Recent content by Sculptured

1. Topology in the Complex Plane

I'm having some trouble understanding the distinction between closed sets, open sets, and those which are neither when the set itself involves there not being a finite boundary. For example, the set { |z - 4| >= |z| : z is complex}. This turns out to be the inequality 2>= Re(z). On the right...
2. Dielectric with net charge

Let's say we have a sphere, radius R, which has a uniform volume charge density. Then we wrap it around with a dielectric with a frozen in polarizability of kz in the z direction. This dielectric goes from R to a radius A. The total surface bound charge on either surface of the dielectric is...
3. Helium Balloon and Compton Generator

From Classical Mechanics by Taylor: "The Compton generator is a beautiful demonstration of the Coriolis force due to the earth's rotation,.... A narrow glass tube in the shape of a torus or ring (radius R of the ring >> radius of the tube) is filled with water, plus some dust particles to let...
4. Transistor CE Amplifier

Alright, I've been stuck on this problem for my electronics class for far too long. " Design a CE amplifier with a gain of 10 using a 2n3906 transistor and a 20 V power supply. Calculate the required resistor and capacitor values for the circuit, assuming a minimum frequency of 10 Hz." It...
5. Projectile Motion of a cannon

It seems wrong because my discriminant gives the relation of \theta > 70.5 rather than less. Also, 70.5 seems too big because by that angle I would have thought that the decrease in y would have overtaken the increase in x. Lastly, my teacher didn't exactly find my answer to be correct.
6. Projectile Motion of a cannon

A cannon (at the origin) shoots a ball at an angle \theta above the horizontal ground. Neglect air resistance. Let r(t) denote the ball's distance from the cannon. What is the largest possible value of \theta if r(t) is to increase throughout the ball's flight? Taylor, Classical...
7. Nonhomogeneous Power Series Solution

Thanks; makes good sense. I have another question. This time it involves indicial roots. Working with the Frobenius method I find that in the DE: xy'' + 2y' -xy = 0 both of the indicial roots come to give me the same series solution. I used r = 0 to attain it and the book used r = -1. (or so...
8. Nonhomogeneous Power Series Solution

For the fun of it, my DE book threw in a couple of problems involving nonhomogenous second order DE's in the section I'm currently going through. Although I have solved for the complementary solution, any suggestions on how to find the particular solution? For example, the one I'm looking at...
9. Apostol: Archimedes' Method

Ah, right. Stupid of me to miss that result.
10. Apostol: Archimedes' Method

I've started with Apostol's first volume of calculus and I'm already having a problem with the first exercise in the book. Weak. It asks to use Archimedes' method to find the area under a parabola from 0<x<b for a parabola where y = a*b^2 + c. Using the summation of upper/lower rectangles of...
11. HUP inside the atom

Are you asking why they don't spiral in or why they can't exist within the nucleus? I'm going through this right now in my modern physics class and from what I've pieced together, using Bohr's model, is about the latter. The energy of the electron is the kinetic minus the potential (difference...
12. Time Dilation Troubles in SR

As usual, a mathematical representation has helped to see it from another angle. I've identified my troubles, stretching the concept out farther than it applies, and feel good about the concept. Thank you.
13. Time Dilation Troubles in SR

Alright, I'm having a bit of trouble understanding time dilation. My biggest concern is in how an observer in a frame will see all other frames at different relative speeds moving slower. I've been trying to show it mathematically, but haven't had the luck yet.
14. Surface Area Confusion

I'm wondering how you got the expression Sqrt[4r^2 + r^2] dr dtheta. I found the magnitude of the cross product to be Sqrt[4r^4 + r^2]. Integrating over r from 0 to 1 and theta from 0 to 2pi gave me (1/6)(-1 +5*Sqrt[5])*pi. The same answer as when I use x=x, y=y, and z=f(x,y)= 9-x^2 -y^2...
15. Surface Area Confusion

Just had my test on Vector Fields and there was one question which really confused me. It asked to find the surface area of the parabaloid z = 9-x^2 -y^2 which is above the cone z = 8Sqrt[x^2 + y^2]. My memory told me to use the differential in rectangular coordinates and then convert to...