Maths at uni

Original post by ABC789

I don't mind statistics but at the moment at A-level I enjoy pure maths the most.

What you do at A-Level isn't pure maths. It's more of a methods course.

Also, if anyone could give me opinions on what the best universities are to study maths that would also be appreciated.

Cambridge, Warwick, Imperial, Oxford.

Reply 3

Look at some of the stuff in e.g. the new edexcel further pure maths 2, particularly the number theory, group theory, and proof by induction of sequences and series. This is the kind of stuff that maths degrees are made of (also you'll get some more calculus with differential equations and vector calculus).

Also the differentiation by first principles is actually a hand wavy formulation of the fundamental theorem of calculus.

Reply 4

Original post by MajorFader

Are you doing a maths degree? What a level subjects did you do?

I'm not doing a maths degree, but I'm quite familiar with the content :tongue:

My educational background is a mess and so not at all useful for a comparison (I didn't do A-level for that matter anyway :smile:

) !

But as other posters stated, it's a huge difference in style of how it's "done"; in fact you'll probably realise that you weren't actually "doing" maths at A-level, you were just "using" maths :biggrin:

Reply 5

Original post by artful_lounger

Also the differentiation by first principles is actually a hand wavy formulation of the fundamental theorem of calculus.

What do you mean by this...?

(edited 7 years ago)

Reply 6

Original post by Zacken

Uh... what?

Well the construction of the derivative part anyway :P

Which you can work backwards from to the definition of the integral, if you were so inclined and careful with your definitions.

My point being, that consideration of how the derivative is built up (and by extension the integral i.e. the FΘC ) is an important result and also by considering it very carefully and being very precise in your definitions this is actually how the rest of maths (well certainly analysis) gets put together.

For the purposes of an A-level student it's sufficient to look at it further and more carefully, I'm not proposing mathematicians start using "differentiation from first principles" as an approach to constructing the integral

Reply 7

Original post by artful_lounger

Well the construction of the derivative part anyway :P

Right, but I don't see how the definition of the derivative is at all a hand-wavy version of the FoC at all. In fact, it's necessary for it.

Reply 8

It's hand wavy in the sense that it doesn't precisely discuss where it's applicable, the necessary conditions for it to be so etc; it obscures this by using a more intuitive sense of the concept of the limit.

I am aware you need the definition of a derivative but this is not really a definition so much as an intuitive derivation of it. Hence handwavy.

Equally I loathed "proper" calculus and analysis so...I'll stick with the handwavy concepts xD

Reply 9

MacaronMeg

I was a bit like you as I hated (and still hate) mechanics, so I looked at all the university's (that I was interested in) modules lists to see if they had too many compulsory mechanics courses eg dynamics. I applied to Oxford (as they had less mechanics than Cambridge), Southampton, Bath, Exeter and Reading. Some have one compulsory mechanics type module first year but after that they are all optional.

Reply 10

Original post by artful_lounger

I think we may be talking about different things: to me, differentiation from first principles refers to the definition of a derivative. That is, a real-valued function

f: A \to \mathbb{R}

is differentiable at

a

if the limit if

\lim_{x \to a} \frac{f(x) - f(a)}{x-a}

exists (and we call it

f'(a)

), where limit refers to

\forall \epsilon>0 \exists \delta > 0 \forall x \in A \, \, 0 < |x-a| < \delta \Rightarrow \left|\frac{f(x) - f(a)}{x-a} - f'(a)\right| < \epsilon

.

I can't see the link to FoC. It might not be explained so rigorously at A-Level (which is what you might have been talking about) but then it's a hand-waving of limits, not FoC.

Reply 11

Also OP, if you're interested in maths at uni, the age old suggestion of finding a copy of Spivak's Calculus (not calculus on manifolds, stay away from that until you've got at least a year of uni maths under your belt lol) and working through that isn't a bad idea. I'd look in your local library possibly, or maybe see if you can get a copy off ebay or something for ~£20ish or under. There are always the dark corners of the internet as well...

It should be challenging, but theoretically accessible, and is the kind of thing maths students will encounter. If anything, even if you can't necessarily do all (or any) of the problems, you'll have a chance to start thinking about things in a more formal way as you would in an undergrad maths course (although it works through from your intuition of calculus as has been developed in A-level/equivalent to developing the formal statements).

In fact most maths students (at least, once they get to the PhD level) seem to be of the opinion that attempting all the problems will make you a much better mathematician and it's generally quite popular due to the style of it's writing.

(edited 7 years ago)

Reply 12

reunion.

em planning for liverpool tbh tis good

Reply 13

Original post by Zacken

I think we may be talking about different things: to me, differentiation from first principles refers to the definition of a derivative. That is, a real-valued function

f: A \to \mathbb{R}

is differentiable at

a

if the limit if

\lim_{x \to a} \frac{f(x) - f(a)}{x-a}

exists (and we call it

f'(a)

), where limit refers to

\forall \epsilon>0 \exists \delta > 0 \forall x \in A \, \, 0 < |x-a| < \delta \Rightarrow \left|\frac{f(x) - f(a)}{x-a} - f'(a)\right| < \epsilon

.

I can't see the link to FoC. It might not be explained so rigorously at A-Level (which is what you might have been talking about) but then it's a hand-waving of limits, not FoC.

FΘC necessarily invokes the definition of a derivative in order to define the integral. I feel like we're going in circles and getting a bit off topic though, but certainly everything you've said has seemed correct to me so I will defer to your knowledge which undoubtedly dwarfs my own :smile:

Reply 14

Also I can confirm Southampton doesn't have a lot of mechanics, you do about ~1/3 of a first year module of mechanics, and then you can pretty much drop it for the rest of the course (barring a bit of fluid mechanics in second year, which is a fairly decent way to introduce vector calculus anyway since otherwise the concept of grad div curl are kind of ambiguous and EM isn't as intuitively obvious in the physical implication of them).

You would have to do at least 3 modules of stats then though, so your mileage may vary...you may prefer the treatment of mechanics in a maths degree (i.e. derived as differential equations and through vector calculus) as opposed to "formula collecting" as in A-level.

Reply 15

dugdugdug

Original post by ABC789

There are broadly two branches of pure maths at uni - analysis and algebra.

An example of analysis would be calculus, in particular, google Riemann Integrals. If that's too easy, proceed to Ito and Stratonovich Integrals.

Algebra would be vectors and matrices, specifically proving theorems (check out fundamental theorem of algebra).

It does have its uses. In stats, you have something called degrees of freedom. That stems from the Rank Nullity Theorem in algebra.

Reply 16

RichE

Original post by dugdugdug

There are broadly two branches of pure maths at uni - analysis and algebra.

Geometry, number theory, logic, set theory, topology, ...

Reply 17

Original post by RichE

Geometry, number theory, logic, set theory, topology, ...

Topology is kind of a branch of analysis, logic and set theory aren't commonly done as subjects unto themselves so much as tools to develop the necessary theorems in other subjects.

Number theory is separate, although usually not developed extensively except through optional modules usually (typically you might do a single number theory module vs 2-4 in algebra and analysis).

Geometry is very variable; basic elements of euclidean geometry tend to be covered alongside linear algebra (in the main algebra sequence) while differential geometry. algebraic geometry etc are a) optional type modules that may or may not be offered and b) often more algebra or analysis leaning.

I mean yes those are all things that come up in the course of a maths degree but the original statement that it's primarily algebra/analysis isn't incorrect or misleading.

Reply 18

RichE

Original post by artful_lounger

Topology is kind of a branch of analysis, logic and set theory aren't commonly done as subjects unto themselves so much as tools to develop the necessary theorems in other subjects.

If you're saying that then I suspect you've only met elements of general point-set topology.

Number theory is separate, although usually not developed extensively except through optional modules usually (typically you might do a single number theory module vs 2-4 in algebra and analysis).

Geometry is very variable; basic elements of euclidean geometry tend to be covered alongside linear algebra (in the main algebra sequence) while differential geometry. algebraic geometry etc are a) optional type modules that may or may not be offered and b) often more algebra or analysis leaning.

I mean yes those are all things that come up in the course of a maths degree but the original statement that it's primarily algebra/analysis isn't incorrect or misleading.

You're of course welcome to your opinion but I do think the statement is incorrect. It's a view that aligns with how a first year might be set up but would become more false during the degree. That there may well be more algebra and analysis is a function of what's missing in school maths and what's deemed useful material for instilling an appreciation of rigour and abstraction, not about some primacy of algebra and analysis within pure maths.

Reply 19