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.
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...
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...
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.
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.
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...
@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...
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...
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.
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.
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...
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...
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.
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...
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?