cauchy as follows:

"For a sequence [itex](x_n)_n[/itex], if for each positive real [itex]\epsilon[/itex], there is a positive integer N such that for all [itex]n>N[/itex], [itex]|x_n-a|<\epsilon[/itex]; then a is the limit of the sequence [itex]x_n[/itex]."

This definition is correct but it does not give the intuition you need for understanding what limit is. Intuitively explained, and most likely the way Khan Academy would explain it, is as follows:

"Some sequences of numbers grow without bound, for example 1,2,3,4... does so. It does not approach any real, it grows constantly. However, let's take the sequence 1/n. The first few terms are 1,1/2,1/3,1/4... We can clearly say this does not grow constantly. Obviously this is never going to be negative for a positive n, so we can say it is always greater than 0. But the sequence gets closer and closer to zero. There is no other rational or irrational it gets close to that is bigger than zero. But, from experience, we know that this should be getting close to something. It isn't greater or less than zero, so it is zero. We say this sequence has a limit of zero as n approaches infinity."

 

[Kaffee]

Because Khan Academy explains it intuitively. Given your age, it is likely you will find it better than a textbook that gives everything rigorously. As an example, take limits. A textbook is likely to use the epsilon-delta definition of Cauchy as follows:

"For a sequence [itex](x_n)_n[/itex], if for each positive real [itex]\epsilon[/itex], there is a positive integer N such that for all [itex]n>N[/itex], [itex]|x_n-a|<\epsilon[/itex]; then a is the limit of the sequence [itex]x_n[/itex]."

This definition is correct but it does not give the intuition you need for understanding what limit is. Intuitively explained, and most likely the way Khan Academy would explain it, is as follows:

"Some sequences of numbers grow without bound, for example 1,2,3,4... does so. It does not approach any real, it grows constantly. However, let's take the sequence 1/n. The first few terms are 1,1/2,1/3,1/4... We can clearly say this does not grow constantly. Obviously this is never going to be negative for a positive n, so we can say it is always greater than 0. But the sequence gets closer and closer to zero. There is no other rational or irrational it gets close to that is bigger than zero. But, from experience, we know that this should be getting close to something. It isn't greater or less than zero, so it is zero. We say this sequence has a limit of zero as n approaches infinity."

But isn't part of attaining actual skill in mathematics learning how to read and understand such definitions?
Don't get me wrong, I DO intend to continue learning high level math. And I can see the advantage in learning with more understandable notation, but if I don't ever make the effort to understand the notation used in a rigorous definition, I won't ever come to be able to understand it, will I?

 

[Millennial]

You got me wrong. Rigour is indeed a necessity of formal mathematics; but what I meant was that the intuitive definitions is not based on the rigorous one, actually the converse is true. The limit is defined as it is because it matches what we need as limit and what satisfies our intuition. Once you have the sense of what limit is, you should be able to understand the rigorous definition much easily and see where it comes from as well.

 

[Kaffee]

You got me wrong. Rigour is indeed a necessity of formal mathematics; but what I meant was that the intuitive definitions is not based on the rigorous one, actually the converse is true. The limit is defined as it is because it matches what we need as limit and what satisfies our intuition. Once you have the sense of what limit is, you should be able to understand the rigorous definition much easily and see where it comes from as well.

Alright. I see what you're saying now. I'll take your advice into consideration.

But the understand is more important because you can work on as many problems as you like, but the thing is to know how to derive things. This is why I said what I said because deriving calculus-based models and analyzing them to get results depends on how something actually changes with respect to the other variables: if you don't learn this, then all the textbook problems in the world won't make an ounce of difference.

Do you know of any books I can use to get more on this?

 

[chiro]

Do you know of any books I can use to get more on this?

Well in terms of deriving a lot of the standard calculus results for areas, volumes, arc-lengths and vector calculus, there are a lot of books like Stewart and Apostol. I used Stewarts book a very long time ago, but the key thing is to focus on proofs that are based on the intuitive version of calculus.

In terms of calculus you have two versions: the first one is the intuitive calculus that people take for granted (even math majors) unless you are in an honors course which is different.

The intuitive calculus basically says "This is the definition of the derivative and the integral" and we assume that we are dealing with functions that have differentiability (and as a result continuity) across the region of intergration or for the region for differentiation.

From this you get the standard results for integration and differentiation which include things like substitution and integration by parts for integration and things like the chain rule, product rule, quotient rule for differentiation.

From this you prove the arc-length, volume rotation, bounded areas, and other formulas.

You can see all of this in a book like Stewarts Calculus or the book by Apostol.

Note that all of these deal with integration and differentiation with respect to one variable. This is important to note because the multivariable calculus proofs are a lot harder and involve what is known as linear algebra and a lot more advanced mathematics.

You can prove the basic theorems for differentiability by using the limit definitions and integration proofs are simple application of the fundamental theorem of calculus. You can read the books for examples of this.

For multi-variable calculus, the proofs are a lot more involved. Spivak has a book called "Calculus on Manifolds" that goes through this in a very rigorous manner and this is the book for this kind of treatment. Books like Stewart goes through some multi-variable stuff and has plenty of exercises, but Spivak is better if you want to go to the rigorous stuff.

An updated version of Stewart:

https://www.amazon.com/dp/0495011665/?tag=pfamazon01-20

Spivaks book:

https://www.amazon.com/dp/0805390219/?tag=pfamazon01-20