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...
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...
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...
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...
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.
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...
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...
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...
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...
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...
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.
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.
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...
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...
The Fundamental Line Theorem is used when the vector field you are integrating the curve over is conservative (curl = 0). A conservative vector field is the gradient of some function so this theorem allows you to just find the original function and take the difference between any two points in...
My attempts to derive the height as a function of radius of the spheres packing the unit cell have failed. The best attempt so far was to create a diagonal line from the top atom to one of the middle ones. With that distance I thought I could use trig to find the component parallel to the height...
As the title implies, I am having a problem with the way this problem is done.
problem: "An insulating sphere with radius a has a uniform charge density p. The sphere is not centered at the origin but at vector(r) = vector(b). Show that the electric field inside the sphere is given by E =...
Alright, after reading the posts I was able to do a few things and hope that I got it right. I distributed (1/a) and then found the LCD of the two fractions. Took out the (1/r) that was in the fraction, distributed a in the denominator, factored out a squared in the denominator, canceled the a...
Well the intended result is necessary because without it Coulumb's Law would not hold. Unless this means I need to work with the rest of the equation.
The full final equation: F = ((q*Q)/a*4*pi*epsilon-naught)*(original equation)
where q is a small fixed charge, Q being the large fixed...
After getting through the set up and integration of this problem I am stuck on using the >> operator. My objective is to show what happens when r, a constant, is >> than a, another constant.
The original equation: (1/r)-(1/(a + r)
The intended restul: (1/r^2)
My initial thought was that...
Although the book (Elementary Linear Algebra by Larason/Edwards/Falvo) gives the definition of a determinant of a 2x2 matrix in the first section dealing with the determinant, it leaves me with nothing. After all, my teacher seems to already assume that because we have taken precalculus before...
My lack of any shred of idea about the concept of the determinant is really beginning to bug me. Especially now that I have begun my Linear Algebra class. Is there anybody available to help explain (or link me to a good explanation) to me the idea behind it and what the value actually stands...