Math Learning: VIC AUS High School Grad Looking for Resources

[GwtBc]

Hi. I've recently finished high school and am planning to study science/physics, hence I'll be doing a lot of calculus courses but I'm looking to do more than that. I was looking to find some good sources, whether they be online or in textbook format or whatever. As for where I'm at currently, I did units Specialist mathematics and Mathematical Methods during my final year of high school in VIC, AUS. For those who don't know what that entails (which I'm assuming will be most people) here is a quick run down of what I've done:

•Calculus: Differentiation and integration. Polynomials, natural logs and powers of e, trig functions, inverse functions, chain rule, product rule, quotient rule, second derivatives (and their relationship with the original function), change of variable method, partial fractions, partial differentiation, solids of rotation

-Algebra: Not really sure what criteria there are to tick under algebra, but I think based on the other stuff I've done you can take a good guess at where I am with it

-Complex numbers: Polar and rectangular (Cartesian form), general algebra, division and multiplication in Cartesian and polar forms, De Moivre's theorem, finding the nth root of a complex number, Argand diagrams

-Graphing: Polynomials, exponential and logarithmic, trig, circles and ellipses, Modular functions, hybrid functions, inverse functions, asymptotic behaviour, domain and range

-Trig: identities relating sin,cos,tan,cosec,sec,cotan. graphs of and application of inverse trig functions. (and obviously all the base material you need to cover up to this point)

-Vectors: Algebra, scalar and vector resolutes, vector proofs, cross and dot products

-probability: So I know set notation, but a lot of the probability stuff covered in the methods course have escaped me since, because I did it a year earlier and back then I thought it was boring and I didn't pay much attention.

-Matrix: Comfortable with the basics of algebra, but not much besides.

I'm interested in all areas of maths. More calc, discrete maths, probability, matrices... anything really.

Any help is appreciated. :)

 

[jedishrfu]

You could check the Mathispower4u.com website to see if there's any math there that you haven't covered. Its a collection of 10 minute videos on math from Algebra thru Calculus III, Linear Algebra and Differential Equations. Khan's Academy covers a lot of high school and undergrad math too.

Also there's the MIT courseware site for more advanced courses presented in hour long lecture format.

 

[micromass]

It's not clear to me what you want exactly. It seems to be that any math is ok for you. Is this the case?
Maybe you should tell us exactly why you want to study extra math. And what your exact goal is at university. Because I can recommend stuff like abstract algebra now and realistically you could do it, but it might not be what you're looking for.

 

[GwtBc]

It's not clear to me what you want exactly. It seems to be that any math is ok for you. Is this the case?
Maybe you should tell us exactly why you want to study extra math. And what your exact goal is at university. Because I can recommend stuff like abstract algebra now and realistically you could do it, but it might not be what you're looking for.

Yeah anything is fine really. My ultimate long term goal would be to get a PhD in Physics I guess , but I'm not just interested in maths that relates to that. As cheesy as it sounds, I just want to learn new maths, but everywhere I look it's either too preliminary or it's written for PhD students. I suppose if I were to highlight areas of special interest (besides calculus obviously) it would be matrices (and other areas that relate to computer science) and also geometry.

 

[micromass]

Yeah anything is fine really. My ultimate long term goal would be to get a PhD in Physics I guess , but I'm not just interested in maths that relates to that. As cheesy as it sounds, I just want to learn new maths, but everywhere I look it's either too preliminary or it's written for PhD students. I suppose if I were to highlight areas of special interest (besides calculus obviously) it would be matrices (and other areas that relate to computer science) and also geometry.

Interesting. OK, here is a list of topics you can do and books you can use to study them:

Discrete Mathematics
This is a very accessible part of mathematics in that it does not require many prereqs at all. This doesn't mean it's easy. But this is very very applicable to computer science. A very good book on the topic is Grimaldi: https://www.amazon.com/dp/0201726343/?tag=pfamazon01-20

Linear Algebra
Another topic which has an amazing number of applications. It is definitely applicable in computer science for computer graphics, or stuff like numerical calculations. It is also applicable in physics. You can't understand quantum theory without really knowing the theory of linear algebra. Linear algebra is the only type of math that you can't know enough of. If you know the basics of matrices, then you can go and study the more theoretical linear algebra of vector spaces.
A very good book is Treil's linear algebra done wrong: https://www.math.brown.edu/~treil/papers/LADW/LADW.html This lacks a lot of problems, so be sure to complement it with a problem book like Schaum: https://www.amazon.com/dp/0071794565/?tag=pfamazon01-20
Another interesting book is MacDonald, which combines linear algebra with a very interesting formalism of geometric algebra. This is a very useful formalism in physics as it simplifies a lot of reasoning. Sadly, it doesn't seem to be too popular in physics circles: https://www.amazon.com/dp/1453854932/?tag=pfamazon01-20

Infinitesimal calculus
When you study physics, you will definitely notice that calculus as done by mathematicians and those as done by physicists differ a lot. Physicists use expressions like $dx$ as if they were actual very very small numbers. On the other hand, in calculus for mathematicians it is very often stressed that $dx$ has no separate meaning, and that $\frac{dy}{dx}$ is not a quotient. This is a strange situation. The invention and rigorization of infinitesimals solve a lot of these issues. The book by Keisler covers all of calculus, but in the language of infinitesimals. It shows how you could mathematically make sense of $dx$ and see $\frac{dy}{dx}$ as fractions. If you feel uneasy with a lot of the physicist arguments, this is probably a book you should go through. https://www.math.wisc.edu/~keisler/calc.html

Abstract Algebra
This is a more mathematically advanced topic, and at face value it doesn't seem to have many applications. It does have quite some applications, for example in chemistry (for the symmetry of atoms), computer science (for finding efficient codes), and even in parts of theoretical physics. A good and quite elementary book is Pinter: https://www.amazon.com/dp/0486474178/?tag=pfamazon01-20
Another book which focuses only on group theory, but done very geometrically is Armstrong: https://www.amazon.com/dp/0387966757/?tag=pfamazon01-20

Euclidean Geometry
Well, this will probably be quite useless for most purposes, but I can't resist since it is so beautiful. All the famous physicists and mathematicians before the 20th centuries went through Euclid's Elements. Newton found Euclid so beautiful that he based his Principia on the style of Euclid. It is a sad state of affairs that Euclid does not seem to be read anymore now, because it hardly aged and it still contains a lot of obvious beauty. I recommend Euclids elements "redux" http://starrhorse.com/euclid/
It might be interested to combine Euclid with a somewhat more modern book. Hartshorne is quite mathematically advanced, but still contains a lot of good stuff. https://www.amazon.com/dp/1441931457/?tag=pfamazon01-20

 

[GwtBc]

Interesting. OK, here is a list of topics you can do and books you can use to study them:

Discrete Mathematics
This is a very accessible part of mathematics in that it does not require many prereqs at all. This doesn't mean it's easy. But this is very very applicable to computer science. A very good book on the topic is Grimaldi: https://www.amazon.com/dp/0201726343/?tag=pfamazon01-20

Linear Algebra
Another topic which has an amazing number of applications. It is definitely applicable in computer science for computer graphics, or stuff like numerical calculations. It is also applicable in physics. You can't understand quantum theory without really knowing the theory of linear algebra. Linear algebra is the only type of math that you can't know enough of. If you know the basics of matrices, then you can go and study the more theoretical linear algebra of vector spaces.
A very good book is Treil's linear algebra done wrong: https://www.math.brown.edu/~treil/papers/LADW/LADW.html This lacks a lot of problems, so be sure to complement it with a problem book like Schaum: https://www.amazon.com/dp/0071794565/?tag=pfamazon01-20
Another interesting book is MacDonald, which combines linear algebra with a very interesting formalism of geometric algebra. This is a very useful formalism in physics as it simplifies a lot of reasoning. Sadly, it doesn't seem to be too popular in physics circles: https://www.amazon.com/dp/1453854932/?tag=pfamazon01-20

Infinitesimal calculus
When you study physics, you will definitely notice that calculus as done by mathematicians and those as done by physicists differ a lot. Physicists use expressions like $dx$ as if they were actual very very small numbers. On the other hand, in calculus for mathematicians it is very often stressed that $dx$ has no separate meaning, and that $\frac{dy}{dx}$ is not a quotient. This is a strange situation. The invention and rigorization of infinitesimals solve a lot of these issues. The book by Keisler covers all of calculus, but in the language of infinitesimals. It shows how you could mathematically make sense of $dx$ and see $\frac{dy}{dx}$ as fractions. If you feel uneasy with a lot of the physicist arguments, this is probably a book you should go through. https://www.math.wisc.edu/~keisler/calc.html

Abstract Algebra
This is a more mathematically advanced topic, and at face value it doesn't seem to have many applications. It does have quite some applications, for example in chemistry (for the symmetry of atoms), computer science (for finding efficient codes), and even in parts of theoretical physics. A good and quite elementary book is Pinter: https://www.amazon.com/dp/0486474178/?tag=pfamazon01-20
Another book which focuses only on group theory, but done very geometrically is Armstrong: https://www.amazon.com/dp/0387966757/?tag=pfamazon01-20

Euclidean Geometry
Well, this will probably be quite useless for most purposes, but I can't resist since it is so beautiful. All the famous physicists and mathematicians before the 20th centuries went through Euclid's Elements. Newton found Euclid so beautiful that he based his Principia on the style of Euclid. It is a sad state of affairs that Euclid does not seem to be read anymore now, because it hardly aged and it still contains a lot of obvious beauty. I recommend Euclids elements "redux" http://starrhorse.com/euclid/
It might be interested to combine Euclid with a somewhat more modern book. Hartshorne is quite mathematically advanced, but still contains a lot of good stuff. https://www.amazon.com/dp/1441931457/?tag=pfamazon01-20

Thanks for the help! I think I should work on my use of infinitesimals in calculus first, since there's a lot there that I still don't know about. Will definitely check out all of these.