I want to apply to Cambridge as well as for the Gates Cambridge scholarship. However, I am unsure of which programs I am eligible for as an American student with a 4 year bachelor of science in physics from a strong liberal arts college. I believe that in Europe, a bachelor's degree is three...
Homework Statement
When carbon dioxide "dissolves" in water, essentially all of it reacts to form carbonic acid, H2CO3:
CO2(s) + H2O(l) <--> H2CO3(aq).
The carbonic acid can then dissociate into H+ and bicarbonate ions,
H2CO3(aq) <--> H+(aq) + HCO3-(aq).
Consider a body of otherwise pure...
Homework Statement
Consider a system of 2 large, identical Einstein solids. Each solid has N=10^23 oscillators, and the total energy units in the combined system is 2N.
a) Assuming that all of the microstates are allowed, compute the entropy of this system. This is the entropy over long time...
Aha. I think I get it. Is this what you were getting at (note, I have slightly altered the notation):
Assume x is a local minimum of f.
Define g(h) = f(x + h e_i) considering small values of h (so that |h| < r)
Note that g(0) = f(x). So,
g(0) \leq g(h) \forall |h| < r
That is, g...
My tex code got screwed up and then I had to step away from the computer so I deleted it. Anyway:
g(0) = f(\hat{x})
g(t) = f(\hat{x} + t e_i) where t \neq 0
So g(t) \geq g(0) for all t.
But how is this different from what I originally had, which is that f(x + te_i) \geq f(x) ? I'm...
Well, the minima of g would occur where
g'(t) = \frac{dF}{dt}(\hat{x} + t e_i) = 0 I suppose, but I'm not sure how to employ that. Can you give me a little more of a hint? I'm not seeing what we can say about the derivative of F with respect to t, I guess.
Homework Statement
Suppose F: Rn --> R has first order partial derivatives and that x in Rn is a local minimizer of F, that is, there exists an r>0 such that
f(x+h) \geq f(x) if dist(x, x+h) < r. Prove that
\nabla f(x)=0.
Homework Equations
We want to show that fxi(x) =0 for i = 1,...,n
So...
Homework Statement
A 21 cm spectral line corresponds to the flipping of the electron in a hydrogen atom from having its spin parallel to the spin of the proton to having it anti-parallel. Find the internal magnetic field experienced by the electron in the hydrogen atom.
Homework Equations...
Homework Statement
Let O be an open subset of R^n and suppose f: O --> R is continuous. Suppose that u is a point in O at which f(u) > 0. Prove that there exists an open ball B centered at u such that f(v) > 1/2*f(u) for all v in B. Homework Equations
f continuous means that for any {uk} in O...