[Torus]

Hello,


I'm in year12 and I'll be finishing further maths a-level this June. I hope to do maths at uni and I want to learn something new from next September. So I was wondering if it's possible to self teach some 1st year university maths?

I don't know how to go on about it; where should I start? Which branch of mathematics do people learn first when they go to uni?
Could you recommend me some introductory textbooks?

Please don't recommend doing BMO/STEP instead. 1 year is a long time and I want to start learning something new. (I'll probably do some BMO problems anyway for fun! )


Thanks in advance.

[Brit_Miller]

Look up what your first year in uni will involve - it differs from uni to uni. Or chuck the maths department an e-mail and ask what you should learn.

[james22]

Do more maths modules? Thats what I did and although the content isn't university level you may find it interesting.

Otherwise, if your really keen and want to dive right in, have a look at the lecture notes for Oxfords fist year: http://www.maths.ox.ac.uk/courses/material

[Bobifier]

The obvious next step is university level Maths, and there are a lot of free lecture notes around the internet. Look for Analysis and Algebra mainly. Warwick is among the best universities for Maths, and a list of their modules can be found at http://www2.warwick.ac.uk/fac/sci/ma...ad/ughandbook/. You can't access the extra learning materials, but the module names and descriptions should tell you where to look.

If you want textbooks, I would reccomend Excursions into Mathematics, which studies topics that you probably won't encounter at university in depth and from the ground up, and in a rigorous style. However, it is quite expensive. If you want textbooks specific to the main areas of Maths, you can look on www.abebooks.co.uk which usually has a lot of cheap textbooks on it. Search Amazon for the reputable textbooks, and then search for the cheap versions on this website. Older textbooks tend to be cheaper, so look for classic textbooks.

[Brit_Miller]

(Original post by james22)
Do more maths modules? Thats what I did and although the content isn't university level you may find it interesting.

Otherwise, if your really keen and want to dive right in, have a look at the lecture notes for Oxfords fist year: http://www.maths.ox.ac.uk/courses/material

For the erotic side of maths.

Edit:I got negged for humour.

[SubAtomic]

These were recommended in another thread recently

here

this on linear algebra here

And I uploaded this pdf

[electriic_ink]

(Original post by Bobifier)
The obvious next step is university level Maths, and there are a lot of free lecture notes around the internet. Look for Analysis and Algebra mainly. Warwick is among the best universities for Maths, and a list of their modules can be found at http://www2.warwick.ac.uk/fac/sci/ma...ad/ughandbook/. You can't access the extra learning materials, but the module names and descriptions should tell you where to look.

Here are the Analysis I workbooks:

http://homepages.warwick.ac.uk/~masdbl/w1.pdf
http://homepages.warwick.ac.uk/~masdbl/w2.pdf
....
http://homepages.warwick.ac.uk/~masdbl/w10.pdf

As the first workbook explains, Analysis is Maths done properly. You might find it bit difficult to teach yourself it though

[nuodai]

If you want to learn first-year uni-level maths, then "Calculus" by Spivak, "Algebra and Geometry" by Beardon, "Probability: An Introduction" by Grimmett & Welsh and "Mathematical Methods for Physics and Engineering" by Riley, Hobson and Bence will probably cover most first-year material. Maybe add a bit of statistics (i.e. beyond probability) and mechanics here and there, but this is definitely enough to keep you going.

If you want to take it a bit lighter (which I'd recommend if you're still doing other A-level subjects) then introductory books like things like "A Concise Introduction to Pure Mathematics" by Liebeck are pretty good.

[sebastienlewis]

(Original post by SubAtomic)
These were recommended in another thread recently

here

this on linear algebra here

And I uploaded this pdf

Who made the document? If you made it, well done, it's very good.

[Bobifier]

I have always found http://tutorial.math.lamar.edu/Classes/DE/DE.aspx to be very helpful. They cover at least as far as my second year analysis stuff (I know this because my searches are still occasionally redirected to them) and I have always found them to be helpful when I have ended up at them. I think they mostly cover Analysis, but he has other topics there as well. Look at the 'class notes' section at the top. A warning: they claim that using the notes is not a substitute for attending lectures. As you progress to university you will find that most published lecture notes claim this, but it is all lies

[Agrippa]

(Original post by nuodai)
If you want to learn first-year uni-level maths, then "Calculus" by Spivak, "Algebra and Geometry" by Beardon, "Probability: An Introduction" by Grimmett & Welsh and "Mathematical Methods for Physics and Engineering" by Riley, Hobson and Bence will probably cover most first-year material. Maybe add a bit of statistics (i.e. beyond probability) and mechanics here and there, but this is definitely enough to keep you going.

If you want to take it a bit lighter (which I'd recommend if you're still doing other A-level subjects) then introductory books like things like "A Concise Introduction to Pure Mathematics" by Liebeck are pretty good.

This, definitely.

I'd start with Spivak and Beardon. Spivak basically covers the material in a first year Analysis course - broadly speaking, it aims to put calculus on a firm, rigorous footing. If I remember correctly (I left my copy at home...) it starts off with formalising the ideas of limits and infinite sequences/series before (after some work) defining differentiation, proving various results like the product/chain rules and moving on to integration, getting to the Fundamental Theorem of Calculus and beyond.

Beardon is an introduction to linear algebra and groups - it's a very good foundation book. It covers groups, particularly permutations, as well as group actions, applications to symmetry of regular solids and Möbius transformations. The other major topic area is vector spaces and linear transformations (matrix algebra, eigenvalues and eigenvectors, etc)

Amongst some of the results proved are:
- The fundamental theorem of algebra
- The Cayley-Hamilton theorem (relating polynomial operations of linear transformations)
- Lagrange's theorem (the first fundamental theorem in group theory)
all of which (and much more) you'll cover in a first-year uni course.

They're definitely worth forking out a bit for rather than trying to use university lecture notes, which whilst often good at covering the pure definitions/theorems, are rarely good at actually explaining what's going on.

[Brit_Miller]

(Original post by nuodai)
If you want to learn first-year uni-level maths, then "Calculus" by Spivak...

That really is a fantastic book, I bought it after yourself or someone else mentioned it on the forum. Really good for definitions and gives you lots of questions to solve.

[Torus]

Thank you for all the recommendations!



By the way, does anyone know anything about Open University course? (pure mathematics)

[sebastienlewis]

(Original post by Torus)
Thank you for all the recommendations!



By the way, does anyone know anything about Open University course? (pure mathematics)

MITOpenCourseware may be a good substitute:

http://ocw.mit.edu/courses/mathematics/