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1. Multiple integration + Centroid Help.

You first integrate with z, that gives you rz in the inside. Using Fubini you get (meaning evaluate the integral) r\sqrt{6-r^2} - r(r^2) and now continue with r and theta.
2. Proof that Log2 of 5 is irrational

\log_2 5 = a/b so 2^{a/b} = 5 \implies 2^a = 5^b. Now use unique factorization.
3. What is Mathematics?

I will begin by saying how much I dislike it when non-mathematical philosophers think about math. There are many threads in the mathematics forum like this and they belong in the philosophy forum, so one good thing about this thread is that it at least is in the philosophy subforum. I am not...
4. A quick Limit

Non-zero constant. Not to offend you but that is what I like to call a "physics-type mistake".
5. Polynomials and function space over fields

Here is a little something to add to what matt-grime was lecturing about. Theorem: Let |F|= \infty and f(x) be a polynomial if f(\alpha)=0 for every \alpha \in F then f(x)=0. Proof: Let \deg f(x) = n (assuming f(x)\not = 0) then f(x) can have at most n zeros. But it clearly does not for...
6. A quick Limit

\frac{1}{n+n}\leq \frac{1}{n+\sqrt{n}}
7. No topology sub-forum?

Place Number theory with Algebra. That makes it better.
8. Residue theorem

In what contour? And what is k? Note if k is zero then it becomes 1/e^z which is analytic. But if k!=0 then there is a singularity which depends on the contour.
9. Lebesgue vs Riemann integral

The delta function cannot be used for a conterexample about theorems about intergration. Because it is not a "function" eventhough it is called a function, rather it is a Schwartz distribution so the known results about integration might not apply to it.
10. A very hard prove

Yes he did. I should be more careful lest someone accuse me of stealing answers. I deleted my post otherwise it does not look good.
11. Possible to teach yourself?

Most of what I know is self-taught, so there really is nothing amazing about teaching youself. All you need is motivation.
12. Galois theory text for second semester undergrad algebra

I really like the algebra book by Blair. It has Galois theory in it. It is better to use Morandi but he is GTM.
13. Example of f integrable but |f| not integrable

If f is a bounded function on [0,1] which is integrable then |f| is integrable on [0,1]. It is a simple theorem to prove.
14. Books on Analysis?

How well do you know undergraduate analysis? Do you know the following topics: Completeness property Limits of sequences and delta/epsilon, Cauchy sequences Subsequences Limsups and Liminfs Continous functions, delta/epsilon definition, sequence definition Properties of continous functions...
15. Proving a polynomial is constant

@ehrenfest. It really is not so hard. If n is a number in {2,3,...,100} it not a prime number then we can write n = p*m where p is a prime number. So for any n there exists a smallest possible prime divisor. Given any n the smallest prime divisor is 7 because it cannot be 11 because if it were...
16. Proving a polynomial is constant

I call the "basic" pigeonhole to be the one that says that there exists at least one hole having two pigeons. The "strong" one is the generalized argument. I am not sure if that is how it is officially called but that is how I refer to it. Here is a problem to try for you to solve: "Let S be...
17. Soluble groups?

Do you agree with me it is just a difference in language?
18. Differentiation question

Just use mean-value theorem.
19. Measure theory and number theory?

No. Field theory in particular over finite fields and rational numbers. And Galois theory are used a lot in number theory. That is probably the largest area of algebra that is used in number theory. And p-adic analysis.
20. Limit of a product cos

You can also use a 'collapsing product'. Note, A_4 = \cos \frac{x}{2} \cos \frac{x}{4} \cos \frac{x}{8} \cos \frac{x}{16} Then, \sin \frac{x}{16} A_4 = \cos \frac{x}{2} \cos \frac{x}{4} \cos \frac{x}{8} \cos \frac{x}{16} \sin \frac{x}{16} So, 2A_4\sin \frac{x}{16} = \cos \frac{x}{2} \cos...
21. AMGM(n) help

This is the famous Cauchy proof of Am-Gm. Post what you dd so far.
22. He Unreasonable Effectiveness of Pure Mathematics

Write the paper on Georg Riemann and his ideas of Differential Geometery. When he first discovered it as a pure area of mathematics it had little effect. 70 years later E'nstein found how to apply it.
23. Proving a polynomial is constant

I did the problem backward, I am sorry. But the way it should be as Ehrenfest posted is correct if you apply my argument. Okay, it is very simple. If you have 6 pigeonholes and 32 pigeons then there is a pigeonhole that has at least 6 pigeons. In general given h pigeonholes and p pigeons then...
24. No topology sub-forum?

Eventhough Topology and Analysis are similar I consider them seperate. I would put Topology under Geometry section.
25. Proving a polynomial is constant

It is a nice problem. First, you need to know that is a polynomial of degree n takes the same value (n+1) times then it must be be a constant polynomial. The problem says P(x) is a polynomial with integer coefficients so it means P(x) is an integer whenever x is an integer. We know that...
26. Reimann Integration, squares and cubes of functions

If f is continous on a compact interval and g is integrabl on that interval then f\circ g is also integrable. That means if f^2 is integrable then (f^2)^{1/2} = |f| is integrable. And similarly (f^3)^{1/3} =f is integrable.
27. Lebesgue vs Riemann integral

It is impossible to construct a non-Lebesaure measurable set which does not involve the axiom of choice. That is a nice advantage.
28. Measure theory and number theory?

There is also algebraic number theory, but algebra and number theory are related as soon as you start learning about them. However, saying that that "...is concerned with abstract sets and quantifiable properties..." is not a good answer because there are certain topics in number theory that are...
29. Soluble groups?

Maybe "solvable" is American and "soluble" is English. For example, "formula" is American and "formulae" is English. Look at the author's name of the book, is it English?
30. Euler Cauchy equation problem

The characheristic equation here is k(k-1)-2=0. The solution would then be y=c_1|x|^{k_1}+c_2|x|^{k_2} on (-\infty,0)\cup (0,\infty).