# Oxbridge Physics Prelims Revision

Scroll to see replies

seeing that I have a maths exam tomorrow, I should probably do some revision (although I feel pretty hopeless at trying to learn stuff which I cant really do anyway...you could say I was lucky I had questions I could manage on monday but still). Just a quick question:

Find the derivative with respect to a of:

$\int_{0}^{a^2} \frac{\sin ax}{x} dx$

I know you can flip the order of integration/differentiation hence write:

$\int \frac{\partial}{\partial a}\frac{\sin ax}{x} dx = \int \cos ax dx$

but how do you change the limits in this case? (since the limits are themselves functions of a)

cheers!
Willa
seeing that I have a maths exam tomorrow, I should probably do some revision (although I feel pretty hopeless at trying to learn stuff which I cant really do anyway...you could say I was lucky I had questions I could manage on monday but still). Just a quick question:

Find the derivative with respect to a of:

$\int_{0}^{a^2} \frac{\sin ax}{x} dx$

I know you can flip the order of integration/differentiation hence write:

$\int \frac{\partial}{\partial a}\frac{\sin ax}{x} dx = \int \cos ax dx$

but how do you change the limits in this case? (since the limits are themselves functions of a)

cheers!

I get sin(a^3)/a

But am in a rush so will have to explain later

Try setting up the integral the integral I(a,x) = Int[t:0_>x] sin(at)/t dt

Work out I's partial derivatives wrt a and x

Then use the chain rule to work out the derivative of I(a,a^2)
RichE
I get sin(a^3)/a

But am in a rush so will have to explain later

Try setting up the integral the integral I(a,x) = Int[t:0_>x] sin(at)/t dt

Work out I's partial derivatives wrt a and x

Then use the chain rule to work out the derivative of I(a,a^2)

*gasp* RichE must have got something wrong!!! That's exactly the answer I handed in back ages ago when I did this question and it is wrong. Because I said you do the integral on the cos ax between a^2 and 0, and you'll get your 1/a *sin a^3. But that's definitely wrong. I've got a note scribbled next to it saying "You must change the limits -> They are functions of a"....but I cant remember how to apply that!

And sorry I dont have the answer written down i'm afraid, but that answer you just gave is definitely wrong! (i think)
Willa
*gasp* RichE must have got something wrong!!! That's exactly the answer I handed in back ages ago when I did this question and it is wrong. Because I said you do the integral on the cos ax between a^2 and 0, and you'll get your 1/a *sin a^3. But that's definitely wrong. I've got a note scribbled next to it saying "You must change the limits -> They are functions of a"....but I cant remember how to apply that!

And sorry I dont have the answer written down i'm afraid, but that answer you just gave is definitely wrong! (i think)

Well I certainly didn't get my wrong answer that way! If you'd read my hint you would have realised this. Though as ever your response is just dripping with gratitude at the time I spend on these questions.

Checking my working now -

I(a,x) = Int[t:0->x] sin(at)/t dt

I_2(a,x) = sin(at)/t

I_1(a,x) = Int[t:0->x] pd/pda (sin(at)/t) dt = Int[t:0->x] cos(at) dt

= sin(ax)/a

[this is where my error came from previously, I'd got this bit as -sin(ax)/a]

You are interested in F(a) = I(a,a^2)

So F'(a) = I_1(a,a^2) + 2a I_2(a,a^2) =

sin(a^3)/a + 2a sin(a^3)/a^2 = 3sin(a^3)/a

Which I hope is ok - I have never made any claims of infallibility - but frankly with your attitude to my posts I don't know why I f*cking bother
Willa
is this simon or chloé?

It was me (Chloé). Simon and two of the other Lincoln Physicists made me drink a bottle of cava and 2 pimms very quickly on an empty stomach straight after I'd finished. He then dragged me back to college on the bus (apologies to anyone who was on it at the same time as me) and left me in my room for an hour while he went to sing with OOTB. I was then sick and thankfully that sobered me up somewhat (I came to the computer when my friend rang my internal number and discovered Simon had been using it and then somehow I found TSR!). Have spent 3hours this afternoon punting and we're off to Pizza Hut soon to eat and am now completely sober (thank god as meeting parents at 7pm for theatre!!!).

Well done to everyone who's finished and good luck to everyone (Willa & Camford) who still have them to go
RichE
Well I certainly didn't get my wrong answer that way! If you'd read my hint you would have realised this. Though as ever your response is just dripping with gratitude at the time I spend on these questions.

Checking my working now -

I(a,x) = Int[t:0->x] sin(at)/t dt

I_2(a,x) = sin(at)/t

I_1(a,x) = Int[t:0->x] pd/pda (sin(at)/t) dt = Int[t:0->x] cos(at) dt

= sin(ax)/a

[this is where my error came from previously, I'd got this bit as -sin(ax)/a]

You are interested in F(a) = I(a,a^2)

So F'(a) = I_1(a,a^2) + 2a I_2(a,a^2) =

sin(a^3)/a + 2a sin(a^3)/a^2 = 3sin(a^3)/a

Which I hope is ok - I have never made any claims of infallibility - but frankly with your attitude to my posts I don't know why I f*cking bother

Yes I realised that you hadnt used my incorrect method, I just wanted to save myself the time of trying to understand the method you have given when it may well be wrong! As it turns out it wasnt the method that was wrong just the working, so I'll have a crack your working. And look, I'm sorry if you dont think I appriciate this help, cos I really really really honestly do. My attitude towards your responses is quite confrontational simply because: 1) I am short of time to try and understand, 2) I honestly dont understand, and 3) I know that you have to be right because you always are! That's what's so damn frustrating, that you present what is obviously the right method to answering a question every time, and yet I am incapable of understanding it.
I think we all know on here how much the methods adopted at cambridge and oxford differ....whenever i have posted my reasoning on here to a question, it has often greatly differed from that of oxford physicists. Likewise for maths, I think the way I approach maths is a very different way to the way you approach maths. I'm sorry if you dont think I appriciate the effort because I really do (and I do want to partly blame the exam stress on this), it's just I want to try and understand your reasoning, I'm not a man who is happy to just learn "this is how to solve this type of question"...I need to understand WHY you are doing something on each line, and I have found your answers to physics problems a tad to self-consistent for me to handle (and hence why I challenge the reasoning, cos I dont understand it).
Take that whole relativity stuff from a week back. From your reasoning all I could deduce was a sort of circular consistency: The lorentz transformation is correct, hence you have this change in time when measuring something, hence the lorentz transformation is correct. If that isnt what your working showed then clearly I misunderstand your working, but that is what I got from what you posted. LM started posting his working, which to me was constructed as follows: One observer sees lengths contracted in the other frame, hence lorentz transformation is correct. Unfortunately his reasoning failed, but that was the type of working I was after: working from something I understand (lorentz contraction for observers watching objects stationary in other frames of ref) up to what i didnt understand (the lorentz transformation).
Take also the rotation question i asked. All I got of your explanation was: It's rotating so you cant use linear mechanics. But then we did use linear mechanics to solve the second part, so I didnt understand that explanation either.

With this question I just asked I'm sorry I jumped on the fact that you had given a wrong answer, it was just as I said, because it takes me a great deal of time to understand your methods, and so it seems pointless to try and understand a method which it looks as though gives incorrect results. As I said though, ultimately you are always right, I just rarely understand why.

can you forgive me?

and now I will have a crack at your method, thanks
Hoofbeat
It was me (Chloé). Simon and two of the other Lincoln Physicists made me drink a bottle of cava and 2 pimms very quickly on an empty stomach straight after I'd finished. He then dragged me back to college on the bus (apologies to anyone who was on it at the same time as me) and left me in my room for an hour while he went to sing with OOTB. I was then sick and thankfully that sobered me up somewhat (I came to the computer when my friend rang my internal number and discovered Simon had been using it and then somehow I found TSR!). Have spent 3hours this afternoon punting and we're off to Pizza Hut soon to eat and am now completely sober (thank god as meeting parents at 7pm for theatre!!!).

Well done to everyone who's finished and good luck to everyone (Willa & Camford) who still have them to go

Yeah you didnt look too good. I was gonna say hi and chat and so on but you didnt look at all well so I thought I would leave it. Hope you are feeling better/ less drunk now

Howd exam go? I think It was really good. Im pretty much sure I nailed it. I finished with 10 mins to spare. I love quantum! I just wish theyd been like that last week.
F1 fanatic
Yeah you didnt look too good. I was gonna say hi and chat and so on but you didnt look at all well so I thought I would leave it. Hope you are feeling better/ less drunk now

I really did feel rough. I don't drink wine/champagne usually (I don't have expensive tastes...I prefer stuff like Archers, Vodka, Bacardi etc etc. I am indeed feeling much better now (having to punt sobered me up loads). I'm not a big drinker, and I haven't had a proper drink in ages and on a completely empty stomach (I didn't eat much at hall the night before) I was a goner very quickly. Remind me never to go drinking alone with guys....they're mean (luckily, Simon looked after me and got me back here safely).

Howd exam go? I think It was really good. Im pretty much sure I nailed it. I finished with 10 mins to spare. I love quantum! I just wish theyd been like that last week.

It wasn't amazing for me but then I'm not really suprised as I could never do the topic. I think I possibly managed to get the 20marks needed to pass. Now all we can do is sit and wait for our results...well actually go and have lots of fun! YAY!
Hoofbeat
It wasn't amazing for me but then I'm not really suprised as I could never do the topic. I think I possibly managed to get the 20marks needed to pass. Now all we can do is sit and wait for our results...well actually go and have lots of fun! YAY!

hey everyone, glad they're finally over - a bit of an anticlimax for me as I got really drunk last thursday, just felt exhausted today. Anyone know when we actually get our results?
Willa

As it turns out it wasnt the method that was wrong just the working, so I'll have a crack your working.

Well I've been guilty of similar recently so won't take issue, and I understand the exam stress thing. Further you shouldn't be annoyed if you feel I understand maths well because I'm older and it's my area. There are plenty of physics questions you ask I can't comprehend let alone answer.

My working on this particular though involves only the chain rule and differentiation under the integral sign (the rule which you mentioned).

Willa

I think we all know on here how much the methods adopted at cambridge and oxford differ....whenever i have posted my reasoning on here to a question, it has often greatly differed from that of oxford physicists.

Undoubtedly the case

Willa

I need to understand WHY you are doing something on each line, and I have found your answers to physics problems a tad to self-consistent for me to handle (and hence why I challenge the reasoning, cos I dont understand it).
Take that whole relativity stuff from a week back. From your reasoning all I could deduce was a sort of circular consistency: The lorentz transformation is correct, hence you have this change in time when measuring something, hence the lorentz transformation is correct.

I never claimed this route. The way SR is taught to the mathematicians in Ox, everything revolves around the Lorentz transformation. This is the right change of co-ordinates between observers based on how distances/time are measured and assuming the constancy of light in each frame - i.e. based on physical laws. From that I deduced the Lorentz contraction (which had previously been no part of my assumptions). Your misconception seemed to be one of physics, not of maths, in that you did not acknowledge that the observer was measuring the rod's length between events that weren't simultaneous in the rest frame of the rod and this was what was missing from LM's calculation. This is again an important physical aspect of SR, that simultaneity isn't absolute and the interaction of the space and time co-ordinates is a crucial difference in this theory from that of Galilean relativity.

I am repeating all this I am afraid, but you said you wished to see the why behind my working - I didn't just give you the equations, I gave you their interpretations as well.

"LM started posting his working, which to me was constructed as follows: One observer sees lengths contracted in the other frame, hence lorentz transformation is correct."

On what basis can we say this? Where is the reasoning? The Lorentz-map is based on how distances/times are calculated experimentally. Look up the appropriate space-time diagrams in any book on SR.

"Take also the rotation question i asked. All I got of your explanation was: It's rotating so you cant use linear mechanics."

The reason I gave for that was that generally the system wasn't free to move in all directions, just free to rotate - and I claimed that that was at best an educated guess.

If you don't like my answers fine. They are a mathematician's view of things - if they actually contribute negatively to your understanding then I will give up trying to help.
Willa
seeing that I have a maths exam tomorrow, I should probably do some revision (although I feel pretty hopeless at trying to learn stuff which I cant really do anyway...you could say I was lucky I had questions I could manage on monday but still). Just a quick question:

Find the derivative with respect to a of:

$\int_{0}^{a^2} \frac{\sin ax}{x} dx$

I know you can flip the order of integration/differentiation hence write:

$\int \frac{\partial}{\partial a}\frac{\sin ax}{x} dx = \int \cos ax dx$

but how do you change the limits in this case? (since the limits are themselves functions of a)

cheers!

i personally would just do integration by parts. take u=sinax and dv=1/x, then you get du=acosax and v=lnx, then you do the uv - &#8747;v.du and thats sinax.lnx - a&#8747;lnx.cosax.dx. Then you do integration by parts again, but let u=lnx this time and it probably works out eventually. If you do it enough times (maybe four?) you end up with taking away the original integral which you can just take over to the other side. Or something.

Can I just ask, was that this term's work?

edit: does integration by parts even work for that?

*resolves not to answer any stupid integration questions tomorrow*
Lozza
i personally would just do integration by parts. take u=sinax and dv=1/x, then you get du=acosax and v=lnx, then you do the uv - &#8747;v.du and thats sinax.lnx - a&#8747;lnx.cosax.dx. Then you do integration by parts again, but let u=lnx this time and it probably works out eventually. If you do it enough times (maybe four?) you end up with taking away the original integral which you can just take over to the other side. Or something.

Can I just ask, was that this term's work?

edit: does integration by parts even work for that?

*resolves not to answer any stupid integration questions tomorrow*

No you dont do integration by parts on it I'm afraid. I know it should be something to do with changing the limits, I cant see that directly from richE's working though, I'll keep trying to crack it.

And this term we did matrices and limits and series - questions which I will be avoiding if I can.

I am hoping for a nice surface/volume integral question tomorrow, with another fourier series question, then a nice vector question or two, perhaps some differential equation stuff (but they had two in the last paper so dunno if it will appear). I tend to avoid the really pure stuff which I cant visualise: taylor series etc I just cant stand. I am very much a visual learner - I want to be able to visualise the mathematics.
Ansorge mentioned that stuff in Michaelmas' lectures. Frankly, I can't be bothered with that kind of thing. plus I can't remember how you change the limits either.
Lozza
i personally would just do integration by parts. take u=sinax and dv=1/x, then you get du=acosax and v=lnx, then you do the uv - &#8747;v.du and thats sinax.lnx - a&#8747;lnx.cosax.dx. Then you do integration by parts again, but let u=lnx this time and it probably works out eventually. If you do it enough times (maybe four?) you end up with taking away the original integral which you can just take over to the other side. Or something.

If you do integration parts twice (for any product) and swap the roles of the u,v functions you'll just get back to showing 0=0. You have to persist with one of the functions - differentiating it out (like with a poly) or getting back to somewhere you've been before (like e^x sinx)
RichE I dont get this bit in your working:

You are interested in F(a) = I(a,a^2)

So F'(a) = I_1(a,a^2) + 2a I_2(a,a^2) =

How are you relating I to I_1 and I_2?
cheers
Willa
RichE I dont get this bit in your working:

How are you relating I to I_1 and I_2?
cheers

I_1 means differentiation wrt the first variable and I_2 wrt the second - which you can calculate one by plain differentiating an integral, and the other by differentiating under the integral sign

Then I apply the chain rule

d/dt (F(f(t),g(t)) = f'(t) F_1(f(t),g(t)) + g'(t) F_2(f(t),g(t))
ahhh that makes things a little clearer, thanking you!

You want

I(a) = Int[x:0->a^2] sin(ax)/x dx

If you make the substitution t = ax you get

I(a) = Int[t:0->a^3] sint/t dt

<with a little working>

Now if J(s) = Int[t:0->s] sint/t dt then J'(s) = sins/s

I(a) = J(a^3)

So I'(a) = 3a^2 J'(a^3) = 3a^2 sin(a^3)/a^3 = 3 sin(a^3)/a

This way at least avoids the use of differentiating under the integral sign and of the two-variable chain rule
oh i really dont get this. i'm hoping for complex numbers, matrices, gaussian elimination, vectors, a nice partial differentiation q with some stationary values and maybe a power series. i dont care any more. my dad told me he didn't send me to cambridge to be a geek. he said if i get a 1st this year great, if not, then i've had a lot of fun, made loads of friends, done loads of extra curric stuff etc. i'll get a 1st in 3rd year. *determined*
well i have been personally having problems in exams of for some bizarre reason suddenly imaging scenes from "The Office". In chemistry for example, I couldnt get the scenes from episode 3 of series one out of my head..."I bloody hate him sometimes clever and funny...that's probably why we get on actually....similar!"