Girls vs boys maths challenge

Here's another one:
Show that the triangle that maximises a given area is equilateral.
I haven't really looked at this question, so sorry if it turns out to involve variational calculus or something .

Or the arguably non-existent r=0
Here's another one:
Show that the triangle that maximises a given area is equilateral.
I haven't really looked at this question, so sorry if it turns out to involve variational calculus or something - it sounds vaguely relevant and a little bit harder level
You might or might not find it helpful to know Heron's formula for the area of a triangle: that if $s=\dfrac{a+b+c}{2}$, then $A = \sqrt{s(s-a)(s-b)(s-c)}$.

Ohh interesting. He researches gen. rel., right? Oh that's awesome. If I've somehow scraped my way in I may well end up drifting towards that department.

Interrogation: What year are you in at the moment then? How are you finding the course and/or coping with it? What areas most interest you? Next year options? etc.. etc.. etc..
Edit: Btw, on the above example, what does 'variational principles' offer? Is this some crazy generalisation using imaginary/non-euclidean metric spaces? Can one not use the three techniques that spring to mind? (leaving it for the younger ones though, of course, unless you make us look up variational principles to try and 'wing' a proof).

It doesn't. Just Lagrange multipliers. In fact I'm pretty sure that even that can b avoided.

Pretty much every applied example sheet we get, he says "Oh, I wrote that question five years ago" or similar on at least two questions :P I'm not sure if there actually is a maths department at all - it's just Siklos and about fifty figureheads…

do you know much about applying to cambridge for postgrad? I was thinking of doing that for my Masters or something like that but I don't know how it works :/

Have you completed/are you shortly to complete an undergrad course? If not, don't worry about this yet - know only that it is pretty common to take the Masters at Cambridge without having studied at Cambridge before
If you are in a position where it would be useful to know, I'm afraid my knowledge of Part III admissions is somewhat limited. http://www.maths.cam.ac.uk/postgrad/mathiii/ might be helpful.

No, I'm about to start my first degree in September xD I'm only asking because I'm planning ahead different possible routes I can take. Thanks for the link

It says I need to get a first as one of the requirements, so I suppose I should see how my undergrad degree goes first and if it's going well then start planning for postgrad :L

Yeah, and the masters is also aimed solely at people who want to go on for research - not just for an extra qualification.

E, C and A?

Nooooo, DAMTP is the dark side! Pure is the way forward Yep, he's GR. "Scraped in" is perhaps a bit pessimistic, from the look of it :P

I'm going into the second year, loving the course I didn't do enough work earlier in the year, so it was a big rush before the exams, but I've got a better work ethic/schedule now.

Group theory is a great thing, I love GRM (Groups, Rings and Modules, it's in Part IB but loads of first years take the lectures early). I'm definitely a puremo! [in case you didn't know, mathematicians are "mathmos", and the term is extensible - "appliedmo", "puremo", "Trinmo" for Trinity, etc] I'm avoiding everything applied, apart from quantum, which is as close to compulsory as they come.

Variational principles is just a more complicated way to get the answer - it would be a disgusting method in this case. It's simply a way of extremising $I[x] = \int_a^b f(x,y,y',y'', \dots z,z', \dots) dx$ for a,b,f fixed. That is, you use it to find y, z, etc such that I[x] is extremised. It's a yucky but very general way to do this kind of thing
Nah, we've not really done anything explicitly in non-euclidean metric spaces - the closest we've come is topological spaces, which we've not covered in much depth (that's a IB course too). Almost everything applied has been in $\mathbb{R}^n$ for some n…

Three techniques that spring to mind - I can only think of one plausible one (divide the thing up into two right-angled triangles), but my problem-solving method is: "Generate a possible line of attack. See if line of attack works. Repeat." so I would usually only have one line of attack at a given time, even on such a simple problem. Actually, the answer is clearly at least isosceles by the most rudimentary application of symmetry, so there's several more lines of attack opened up that would simplify a variational calculus proof a lot, too, knowing that a=b.
Bleugh, late at night and incoherent…

It doesn't. Just Lagrange multipliers. In fact I'm pretty sure that even that can b avoided.

Nooooo, DAMTP is the dark side! Pure is the way forward Yep, he's GR. "Scraped in" is perhaps a bit pessimistic, from the look of it :P

I'm going into the second year, loving the course I didn't do enough work earlier in the year, so it was a big rush before the exams, but I've got a better work ethic/schedule now.

Group theory is a great thing, I love GRM (Groups, Rings and Modules, it's in Part IB but loads of first years take the lectures early). I'm definitely a puremo! [in case you didn't know, mathematicians are "mathmos", and the term is extensible - "appliedmo", "puremo", "Trinmo" for Trinity, etc] I'm avoiding everything applied, apart from quantum, which is as close to compulsory as they come.

Variational principles is just a more complicated way to get the answer - it would be a disgusting method in this case. It's simply a way of extremising $I[x] = \int_a^b f(x,y,y',y'', \dots z,z', \dots) dx$ for a,b,f fixed. That is, you use it to find y, z, etc such that I[x] is extremised. It's a yucky but very general way to do this kind of thing
Nah, we've not really done anything explicitly in non-euclidean metric spaces - the closest we've come is topological spaces, which we've not covered in much depth (that's a IB course too). Almost everything applied has been in $\mathbb{R}^n$ for some n…

Three techniques that spring to mind - I can only think of one plausible one (divide the thing up into two right-angled triangles), but my problem-solving method is: "Generate a possible line of attack. See if line of attack works. Repeat." so I would usually only have one line of attack at a given time, even on such a simple problem. Actually, the answer is clearly at least isosceles by the most rudimentary application of symmetry, so there's several more lines of attack opened up that would simplify a variational calculus proof a lot, too, knowing that a=b.
Bleugh, late at night and incoherent…

It doesn't. Just Lagrange multipliers. In fact I'm pretty sure that even that can b avoided.

Phew


Apologies, I posed it badly - I meant to say "the triangle with fixed perimeter that maximises the area is equilateral".
Actually, this is a fairly nice way to extoll the virtues of symmetry
OK, so I've told you that the area of the triangle is that expression. The area is fixed, so let's call it A. (It's the same A as in that expression.) Let's call the side lengths a,b,c.
Now, what are you trying to show? Can you give me a couple of equations that sum up the problem "Prove that the triangle with area A should be equilateral"? (I realise it's quite hard to tell what I mean by that - say if you don't understand!)

I'm very unsure. I love mathematics as an objective art (or because because it is 'the' objective art) but have lately realised that lots of Calculus/Analysis 'apparently' is labelled as 'applied'?! I also love Physics though have never been too keen on Newtonian Mechanics (I feel as though, if I'm going to do ugly maths, then it better damn well mean something!) I do, of course, have no idea which I will opt for at the moment.. hopefully the next year or two will be long enough to decide.

Well I might have cocked up STEP.. and unfortunately there isn't much else..

Awesome. Why not enough work earlier in the year? Social life, etc.. ? Btw, do you ever get to just hang about a chalkboard (with other students) doing random maths? And also, do you meet loads of brilliant mathematicians?

I am yet to be impressed by group theory. It seems that most of the significant applications and links to other areas of maths that can be drawn from it are rather inaccessible (none the less, I persist..). Would love it if you have any examples of it's power and/or importance?

"Trinmo" lmao, please say that's not seen as some sort of 'status'?

Oh god, I need to do so much vectors over the summer..

Actually, come to think of it, perhaps M3/M5-dodgy-limit-calculus could be used? i.e. maximising M.O.I and similar things and making deductions etc.. hmm..

Hold on.. you say 'maximise for a given area'. I don't think that makes sense? Is this subject to the sum of the lengths being constant or something like that?

I thought about dividing it into two right-angled triangles and going from there, but I ended up with a pile of gibberish. Attempted calculus, but I ended up with a pile of gibberish. AS level is going to go reeeeeally well at this rate.

Don't worry - it turns out that the problem is quite hard using GCSE maths. AS and A-level is nothing like this - they'll as good as tell you exactly what to do in each question
I'll give you a step-by-step answer instead - do you know about partial differentiation? (Or any differentiation at all - we can work with that.)

Don't worry - it turns out that the problem is quite hard using GCSE maths. AS and A-level is nothing like this - they'll as good as tell you exactly what to do in each question
I'll give you a step-by-step answer instead - do you know about partial differentiation? (Or any differentiation at all - we can work with that.)

Or the arguably non-existent r=0
Here's another one:
Show that the triangle that maximises a given area is equilateral.
I haven't really looked at this question, so sorry if it turns out to involve variational calculus or something - it sounds vaguely relevant and a little bit harder level
You might or might not find it helpful to know Heron's formula for the area of a triangle: that if $s=\dfrac{a+b+c}{2}$, then $A = \sqrt{s(s-a)(s-b)(s-c)}$.

Hey! Could you possibly explain that partial differentiation please? I've done C1,C2 and have just started C3. I can differentiate e^x, ln x, sin, tan, cos and using the chain, product and quotient rules if that helps. Thanks

Great, that's all we need
OK, so the key point is that the differentiation you've learnt is like:
$f(x) = \log(x) \implies \dfrac{df}{dx} = \dfrac{1}{x}$.
We want a nice consistent way to define differentiation in the case of (for example) $f(x,y) = y \log(x)$.
There's a very simple way we do this:
$\dfrac{\partial f}{\partial y} = \log(x), \dfrac{\partial f}{\partial x} = \dfrac{y}{x}$.
That is, we differentiate as if everything is constant except the variable we're differentiating with respect to. I'll now explain this better :P
Now, the key thing is that, just as a stationary point of $f(x) = x^2$ can be found by differentiating and setting to 0 - that is, $\dfrac{df}{dx} = 2x = 0$ so $x=0$ is the stationary point - so we can find a stationary point of $f(x,y)$ by partial-differentiating and setting to 0. Think of it as: we're making a 3D diagram with x and y on the x and y axes, and with height representing f(x,y) at the point (x,y). Then any stationary point (for example, a minimum - that looks like a bowl on our 3D diagram) must be stationary if we slice through the x-axis or the y-axis.
Example series of pictures:


Now, the key thing is that we can easily do this "slicing" just by setting x or y to be constant (in the example, I've set x=0). But if x is constant, then if we differentiate with respect to y, we don't care about x - it just behaves as if it were (say) 2.
So $\dfrac{\partial}{\partial y}(x^2 + y^2) = 2y$, because the x^2 was constant so when we differentiated, it became 0. $\dfrac{\partial}{\partial x}(x^2 \sin(y)) = 2x \sin(y)$, because y was constant so sin(y) was constant.

OK so far? That's partial differentiation in a nutshell - my next post, if you're with me so far, will be how to solve the triangles question using this.