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1. Definition of topology

Hi, I've been studying topology over the last semester and one thing that I was wondering about is why exactly is topology defined the way it is? For a refresher: given a set X we define a topology, T, to be a collection of subsets of X with the following 3 properties: 1) the null set and X...
2. Change of variables problem

let f be continuous on [0,1] and R be a triangular region with vertices (0,0), (1,0) and (0,1). Show: the double integral over the region R of f(x+y)dxdy = the integral from 0 to 1 over u f(u)du I recognize it is a change of variables problem but I'll be damned if I can create a set of...
3. Lagrange Error

In my BC calc class, we just finished working through most of series and sequences, and as we were reviewing years past free response questions on the topic, and in 2004, they dropped a lagrange error analysis. I've been looking at different explanations, but I'm not getting the concept. It...
4. Induced current in circuit

This one is really getting to me. Find the current through section PQ (the middle column of the circuit) of length a, a =.65 m. The circuit is located in a magnetic field whose magnitude varies with time according to the expression B(t) = .001t. Assume the resistance per length of wire is...
5. Potential difference problem

Two parallel plates having charges of equal magnitude but opposite sign are separated by .12 m. Each plate has a surface charge density of 36 nC/m^2. Determine the potential difference between the plates. I have to admit I am at a bit of a loss on how to answer this (lettings S...
6. Charge to mass ratio of an electron shot through a cathode-ray oscilloscope

I've been working on this problem for awhile but I just can't seem to hack it. (This is problem 76 from Serway 3rd for the first electricity chapter if you happen to have that book and want a visualization) Given: an electron with charge, -e, and mass, m, is projected with speed v, at a right...
7. Derivation of solid sphere moment of inertia

I need to derive the moment of inertia for a solid sphere, but I'm having some trouble. I did the following. I=?r^2dm given density, p= m/V pV=m so pdV=dm and differentiating V wrt r, d(4/3?r^3)dr = 4?r^2 so p4?r^2dr=dm and plugging that in I get I=?r^2(p4?r^2)dr I pull the p4...