Mathematical applications for natural sciences

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yusyus
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Hi,
I will be reapplying for Cambridge next year have been rejected post-interview last year. Last year I talked mostly about calculus/differential equations.
I have been learning a lot of new maths and couldn't really find that many applications to physics and chemistry to these topics that I learned about.

How could I talk about:
Limits (L'Hopitals rule just generally limits)
Groups (Cayley tables Lagrange's theorem)
Number theory
Vectors like the scalar triple product(i know this can help with volumes but how can that help with physics/chemistry?)
I also learned more about matrices like eigenvalues but I guess that's good with vectors?
Co-ordinate systems
random bits of calculus like t-formula, Weierstrass substitutions, arc lengths (particularly interested in this)

Can anyone help with ways to put spins on these so they would match a Physics/Chemistry personal statement?
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InsertNameHero
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Linear Algebra (Matrices, Vectors, Eigenvectors etc..) are used significantly in Quantum Mechanics. You can spin this off for Physics and Chemistry (Quantum theory is a crucial component to chemistry).
Group Theory has applications to symmetry in molecules - sorry that's the extent of my knowledge in this field.
Obviously calculus will be used a lot in both fields but you need to show where this applies. I don't think specific methods of integration are really going to add to the personal statement.
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yusyus
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(Original post by InsertNameHero)
Linear Algebra (Matrices, Vectors, Eigenvectors etc..) are used significantly in Quantum Mechanics. You can spin this off for Physics and Chemistry (Quantum theory is a crucial component to chemistry).
Group Theory has applications to symmetry in molecules - sorry that's the extent of my knowledge in this field.
Obviously calculus will be used a lot in both fields but you need to show where this applies. I don't think specific methods of integration are really going to add to the personal statement.
thanks a lot for this!
I have a book on quantum mechanics so I will definitely give this a read if it links all three subjects.
For the calculus how do you think I could include it? I have heard about how you can integrate surfaces for mechanics to find COM or something? And I'm guessing arc length can be useful in physics I just have to figure out how to include it.

How do you think I could include integration into my PS? I've learned quite a bit about it and can integrate past A-Level maths level (I learned about hyperbolic equations for integration)

Thanks a lot for all of this again!
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marinade
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(Original post by yusyus)
Groups (Cayley tables Lagrange's theorem)
Group theory I would go much more deeply. The photon (light) needs U(1) gauge symmmetry which is the special unitary group of dimension 1. From this much follows. The strong interaction has SU(3) symmetry and so on leading us to :-

The standard model of particle physics is U(1) x SU(2) x SU(3)

(Original post by yusyus)
Vectors like the scalar triple product(i know this can help with volumes but how can that help with physics/chemistry?)
The vector product is an extremely profound concept in natural sciences. The vector product is related to the idea of vectors and pseudovectors. The problem is this isn't really pointed out. So someone will read up on pseudovectors, yeah this sign flipping thing is quite interesting, bit like the vector product, but so what? Wrong! The magnetic field is a pseudovector! This also links in with co-ordinate systems.

(Original post by yusyus)
Co-ordinate systems
This has very profound applications to natural sciences. Discussions about Co-ordinate and co-ordinate free systems are at the very heart of things like Special and General Relativity. In another formulation of mechanics called Lagrangian Mechanics, the stepping away from the cartesian co-ordinate system to symmetrical principles and conservation lets us solve difficult to solve problems in Newtonian Mechanics.


(Original post by yusyus)
random bits of calculus like t-formula, Weierstrass substitutions, arc lengths (particularly interested in this)
It depends what you mean by arc lengths. Finding the arc length to me sounds a bit like the basic problems in the calculus of variations, which is intimately linked to mechanics.
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yusyus
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(Original post by marinade)
Group theory I would go much more deeply. The photon (light) needs U(1) gauge symmmetry which is the special unitary group of dimension 1. From this much follows. The strong interaction has SU(3) symmetry and so on leading us to :-

The standard model of particle physics is U(1) x SU(2) x SU(3)



The vector product is an extremely profound concept in natural sciences. The vector product is related to the idea of vectors and pseudovectors. The problem is this isn't really pointed out. So someone will read up on pseudovectors, yeah this sign flipping thing is quite interesting, bit like the vector product, but so what? Wrong! The magnetic field is a pseudovector! This also links in with co-ordinate systems.



This has very profound applications to natural sciences. Discussions about Co-ordinate and co-ordinate free systems are at the very heart of things like Special and General Relativity. In another formulation of mechanics called Lagrangian Mechanics, the stepping away from the cartesian co-ordinate system to symmetrical principles and conservation lets us solve difficult to solve problems in Newtonian Mechanics.




It depends what you mean by arc lengths. Finding the arc length to me sounds a bit like the basic problems in the calculus of variations, which is intimately linked to mechanics.
Thanks for all of this,
where do you think I could learn more about groups? What I have learned so far is just from A2 Further maths is the next resource just lecture notes?
In terms of the vectors/pseudovectors do you think this is stuff I could learn about in enough depth to talk about it in interview and do you have any ideas about where I could learn about this (In terms of the vector product I basically only know how to calculate it using determinants and to show the vecor that 2 vectors make.
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marinade
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(Original post by yusyus)
Thanks for all of this,
where do you think I could learn more about groups? What I have learned so far is just from A2 Further maths is the next resource just lecture notes?
In terms of the vectors/pseudovectors do you think this is stuff I could learn about in enough depth to talk about it in interview and do you have any ideas about where I could learn about this (In terms of the vector product I basically only know how to calculate it using determinants and to show the vecor that 2 vectors make.
Groups are a very easy topic to understand, just it's really boring on the A-level further maths syllabus. It can be very visual which is a shame as there are some people that would probably find it more interesting than the boring A-level syllabus.

There are many popular science books and on-line resources. So for example on group theory I would recommend symmetry and the monster by Mark Ronan. I think popular science books are just good in general, so another one you might find interesting on the different bits you said is 17 equations that changed the world by Iain Stewart. I would recommend reading Feynman books too actually for most of the stuff talked about six-easy-pieces/not-so-easy-pieces

If you are genuinely interested in vectors then yes you can learn enough. It's a case of reading around. pseudovectors is an incredibly simple thing to understand but as I said gets overlooked. If you understand that swapping a and b around results in the sign of c flipping then you can understand pseudovectors. It's just a departure from the A-level syllabus because the teacher may never have thought about this sort of thing!

The cross product of a vector and a vector is a pseudovector. The cross product of a vector and a pseudovector (either way around) is a polar vector, pseudo crossed with pseudo is a pseudovector. And so on.

http://www.feynmanlectures.caltech.edu/I_52.html scroll down to 52-5 to read a Nobel prizewinner explaining it.

You do have to be careful with some things. It depends what you mean by calculus. At university it's called 'analysis' and this area of mathematics includes many potential applications to physics such as 'special functions' an area of maths that virtually nobody outside maths has any interest in unless there are specific equations used and then an engineer or physicist will say well of course I've heard of Bessel's equations/Helmholtz equations I use them all the time in electromagnetism/the wave equation. But someone else would think eh, is this person lying.
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