Homework Statement
Determine Equilibrium points, limiting cycles, and their stabilities for the following equations
r'=r(r-1)(r-3)
θ'=1
The Attempt at a Solution
So I know one equilibrium point is going to be (0,0) because r=0 is a limiting cycle (I believe), and that is simply a...
Homework Statement
1. The sample space Ω of a certain experiment are the values 1, 2, 3, 4 and
5 and the probability assigned to a possible value w is proportional
to w2.
(a) What is the probability mass function p(w) for this probability
space?
(b) What probability does this mass...
Ok, so I know that the laws of physics say reaching absolute zero temperature is impossible, but suppose we took a box that was perfectly insulated in completely empy space, and I took all the particles out of it to create a vacuum. Now, since there are no particles in the box, then wouldn't...
Ok, so I was just thinking about einsteins famous equation E=mc^2, and I was just wondering, if I were to take, say a rock or piece of metal. If I were to weigh it, and get it's mass, I could compute how much energy it contains. However, if I were to add heat to the rock or piece of metal by...
Homework Statement
A mass of 4 kg is stretches a spring by 1 m. An external force of cos (!t)
N acts on the mass. Assume that the damping constant is nonzero and gravity is 10 ms^-2.
Consider a spring mass system described by the following IVP.
u''+yu'+u = cos(wt)
u(0) = 0
u'(0) = 0
1) Find...
Ok, so I just watched the nova fabric of the cosmos (blew my mind). Anyways, suppose I jumped on top of a beam of light and road it as it traveled out into space. Now, suppose that as I was traveling out there, another person is coming at me in the exact opposite direction on another beam of...
Homework Statement
Consider the hyperboloid x2+y2-z2 = 1 at the point (1,0,0). Take the normal direction i to the surface.
a) Compute the curvature of the circle x2+y2=1 on the hyperboloid (z=0) at the point (1,0).
b) Compute the curvature of the hyperbola x2-z2=1 on the hyperboloid...
I'm just checking my work on this. Given an 8x8 chessboard, you randomly place 8 rooks on the board. What is the probability that no rooks can capture another one. In other words, probability that no 2 rooks are in the same row or column.
My solution is simply 8!/(64 choose 8), but that...
Ok, so this relates to my homework, but I really can't find an answer anywhere, so this is more of a general question. First off, what does a "chart" of a manifold look like? Is it a set, a function, a drawing, a table, what?! I have found so many things about charts, but nothing shows what...
Homework Statement
Charts on the torus T.
Let S1 be the unit circle, and for each value 0 ≤ θ < 2π, let P(θ) be the point on the circle
at angle θ.
Let S1×S1 be the Cartesian product of two circles. The elements of S1×S1
are P(φ), P(θ), where 0 ≤ φ, θ < 2π.Let P(φ, θ) be the point on the...
Thank you, that makes a lot more sense. It also helps with part 2 of the problem (where you can take the limit as theta approaches 3π/2 for MS and you get x>1 or x<1, and you can do the same with MN, but instead have theta approach ∏/2 to get the same thing).
I am a little confused on how to...
Homework Statement
Take the unit circle in the x-y plane with center at (0, 0), bisected by the x-axis. Take two
maps, the first MS from the circle minus the south pole S to the x-axis that take a point P on the circle to the intersection of the line from the south pole (0, −1) through P with...
Ok, so I was just thinking about this today. First off, I was thinking about absolute zero, and how they say it's impossible, but then I thought about what the average temp. of the universe is. According to several internet sources, the current average temp is about 2.73 kelvin. I was...
Ok, so I just looked up the formula for the radii of a torus, and I just explained how I got 1 as the radius of the circle of the tube, and 3 as the radius of the torus itself. As for the Area of coverage, I was able to simplify it down and for the Area you end up just getting the double...
Homework Statement
Consider the parametrization of the torus given by:
x = x(s, t) = (3 + cos(s)) cos(t)
y = y(s, t) = (3 + cos(s)) sin(t)
z = z(s, t) = sin(s),
for 0 ≤ s, t ≤ 2π.
(a). What is the radius of the circle that runs though the center of the tube, and what is
the radius of the tube...