Homework Statement
Find the limit of
\lim_{(x,y)\rightarrow (0,0)}\frac{x^2y^2}{x^3+y^3}
Homework Equations
I'd like to solve this in a rather elementary manner, so preferably only using the squeeze theorem or through proving the limit doesn't exist via multiple path approach.
The...
I was reading the proof that every Vector Space has a basis which invoked Zorn's Lemma. The proof can be found http://mathprelims.wordpress.com/2009/06/10/every-vector-space-has-a-basis/" [Broken].
Now I have an issue specifically with the claim that U := \bigcup_{S\in C}S is an upper bound...
While looking for interesting problems, I was led to a problem involving infinite pulleys.
The problem and solution can be found http://www.feynmanlectures.info/solutions/infinite_pulleys_sol_1.pdf" [Broken]. Now I follow the solution up to the part where he claims \frac{1}{4^{i}M_i} is always...
The definition of the Euler - Mascheroni constant, \gamma, is given as
\gamma = \lim_{n\rightarrow\infty}\sum_{k=1}^{n}\frac{1}{k} - \ln(n)
or equivalently in integral form as \gamma = \int_{1}^{\infty}\frac{1}{\left\lfloor x\right\rfloor} - \frac{1}{x}\ dx
I saw a seeming related integral...
Today I read about the AKS - Primality test in which the simple theorem
For gcd(a, n) = 1, we have
(X - a)^n\equiv X^n - a\ (mod\ n)if and only if n is prime.
was proven. The if direction is quite trivial from the fact that \binom{p}{k}\equiv 0\ (mod\ p) for 1\leq k < p. The other direction...
I was reading a derivation of the rate equation from collision theory and there is one thing which confuses me a bit. In the derivation we assume that a particle will collide with every particle within its "collision volume" but it seems to me that this is a gross overestimate of the actual...
It is true that phi(pq)=(p-1)(q-1) when p and q are distinct primes, but is it true that given phi(pq) = (p-1)(q-1) then p and q have to be distinct primes? It seems intuitive, but I'm having some difficulty proving it.
Also, if this was true, is it possible to generalize this into more than 2...
From what I had read, Euler had originally proved the infinitude of primes through proving his product formula and equating the primorial to the divergent harmonic series.
\sum^{\infty}_{n=1}\frac{1}{n^{s}} = \prod^{\infty}_{p}\frac{1}{1-p^{-s}}
where p ranges through the primes. However, the...
I was reading a book which is a collection of interesting mathematical journal articles. Within the book there was an article which discussed alternating series. In particular, at one point in the article it proves that the series
\sum (-1)^n \frac{\sqrt(2)^{(-1)^{n}}}{n}
diverges. To be a bit...
We often have
lim_{x\rightarrow a}f(g(x)) = f(lim_{x\rightarrow a} g(x))
if f(x) is continuous at g(a).
But then my question arises where g(x)\rightarrow\infty. I am not sure if there is any meaning to continuity at infinity as it seems that continuity is the property of a particular...
Hey everyone!
This has always been a source of confusion for me that everyone seems to play around with dx and dy as if they were variables while in many sources it was stated that they are purely symbolic. For example, in the integral, dx is purely symbolic. If I'm not misunderstanding, dx is...
Say I have a sequence a(n) = 100 + n^2 with n ranging through the positive integers. I want to find the largest number g(n) such that g(n) = gcd(a(n), a(n+1)).
Let g be a number that divides both a(n) and a(n+1)
g | 100 + n^2 and g | 100 + n^2 + 2n + 1 therefore g | 2n + 1. This means that g...
In most of the books I've read, chromosomal cross over has been described as simply "segments of the chromosome breaking off and reattaching", but I have a bit of difficulty with this explanation. Chromosomes are densely packaged DNA that have been wound and wound again. How can DNA which has...
I was using extra virgin olive oil recently and noticed something peculiar. I had always thought that oils will spread when in contact with water, but the olive oil didn't seem to do so.
When the oil entered the water, they were pinched into spheres which quickly rose to the surface. The...
It has always been explained to me that balls curve because a thin layer of fluid adheres to its surface as a boundary layer. This boundary layer retards the flow on one side of the ball and speeds it on the other. And then Bernoulli's principle is applied to make the pressure difference happen...
I've read about wire-grid polarizers from wikipedia. The article says that waves perpendicular to the wires cannot induce electron movement and hence they pass through with minimal energy loss. On the other hand, waves parallel to the wires are reflected by the wires because electron movement is...
In Faraday's ice pail experiment, a positive charged ball is lowered into an ice pail without contact. A charge is registered on the electroscope attached to the outer surface. Next, the ball is allowed to contact the bucket in which the electroscope still registered the same charge. But upon...
I'm a bit confused on the exact workings of Ampere's law.
Firstly, why does the shape of the Amperian loop not matter. Is there a mathematical proof that all Amperian loops are equivalent for the purpose of this law?
Secondly, the law still holds valid in the presence of external magnetic...
When deriving the formula for the magnetic field of a solenoid it is said that the side of the rectangle outside and parallel to the solenoid can be ignored because it is taken far away and the contributions of the field outside is negligible.
But any arbitrary path can be taken and the...
I saw this identity somewhere and have been looking for a derivation but I can't seem to find one. It would be of great help if someone can show me where this comes from.
tan\frac{\theta}{2} = csc\theta - cot\theta
For collisions wouldn't the distance traveled during the collision be the same? The energy lost is from heat, sound, deformation, etc. but wouldn't the work done on each car be the same since during the duration of contact they have to move the same distance (as soon as they move different...
Temperature is defined as the average kinetic energy of a system. But would temperature change if some external force was applied to change the total motion? For example, would a ball experience an increase in temperature if the ball was traveling at a high speed as a whole? The average kinetic...
I just have some confusion about whether or not self-inductance will occur in an AC generator.
The current is alternating in both direction and magnitude rapidly when it is generated and this will in turn cause a changing magnetic field around the conductor. This magnetic field will then...
If a transformer with coil ratios 2:1 creates a voltage twice that of the primary in the secondary coil why is the current halved? I realize that this is a consequence of the conservation of energy, but there must be a physical reason behind this. What is causing the additional impedance that...
I have some trouble understanding how changing flux produces a field. For example, I know that a changing magnetic field will induce an electric field because the flux of the magnetic field is changing. But how does this change in flux create the electric field. I just can't picture it.
Is...
I'm learning about electricity and I've stumbled upon a question which I cannot seem to find the answer to. I realize that I am probably missing some key piece of information or making some form of understanding error but just can't seem to grasp what it is.
My understanding is that if you...
I have a bit of trouble with understanding how electromagnetic fields. More specifically, how the magnetic and the electric component of the field interact. I am aware that a time-varying electric field will generate a magnetic field and vice versa. What I am having trouble understanding is how...
I have trouble understanding why water waves tend to speed up for waters deeper than 1/2 wavelength and slow down for waters shallower. I have read it has something to do with how the particle orbit changes to elliptical in shallow waters.
Why does the particle orbit become a flat ellipse in...
I've been told that longer wavelengths are able to diffract more than short wavelengths. The hyperphysics page on sound wave diffraction also states this fact. Nothing I've read have explained why exactly longer wavelengths are able to diffract more than shorter wavelengths, so that's my...
I need some help on what exactly is happening here at the cellular level.
I understand that incomplete dominance is a blending, for example caused by a functional enzyme producing pigments and a non-functioning enzyme. The result would be less pigments, therefore an apparent mixing.
But what...