Recent content by Parmenides

Hello, A question I can't seem to find a simple answer to is, what happens to the Fermi-Dirac distribution at T grows large? Mathematics suggests that it approaches 1/2, like it does when the energy becomes equal to the Fermi energy. Or, are we not allowed to use the F-D distribution for high...

When considering the grand potential for a photon gas, one encounters an integral of the form: \Sigma = a\int_{0}^{\infty}x^2\ln\Big(1 - e^{-bx}\Big)dx I have never had to integrate something like this before; I was told it is done via contour integration, but I have never used such a method...

Homework Statement Estimate the size of the spherical abberation of a spherical mirror of 1m-diameter and a focal length of 2 meter. (Hint: Calculate the size of the smeared image of a star at the focal point and compare it to the size (in arc-sec) of an extended object)Homework Equations The...

The problem states that a particle moves in a plane under the influence of the following central force: F = \frac{1}{r^2}\Big(1 - \frac{\dot{r}^2 - 2\ddot{r}r}{c^2}\Big) and I am asked to find the generalized potential that results in such a force. Goldstein gives the following equation...

This is a problem from the Goldstein text. It gives two point masses ##m_1## and ##m_2## connected by a string (negligible mass), where ##m_2## is suspended by the string through a hole in a smooth table; ##m_1## rests on the table. It is important to note that ##m_2## only travels in a vertical...

Hello, I'm trying to learn C and am only a couple weeks into it. Suppose I want to write a simple code for Trapezoidal integration and want to have the user input the integration limits via use of scanf. Also, suppose I want to focus on only trig functions such that a common limit is pi. How...

I am attempting to work through a paper that involves some slightly unfamiliar vector calculus, as well as many omitted steps. It begins with the potential energy due to an electric field, familiarly expressed as: U_{el} = \frac{\epsilon_r\epsilon_0}{2} \iiint_VE^2dV =...

Solved. Sort of. This is actually Green's first identity, with ##\nabla_n{a}## being the normal derivative of the scalar function ##a##. As this is in the mathematics section and not physics, I won't bother with the extension to electric fields and potential.

There is a paper in chemical physics by Overbeek in which he describes the electrostatic energy of a double layer as the "energy of the surface charges and bulk charges in a potential field"; the transformation that he provides appears to be a variant of the divergence theorem in which he...

Suppose I was asked if G \cong H \times G/H . At first I considered a familiar group, G = S_3 with its subgroup H = A_3 . I know that the quotient group is the cosets of H, but then I realized that I have no idea how to interpret a Cartesian product of any type of set with elements that aren't...

Do you mean that since every element of H \times G_1 is unique and that every element of H \times G_2 can be defined as (c, f(a)) = (c, b), \forall c \in H, a \in G_1, b \in G_2 such that it is also unique, we have F: (c, a) \rightarrow (c, f(a)) = (c, b) And thus, there is a one-to-one...

Im asked to show that, given the groups H, G_1, and G_2 in which G_1 \cong G_2, that H\times{G_1} \cong H\times{G_2} Because of the isomorphism between G_1 and G_2, their cardinalities (order) are equal, which i think will be of good use when considering their Cartesian product with H. So...

Hello, Near the end of undergraduate physics, we are often told about the difficulty of quantizing the gravitational field and the absurdities that arise from it. However, I've yet to see a mathematical demonstration of where the incompatibilities of QFT and GR arise. Does anybody know where...

Implying that: \lim_{h \to 0}L_1(1) = \lim_{h \to 0}L_2(1) But as ##h \rightarrow 0##, doesn't this imply that ##L_1 = L_2##?

\lim_{h \to 0}\frac{L_1(h) - L_2(h)}{h} = 0 Since our theorem is satisfied, there exist linear mappings ##L_1## and ##L_2## such that ##L_1(h{\bf{x}})## and ##L_2(h{\bf{x}})## = ##h{\bf{x}}_1## and ##h{\bf{x}}_2##, respectively for all ##{\bf{x}}_1,{\bf{x}}_2 \in R##. By linearity, we have that...