I like your approach. I haven't looked it over yet, but it appears quite promising. However, I would like to comment that your picture 4.0a shows you calculating the "inside" toroidal region. The problem my paper considers is the "outside". However, I think only a slight change to you method...
The purpose of the paper I'm writing is to establish exact formulas for the volume in terms of height. The current approximations used in these cases are somewhat inadequate and I'm trying to abolish them completely.
Actually, what I said was "revolving the region inside of the circle and below y_1 about the x-axis "
I've attached a picture showing the shaded region that is revolved about the x-axis. Just use the washer method, using the outer radius R(x) as the circle and the inner radius r(x) as the...
This problem seems to be ridiculously hard in this form, so I've considered that it might be useful to try to find the volume of an entire torus as a function of height. If we want h>0, then the torus must be shifted up so that it is tangential to the xy-plane.
Manipulating the equation of a...
Assume that 0<r<R.
Consider the circle x^2+(y+R)^2=r^2. Obviously, if we revolve this circle about the x-axis, we get a torus whose volume is (\pi r^2)(2\pi R).
Now consider a point y_1 such that R-r<y_1<R. and let (x_1,y_1) be a point on the circle. The volume of the solid formed...
I may have answered my own question, but I don't trust my results at 2 am. Does this make sense?
Claim: \frac{2x+1}{(x+1)^2+y^2}<\frac{1-x}{[(x-1)^2+y^2]^{3/2}}.
Proof of Claim: Notice that along the line x=0, equality holds. However, we are only considering 0<x<1; also, notice that...
Yes, there is an actual problem here. This is a small portion of one of the proofs for a theorem in my thesis on electrostatics and potential theory.
Consider a configuration of three equal, positive point charges arranged in an equilateral triangle, with the charges located at (-1,0)...
Unfortunately, I already know that at no point of the given region will I have critical points, so this method would not be useful. I unfortunately have been down this path before with a similar problem.
Homework Statement
Show that \frac{2x+1}{[(x+1)^2+y^2]^{3/2}}+\frac{x-1}{[(x-1)^2+y^2]^{3/2}} < 0 for 0 < x < 1 and 0 < y < \frac{x}{\sqrt{3}}+\frac{1}{\sqrt{3}} .
Homework Equations
The Attempt at a Solution
I've confirmed by graphing in Maple.
It's easy to see that...
The most I can say is that they are subharmonic.
These are potential functions using the Newtonian kernel. Particularly, I've placed two positive charges on the x-axis at (0,0) and (1,0), with the third positive charge somewhere in the upper half-plane. Choose 0<c<1, and let \vec{v} be the...
This is not a homework problem. I've encountered something in my research in potential theory, and I need to prove the following.
Given:
h(t) = f(t) + g(t),
f '(t) < 0 for all t,
g'(t) > 0 for all t, and g'(t) is monotonically increasing,
f '(0) = 0,
g'(0) > 0,
f '(t) has exactly one...
I've attached my previous post as a pdf for those who don't have LaTeX.
I've been shown that it is impossible to have the potential become constant on a curve of finite length (this follows from a result of Puiseaux), unless the curve is a loop. However, a loop will violate the harmonic...
The only assumption in this problem is that I have n point charges somewhere in the xy-plane. The point charges can be any combination of positive or negative, and there is no restriction on their magnitude (obviously cannot equal 0...). The k-th point charge (q_k) is located at (x_k,y_k)...
This part I can answer, although I'll have to study the rest of your response in greater detail later.
Everything works in R^3 because the potential is a harmonic function (regardless of what combination of positive or negative charges). In the example of equal but alternating even number of...
As I said in my last post, the potential does not have to be zero if the curve does not extend through the whole plane. However, the existence of an equipotential curve is not the ultimate goal of this problem: it is to find a curve along which the force vanishes. Also, the example I gave in...