Recent content by Frillth

Hahaha, of course. Man, it is clear that I haven't slept for a while. Thanks, Hurkyl. Edit: Well, since my first question was pretty stupid, I have a new one to ask. Is it possible to divide up the irrationals from [0,1] into two sets A and B such that each set contains "half" of the...

I just finished a course where we discussed concepts such as Lebesgue integration and Lebesgue measure of sets. Today, I was telling my brother about how the irrationals on the interval [0,1] have measure 1, which is sort of counter-intuitive. Anyway, he proposed the following question. Let A...

I know from my physics course that sound has a particular speed in any medium that it is traveling through, much like light. I have not learned, however, why this is so. I know that sound is simply a pressure wave traveling through air (or water or steel or whatever), but it doesn't seem...

Can I use Biot-Savart to find the magnetic field at some general point (x,y) in the plane, then integrate that over the area of the top loop to get flux?

Homework Statement Homework Equations Φ = M*I Biot-Savart law Ampere's law The Attempt at a Solution I have the formula Φ = M*I, which means that to find the mutual inductance I simply need to find the magnetic flux through the top circuit due to the current in the bottom...

Homework Statement I have the following problem for physics homework: We are supposed to take t=0 as the instant when the corner of the square is just about to leave the magnetic field. Homework Equations Unsure. The Attempt at a Solution Since the magnetic flux will decrease as the...

I did it by induction on n. How would you approach it with your second method?

The volume of that should be (I think): Is there a simple way to evaluate this, or am I doing this completely wrong? By the way, sorry if that's hard to read. I don't know latex, so I just used Mathematica. Edit: Ah, I just realized how to compute that integral. It comes out to n^n/n...

Homework Statement Let S = {x∈R^n : x_i ≥ 0 for all i, x_1 + 2x_2 + 3x_3 + ... + nx_n ≤ n}. Find the n-dimensional volume of S. Homework Equations I'm 95% sure that I'm supposed to use the change of variables theorem here. The Attempt at a Solution So far, I have calculated the values for...

Homework Statement Let B=B(0,r) be an open ball of radius r centered at the origin in R^n. Suppose U is an open subset of R^n containing the closed ball of radius r centered at the origin, f is a function from U to R^n that is differentiable, f(0) = 0, and ||Df(x) - I|| <= s < 1 for all x...

Homework Statement Suppose X ⊂ R^n is a k-dimensional manifold and Y ⊂ R^p is an l-dimensional manifold. Prove that: X × Y = {[x,y] ∈ R^n × R^p : x ∈ X and y ∈ Y} is a (k+l)-dimensional manifold in R^(n+p). (Hint: Recall that X is locally a graph over a k-dimensional coordinate plane...

All right, so from there I take the partials and I get: dV/dy = (-12z^2 + 6z)/(12y + 12z - 6)^2 dV/dz = (-12y^2 + 6y)/(12y + 12z - 6)^2 Setting these equal to zero, I get: y = z = 0 and y = z = 1/2 However, I know that y and z can't be 0, and if y and z are 1/2, then by my earlier...

Homework Statement Find the equation of the plane through the point [1,2,2] that cuts off the smallest possible volume in the first octant. Homework Equations Volume of a pyramid = 1/3Ah The Attempt at a Solution The plane is going to cut out a pyramid with the x-, y-, and z-intercepts, so...

This makes intuitive sense to me, but I'm not sure if it would work as a proof: Since each U is open, there is an open ball around each point in U. But an open ball must have some finite volume, and since the set X is compact, it can be contained by a ball around the origin, and it must thus...

Ah, ok. That makes a lot more sense. Would this proof suffice? We will prove by contradiction. Assume that no finite number of the U's will contain all of X. Let x_k ∈ X be one such element that is cannot be contained in a finite number of the U's. But if x_k ∈ X, then it must be contained...