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Muon's Daily Revision Summary!

This is basically me spilling my thoughts out onto a webpage so people may or may not see them and gain some insight!

Every evening once a day I will write a brief overview of what I have learnt (or not) including mainly maths tips/techniques (A-level, STEP and beyond...) and general reflections about all subjects/revision!

This is not intended to copy all the other "My road to 19A*s" threads (as proper Gs like @Zacken will have that sorted-enjoy your imac :wink: ). I just intend to share the key points from what I have learned to benefit my self and others, and make sure I do my best in exams! I feel so much of achieving your best is due to motivation so hopefully this will get me doing everything I can :smile:

So a bit about me:

A levels/Subjects:
Maths-(C1-4,M1-3,S1-2)(Completed, A*),
Further Maths- (FP1-3,M4-5)
Biology
Physics
STEP I/II/III
EPQ (Group Theory, Completed)

AS subjects:
Biology, Chemistry, Physics, Maths (As/A*)

Offers(Mathematics):
King's College Cambridge - A*A*A 1,1 STEP II/III. Firmed.
Warwick - AA 2 in any STEP (or some other funny conditions :tongue:). Insurance.

Well anyway, so long as someone out there can benefit at least once from this, ill be happy :smile: (I hope this is in the right forum). Feel free to post any comments/feedback.

Scroll to see replies

Sounds terrific :h: Looking forward to this - and wow didn't know you were a Cam Maths offer holder :redface:
Reply 2
Avatar for Marxist
Marxist
10
Original post by EnglishMuon
Hi


Hi
Reply 3
Avatar for Zacken
Zacken
22
Original post by EnglishMuon

Offers(Mathematics):
King's College Cambridge - A*A*A 1,1 STEP II/III. Firmed.
Warwick - AA 2 in any STEP (or some other funny conditions :tongue:). Insurance.


This is exactly my status right now as well! Firmed King's and insure Warwick, let's hope we both end up at our firm! Crazy to think I might be seeing/studying with you at King's. :eek:

Slightly more on topic: would you happen to know of any good resources for learning about polar coordinates beyond the usual A-Level stuff? A blog post or some videos or a good article/paper or such? I was especially interested about dealing with areas of polar curves when they have "loops" or "petals" or intersections of such curves, etc...
Original post by Student403
Sounds terrific :h: Looking forward to this - and wow didn't know you were a Cam Maths offer holder :redface:


Yep :smile: Been hiding in the shadows :wink: But thanks for the feedback!
Original post by Zacken
This is exactly my status right now as well! Firmed King's and insure Warwick, let's hope we both end up at our firm! Crazy to think I might be seeing/studying with you at King's. :eek:

Slightly more on topic: would you happen to know of any good resources for learning about polar coordinates beyond the usual A-Level stuff? A blog post or some videos or a good article/paper or such? I was especially interested about dealing with areas of polar curves when they have "loops" or "petals" or intersections of such curves, etc...

Yeah I know right! Ill see you at Kings then, Im sure :smile: But yeah, im more than happy to find some resources on polar coordinates. I mean I havent yet come across anything yet but maybe something to look at is spherical coordinates if you havent already?
You can derive some of the formulae e.g. integration/diff. yourself.
Consider a point with coordinates (r,θ,ϕ) (r,\theta,\phi) so after a displacement delta in the limit delta tends to 0 we have coordinates (line element) (r+dr,θ+dθ,ϕ+dϕ) (r+dr,\theta+d\theta,\phi+d\phi) and I think then this leads to dr=drr^+rdθθ^+rdϕϕ^ dr=dr\hat{r}+rd\theta\hat{\theta}+rd\phi\hat{\phi} for suitable (unit) vectors r,theta,z hat. We can then use this for deriving other things, but it seems to me like this will all turn into pure analysis. I have a few great analysis books I can recommend in that case, but not on solely polar coordinates!
Reply 6
Avatar for Zacken
Zacken
22
Original post by EnglishMuon
I have a few great analysis books I can recommend in that case, but not on solely polar coordinates!


I've started learning some analysis but haven't really gotten further than epsilon-delta/construction of the reals/convergence/divergence stuff, so if you've got any recommendations for a great analysis book, I'd love that! :biggrin:

Re: spherical coordinates, I'll have a look at some of the stuff next week.
Original post by Zacken
I've started learning some analysis but haven't really gotten further than epsilon-delta/construction of the reals/convergence/divergence stuff, so if you've got any recommendations for a great analysis book, I'd love that! :biggrin:

Re: spherical coordinates, I'll have a look at some of the stuff next week.


A great book in my opinion is Mathematical Analysis by K.G. Binmore- Its an old one but contains some really interesting questions and answers to all of them. Covers a great amount of stuff too. http://www.amazon.co.uk/Mathematical-Analysis-A-Straightforward-Approach/dp/0521288827
(Its unbelievable how much some of these books cost, I just go for second hand nearly always now being the cheapo I am :tongue:)
Original post by Zacken
I've started learning some analysis but haven't really gotten further than epsilon-delta/construction of the reals/convergence/divergence stuff, so if you've got any recommendations for a great analysis book, I'd love that! :biggrin:

Re: spherical coordinates, I'll have a look at some of the stuff next week.


A first course in Mathematical analysis. Pdf online aswell.
Original post by EnglishMuon
A great book in my opinion is Mathematical Analysis by K.G. Binmore- Its an old one but contains some really interesting questions and answers to all of them. Covers a great amount of stuff too. http://www.amazon.co.uk/Mathematical-Analysis-A-Straightforward-Approach/dp/0521288827
(Its unbelievable how much some of these books cost, I just go for second hand nearly always now being the cheapo I am :tongue:)



Why don't you make your school buy it for you? I had my school buy "A first course in Mathematical Analysis" by Burkill, the one physicsmaths recommended. I found that one from the Mathematical Tripos reading list, and they highly recommended it for Analysis I.

Original post by physicsmaths
A first course in Mathematical analysis. Pdf online aswell.
Reply 10
Avatar for Zacken
Zacken
22
Original post by physicsmaths
A first course in Mathematical analysis. Pdf online aswell.


You got the PDF online link? I tried looking for it, couldn't find it.
Reply 11
Avatar for kkboyk
kkboyk
21
Original post by Zacken
You got the PDF online link? I tried looking for it, couldn't find it.



I just found out I have a page bookmarked with the same title :colondollar:
http://users.uoa.gr/~pjioannou/analysis1/burkill_a_first_course_in_mathematical_analysis.pdf
Reply 12
Avatar for Zacken
Zacken
22
Original post by kkboyk
I just found out I have a page bookmarked with the same title :colondollar:
http://users.uoa.gr/~pjioannou/analysis1/burkill_a_first_course_in_mathematical_analysis.pdf


Oh gosh, thank you so much!! :biggrin:
Original post by Insight314
Why don't you make your school buy it for you? I had my school buy "A first course in Mathematical Analysis" by Burkill, the one physicsmaths recommended. I found that one from the Mathematical Tripos reading list, and they highly recommended it for Analysis I.


Good idea, Ill try. However unfortunately my school is in the 2nd most poorly funded region of the country and we're currently sitting on broken wooden stools since they apparently cant afford new ones.
Day 1 Summary
Well an unexpectedly large proportion of my day was spend completing and reviewing STEP III 2008, but the time was needed!

One point that I may not have appreciated enough until today is that for any function f(x) f(x) and any even function
g(x),f(x)∘g(x) g(x), f(x)\circ g(x) is always even. As obvious as it seems, I missed its use in the following example:
d2yndθ2+n2yn=0. \frac{d^2y_n}{d\theta^2}+n^2y_{n}=0. If x=cosθ,yn(1)=1 x=cos\theta, y_{n}(1)=1 and yn(x)=(−1)nyn(−x) y_{n}(x)=(-1)^{n}y_{n}(-x) , deduce y0=1,y1=x y_{0}=1, y_1=x

Spoiler

(some very vague/loose spoilers coming up, but nothing major about STEP III 2008).

Spoiler

I've been looking at a fair few questions on using vector equations to substitute for Cartesian equations, and came across the idea of Gram-Schmidt Orthonormalisation which has been extremely useful in visualising the change of basis for eigenvectors and similar concepts!

This process takes a finite linearly independent set of vectors S={v_{1},v_{2},...,v_{n}} and generates an orthogonal set that spans the same subspace. It effectively just removes the 'like' components of each vector in another's direction via the dot product making them orthogonal, and then normalising them as usual by dividing by the mod. Covered well on Khan academy: https://www.khanacademy.org/math/linear-algebra/alternate_bases/orthonormal_basis/v/linear-algebra-the-gram-schmidt-process
(edited 8 years ago)
Original post by Zacken
This is exactly my status right now as well! Firmed King's and insure Warwick, let's hope we both end up at our firm! Crazy to think I might be seeing/studying with you at King's. :eek:

Slightly more on topic: would you happen to know of any good resources for learning about polar coordinates beyond the usual A-Level stuff? A blog post or some videos or a good article/paper or such? I was especially interested about dealing with areas of polar curves when they have "loops" or "petals" or intersections of such curves, etc...


Yeh polar coodrinates is a topic we/you guys should pay attention too.... Thats all I am saying...


Posted from TSR Mobile
Original post by EnglishMuon
Day 1 Summary
Well an unexpectedly large proportion of my day was spend completing and reviewing STEP III 2008, but the time was needed!

One point that I may not have appreciated enough until today is that for any function f(x) f(x) and any even function
g(x),f(x)∘g(x) g(x), f(x)\circ g(x) is always even. As obvious as it seems, I missed its use in the following example:
d2yndθ2+n2yn=0. \frac{d^2y_n}{d\theta^2}+n^2y_{n}=0. If x=cosθ,yn(1)=1 x=cos\theta, y_{n}(1)=1 and yn(x)=(−1)nyn(−x) y_{n}(x)=(-1)^{n}y_{n}(-x) , deduce y0=1,y1=x y_{0}=1, y_1=x

Spoiler

(some very vague/loose spoilers coming up, but nothing major about STEP III 2008).

Spoiler

I've been looking at a fair few questions on using vector equations to substitute for Cartesian equations, and came across the idea of Gram-Schmidt Orthonormalisation which has been extremely useful in visualising the change of basis for eigenvectors and similar concepts!

This process takes a finite linearly independent set of vectors S={v_{1},v_{2},...,v_{n}} and generates an orthogonal set that spans the same subspace. It effectively just removes the 'like' components of each vector in another's direction via the dot product making them orthogonal, and then normalising them as usual by dividing by the mod. Covered well on Khan academy: https://www.khanacademy.org/math/linear-algebra/alternate_bases/orthonormal_basis/v/linear-algebra-the-gram-schmidt-process


I might do this paper and see what happens tmmrw! I actualky skipped alot in this pper i think only did 5 questions.


Posted from TSR Mobile
Original post by physicsmaths
I might do this paper and see what happens tmmrw! I actualky skipped alot in this pper i think only did 5 questions.


Posted from TSR Mobile


Yeah there are a mixed bag of questions on it. I felt some were just a bit fiddly rather than interestingly difficult, but may well be personal preference. Please let me know any thoughts you have on any of the questions! :smile:
Original post by EnglishMuon
Yeah there are a mixed bag of questions on it. I felt some were just a bit fiddly rather than interestingly difficult, but may well be personal preference. Please let me know any thoughts you have on any of the questions! :smile:


Lol i didnt do this paper as A mock i just seen it now. This is when i thoughtnthe difficulty tripped up so I only didd Q1,6,7,8. and didnt even try the rest haha. Might aswell do the other questions as a mock or something since most recent paper inhavent done alot of the pures.


Posted from TSR Mobile
Day 2 Summary
Completed M4 June 05 since I haven't done one in a while. M4 seems like a straight forwards module where rearranging is the main thing you need to be careful of, but I can't help but feeling the mark scheme methods for relative velocity questions are extremely illogical at times, maybe because I can't remember why the techniques actually used work!
I feel the best way is to re-derive them myself, so here is my attempt (please let me know of improvements/corrections).

So lets start with the obvious case of collision of two particles
A:rA=rA0+vAt,B:rB=rB0+vBt A: r_{A}=r_{A0}+v_{A}t, B:r_{B}=r_{B0}+v_{B}t. If collision occurs, by equating we see rB0−rA0=t(vA−vB)=tAVB r_{B0}-r_{A0}=t(v_{A}-v_{B})=t_{A}V_{B} i.e.AVBi.e. _{A}V_{B} is in the direction of the line joining initial A position to initial B position.

Closest Approach of 2 non colliding particles:
Occurs for
min∣rB−rA∣⇒∣rB−rA∣2=ArB⋅ArB min |r_{B}-r_{A}|\Rightarrow |r_{B}-r_{A}|^2=_{A}r_{B}\cdot_ {A}r_{B} is a minimum.
⇒ddt(∣ArB∣2)=2ArB⋅AvB=0⇒ArB⋅AvB=0\Rightarrow \frac{d}{dt}(|_{A}r_{B}|^2)=2_{A}r_{B}\cdot_{A}v_{B}=0 \Rightarrow _{A}r_{B}\cdot_{A}v_{B}=0

Finding the direction of velocity for closest approach:
image1.jpg
The conclusions of this are that if B passes as close as possible to A, the motion of B must be perpendicular to vB−vA v_{B}-v_{A} or vB⋅(vBvA)=0 v_{B}\cdot(v_{B}v_{A})=0

Solving 'compound variable' Simultaneous Equations in STEP

This is a good technique I used earlier which made life a little easier:
Consider the equations a+bxy=c(x+y),d+exy+f(x+y) a+bxy=c(x+y), d+exy+f(x+y) where a,b,c,d a,b,c,d are constants. Then to solve we could find y in terms of x using the first, sub in and rearrange, however solving simultaneously for xy, and then for x+y and then subbing in usually turns out much neater. (However the STEP question that reminded me of this turned out to provide extremely dumb solutions no matter what path was taken and consisted of nearly all fraction manipulation, however it still saved some valuable time!).
(edited 8 years ago)

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