[QUOTE=Cosa8888;46817017]Alright, I think I understoond how line parametrization works and how did you got to wt=kv+(1-k)u, but I got lost after that, can you explain a little more?[/QUOTE]
suppose [b]u[/b] = (u_1, u_2) etc.
then the equation becomes:
k (v_1, v_2) + (1-k) (u_1, u_2) = t (w_1, w_2)
which is precisely the same as the pair of simultaneous equations:
k v_1 + (1-k) u_1 = t w_1
k v_2 + (1-k) u_2 = t w_2
simply substitute the values and solve the equations.
Wow, it's awesome. Many thanks for your help, you guys have always been really helpful!
[IMG]http://latex.codecogs.com/gif.latex?%5Clim_%7B%28x%2Cy%29%5Cto%280%2C0%29%7D%5Cfrac%7Bx%5E3+x%5E2y%7D%7Bx%5E2+y%5E2+xy%7D[/IMG]
How would you guys go about doing the trig magic/comparison testing?
[QUOTE=JohanGS;46837756][IMG]http://latex.codecogs.com/gif.latex?%5Clim_%7B%28x%2Cy%29%5Cto%280%2C0%29%7D%5Cfrac%7Bx%5E3+x%5E2y%7D%7Bx%5E2+y%5E2+xy%7D[/IMG]
How would you guys go about doing the trig magic/comparison testing?[/QUOTE]
I'm assuming you're working over reals and not complex numbers, otherwise it's not well defined.
If you are, an inequality that should prove useful is
[;x^2 + y^2 + xy \geq \frac{x^2}{2};]
Which isn't too hard [sp]try doing it in reverse if you're struggling[/sp]
Then you can factor the top of the limit, use the inequality on the bottom and you should quickly conclude it tends to zero.
I don't really like these sort of 'multivariable limits', they're usually either undefined or zero/infinity which isn't very exciting. It's also worth pointing out that if you allow complex variables, it certainly isn't defined, since we can find a line for which the bottom approaches zero arbitrarily fast, meaning we can make the limit any value we like!
Like this?
[IMG]http://latex.codecogs.com/gif.latex?%5Cfrac%7Bx%5E3+x%5E2y%7D%7Bx%5E2+y%5E2+xy%7D%5Cleq%202x+2y%3D2r%28%5Ccos%5Ctheta+%5Csin%20%5Ctheta%29%5Cleq%204r%20%5Cunderset%7Br%5Crightarrow%200%7D%7B%5Crightarrow%7D0[/IMG]
[QUOTE=JohanGS;46841596]Like this?
[IMG]http://latex.codecogs.com/gif.latex?%5Cfrac%7Bx%5E3+x%5E2y%7D%7Bx%5E2+y%5E2+xy%7D%5Cleq%202x+2y%3D2r%28%5Ccos%5Ctheta+%5Csin%20%5Ctheta%29%5Cleq%204r%20%5Cunderset%7Br%5Crightarrow%200%7D%7B%5Crightarrow%7D0[/IMG][/QUOTE]
I wouldn't have thought you need to switch to polar coordinates, it seems pretty clear that 2x + 2y -> 0 if x and y -> 0 (it's obviously continuous and you can just set both to zero immediately)
Isn't polar coordinates the better way to do it? I guess I'll develop an eye for it later on, it just feels safer I guess?
[QUOTE=JohanGS;46842803]Isn't polar coordinates the better way to do it? I guess I'll develop an eye for it later on, it just feels safer I guess?[/QUOTE]
I don't think so, I think it's just an unnecessary complication.
With any limit on a continuous function you can just substitute the value directly (this is one definition of continuity!)
TBH I wouldn't even explain that much - you'd have an absurd definition of limits if x + y didn't vanish with x and y.
Should I prepare for a Complex Analysis exam by just looking at tons of contour integrals to absorb as many tricks as possible?
Does anyone have a good compilation of often-used tricks for contour integrals? For dirty shit like
[IMG]http://upload.wikimedia.org/math/a/9/c/a9c2436a733d4f41b6b9b74c8335e84a.png[/IMG]
where everything is fine if you happen to choose f(z) as the [I][URL="http://en.wikipedia.org/wiki/Methods_of_contour_integration#Example_.28V.29_.E2.80.93_the_square_of_the_logarithm"]square[/URL][/I] of the logarithm.
It seems like such a rabbit out of the hat trick, and it terrifies me to get an exam with this stuff. I've always been bad in doing substitutions in integrals too, and only got a little bit better by just picking up some tricks along the way...
I recently thought "Let's put this complex analysis to good use and apply the Cauchy integral theorem to prove that the Euler integral equals the square root of pi" and then I got stuck and [URL="http://www.math.uconn.edu/~kconrad/blurbs/analysis/gaussianintegral.pdf"]looked it up[/URL] (method 8) and up until 1940 or something it was deemed impossible. Talk about choosing bad methods to solve an integral :v:
I have big problems with analysis branch of math, I feel like it's too hard.
Meanwhile I have 10/10 in linear algebra.
I have question about rational functions r(x):
What happens when you have poles and zeros that are in same spot, same x?
I'll assume by x you mean rational functions over the real numbers.
p and q are polynomials, r(x) = p(x)/q(x), and there is a c such that both p(c) and q(c) are zero.
r(c) is undefined. The limit of r(x) as x approaches c might be defined, but this depends on p and q.
p and q are differentiable everywhere so by L'Hôpital's rule this limit is equal to the limit of p'(x)/q'(x).
[QUOTE=Krinkels;46909950]I'll assume by x you mean rational functions over the real numbers.
p and q are polynomials, r(x) = p(x)/q(x), and there is a c such that both p(c) and q(c) are zero.
r(c) is undefined. The limit of r(x) as x approaches c might be defined, but this depends on p and q.
p and q are differentiable everywhere so by L'Hôpital's rule this limit is equal to the limit of p'(x)/q'(x).[/QUOTE]
More generally, given a rational function r(x) = p(x)/q(x) over some field (reals/complex etc.)
Suppose p(a) = q(a) = 0 (i.e. we have both a 'zero' and a 'pole' at a)
The remainder theorem tells us that
p(x) = (x-a)p_1(x)
q(a) = (x-a)q_1(x)
Thus r(x) = ((x-a)p_1(x))/((x-a)q_1(x)) = p_1(x)/q_1(x)
Then if p_1 and q_1 both still have a zero at a, you can repeat to get p_2, q_2 etc.
Ultimately:
- p_k has a zero but q_k doesn't: r has a zero at a
- q_k has a zero but p_k doesn't: r has a pole at a
- neither has a zero: r(a) = p_k(a)/q_k(a) and doesn't have a zero [i]or[/i] a pole at a
So quick question with math. I've been out of HS for about 3 years and decided to look into engineering. Not sure what field or anything, and well I went in to take a placement test to see how much I've forgot in the past 3 years. Unfortunately I've apparently forgotten a lot of Algebra and it placed me in like math 91 or Algebra 1. I'm just about to turn 22, I've already gotten myself an associates in arts. Do I sound like I got a chance at all to get myself into engineering?
So the question basically is would it be the best option to just go back and relearn from step one?
Just revise it yourself. I revised my HS maths in a few weeks (barely had an idea what a matrix or differential equation was :v:) before starting out my first year of physics and I managed with only 2 or 3 resits (out of 14) (those being Mechanics and Calculus). It's tough, but doable if you're motivated.
[I]Mathematical Aspects of Quantum Field Theory[/I], by de Faria and de Melo. Very neat book. Talks about the cool mathematical details of classical mechanics and quantum mechanics, as well as obviously quantum field theory.
"The central idea of Yang-Mills theory is that there is a background field (such as the electromagnetic field) which is given by a connection A defined on a principal bundle over spacetime. The structural group of this bundle represents the internal symmetries of the background field. The possible interactions – also called couplings – of the background field with, say, particles such as photons or electrons, are dictated by the representations of the group. Each particle field turns out to be a section of the associated vector bundle constructed from the principle bundle with the help of a given representation of the group."
Concise, precise, interesting, and descriptive. Neat.
[QUOTE=Branflakes;46913641]So quick question with math. I've been out of HS for about 3 years and decided to look into engineering. Not sure what field or anything, and well I went in to take a placement test to see how much I've forgot in the past 3 years. Unfortunately I've apparently forgotten a lot of Algebra and it placed me in like math 91 or Algebra 1. I'm just about to turn 22, I've already gotten myself an associates in arts. Do I sound like I got a chance at all to get myself into engineering?
So the question basically is would it be the best option to just go back and relearn from step one?[/QUOTE]
All universities offer a precalc math course, and usually for credits too. I'd highly suggest you to take that before diving into Calc 1 if you're no confident about your precalc skills.
I signed up for a precalc course and my first class was yesterday, the first thing the prof did was hand out a review sheet that was a refresher of college level algebra. Out of the 20 some odd questions I could only remember one formula and the rest looked like a different language. I still can't help but find it funny that I used to be amazing at this, but I've just lost it over time and just gotta relearn it.
Could anyone give some hints how I would prove this, I think it's called a Holder-type inequality.
[IMG]http://quicklatex.com/cache3/ql_99a5c0c06e0375ce984d1ea9abb34c9e_l3.png[/IMG]
[URL="http://mathworld.wolfram.com/MatrixNorm.html"]Background info.[/URL]
Maybe I should square the 2-norm and somehow use these properties (the 1- and infinity norm are related through a matrix transpose):
[IMG]http://quicklatex.com/cache3/ql_02c3abe54f7b894ff99a6bf677c62c2d_l3.png[/IMG]
[QUOTE=Number-41;46861146]Should I prepare for a Complex Analysis exam by just looking at tons of contour integrals to absorb as many tricks as possible?
Does anyone have a good compilation of often-used tricks for contour integrals? For dirty shit like
[IMG]http://upload.wikimedia.org/math/a/9/c/a9c2436a733d4f41b6b9b74c8335e84a.png[/IMG]
where everything is fine if you happen to choose f(z) as the [I][URL="http://en.wikipedia.org/wiki/Methods_of_contour_integration#Example_.28V.29_.E2.80.93_the_square_of_the_logarithm"]square[/URL][/I] of the logarithm.
It seems like such a rabbit out of the hat trick, and it terrifies me to get an exam with this stuff. I've always been bad in doing substitutions in integrals too, and only got a little bit better by just picking up some tricks along the way...
I recently thought "Let's put this complex analysis to good use and apply the Cauchy integral theorem to prove that the Euler integral equals the square root of pi" and then I got stuck and [URL="http://www.math.uconn.edu/~kconrad/blurbs/analysis/gaussianintegral.pdf"]looked it up[/URL] (method 8) and up until 1940 or something it was deemed impossible. Talk about choosing bad methods to solve an integral :v:[/QUOTE]
I'd be tempted to use a hyperbolic substitution, x = sinh(u), to get the denominator to be cosh^4(u), and then given the expression for sinh in terms of the exponential function the log would wind up disappearing. I'm not sure if that would help or just send you in circles, though (don't have any paper handy to give it a shot).
Probably a little late now, but just a thought.
Yeah exercises was pretty shit but I nailed theory so I hope they kinda balance out.
I just need time to try different things when I have to solve an integral. If there's a time limit you have to pick the right method from the beginning because otherwise you jeopardize the other exercises. Fucking hate that exam so much.
[QUOTE=Number-41;46960038]Could anyone give some hints how I would prove this, I think it's called a Holder-type inequality.
[IMG]http://quicklatex.com/cache3/ql_99a5c0c06e0375ce984d1ea9abb34c9e_l3.png[/IMG]
[URL="http://mathworld.wolfram.com/MatrixNorm.html"]Background info.[/URL]
Maybe I should square the 2-norm and somehow use these properties (the 1- and infinity norm are related through a matrix transpose):
[IMG]http://quicklatex.com/cache3/ql_02c3abe54f7b894ff99a6bf677c62c2d_l3.png[/IMG][/QUOTE]
Given the definition of the 2-norm, suppose t is an eigenvalue with eigenvector x.
Now we have that tx = A^T Ax
Take the 1 or infinity norm of that equation and it should fall out fairly quickly.
P.S. Your definitions don't look [i]quite[/i] right (why the i=/j complication?)
Trying to understand something here, someone correct me if I'm wrong:
If we want to build something like the Mobius strip out of a vector bundle (my book calls it the Mobius bundle) it's gonna have to look a little wonky. Something like an infinite hollow cylinder with a hard-to-visualize twist, since the fibers are required to have the structure of a vector space, right? But we can build a Mobius strip out of a fiber bundle just fine since we only need the fibers to be topological spaces, so we can have them look like [0,1].
Just trying to see if I have this intuition right.
[QUOTE=Joey90;46969375]Given the definition of the 2-norm, suppose t is an eigenvalue with eigenvector x.
Now we have that tx = A^T Ax
Take the 1 or infinity norm of that equation and it should fall out fairly quickly.
P.S. Your definitions don't look [i]quite[/i] right (why the i=/j complication?)[/QUOTE]
Well shit now it looks very obvious :v:
Also I just got it from my syllabus (Finite Difference Methods), pretty sure it's correct because quite a few stability/convergence depend on it.
[QUOTE=Number-41;46972527]Well shit now it looks very obvious :v:
Also I just got it from my syllabus (Finite Difference Methods), pretty sure it's correct because quite a few stability/convergence depend on it.[/QUOTE]
Yeah, it's the 'trivial' step of applying the norm to some statement - and yet it can be incredibly hard if you don't think of it.
What is the norm of -I? If we don't have |a_ii| then we'll get a negative value!? (And once you have |a_ii| why separate it out)
[QUOTE=JohnnyMo1;46972105]Trying to understand something here, someone correct me if I'm wrong:
If we want to build something like the Mobius strip out of a vector bundle (my book calls it the Mobius bundle) it's gonna have to look a little wonky. Something like an infinite hollow cylinder with a hard-to-visualize twist, since the fibers are required to have the structure of a vector space, right? But we can build a Mobius strip out of a fiber bundle just fine since we only need the fibers to be topological spaces, so we can have them look like [0,1].
Just trying to see if I have this intuition right.[/QUOTE]
I think you're right, though it's much easier to consider it as a 'normal' mobius strip which extends infinitely outwards (don't try and embed this in R^3! - I don't think you get any better intuition by doing so.)
As you say, it needs to be 'infinitely wide' as [0,1] isn't a vector space (but R is).
It'd be nice distinguish, e.g. say a mobius strip is finitely wide and a mobius band is infinitely wide, but I don't know of any such convention.
Oh fuck I confused the logarithmic norm (which isn't a norm) with the usual [URL="http://mathworld.wolfram.com/MatrixNorm.html"]matrix norm[/URL], they're very similar, I'm sorry.
[QUOTE=Joey90;46974392]I think you're right, though it's much easier to consider it as a 'normal' mobius strip which extends infinitely outwards (don't try and embed this in R^3! - I don't think you get any better intuition by doing so.)
As you say, it needs to be 'infinitely wide' as [0,1] isn't a vector space (but R is).
It'd be nice distinguish, e.g. say a mobius strip is finitely wide and a mobius band is infinitely wide, but I don't know of any such convention.[/QUOTE]
Thanks. The book I was reading has an image of the Mobius vector bundle... but it only shows the strip and doesn't mention what it cuts off. Pretty annoying.
i'm not looking forward to linear algebra next semester
cal I-III were a breeze but I'd be lying to you if I said I retained any knowledge 5 hours after finishing the final and on my way to shotgunning a keystone
[editline]24th January 2015[/editline]
people tell me that not retaining knowledge is normal though, and you only ever really commit to memory what you'll be practicing on your job. some guys told me that they have their textbooks from college on hand to reference incase they get stuck, because they hardly remember a thing
but then there's people who seem to have absorbed all the maths and sciences over the course of their degree so well, and I feel like I'm going to make a shit engineer lmao
perhaps I'm not learning the right way
If you get it once, second time will be a breeze. Of course you will need to re-learn stuff, but it will be faster - so you actually do retain some of the knowledge :).
Doing a Dynamics course @ university as part of my degree, the lecturer has spent the last two weeks trying to explain ODE's using different variations of dy/dt + p(t) = q(y) or whatever. Does anyone know of any online material or whatnot that actually explains what the fuck he is talking about, or anything that's quite readable on differentials and their non linear / linear / second order forms?
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