# Search results

1. ### Probability distribution function / propagator

I have the following equation: \frac{1}{2N^{2}}\int_{s} \int_{p} \left\langle (\textbf{R}_{s} - \textbf{R}_{p} )^{2} \right\rangle which describes the radius of gyration of a polymer. (the term being integrated is the average position between beads p and s) This is shown to be...

3. ### Calculus question

Homework Statement I have the following expression: cos\theta = \frac{\vec{a}\cdot \vec{b}}{ab} (this is simply taken from a dot product rule for two vectors) However I need to find \nabla_{\vec{r}i} \theta Is there a way I can do it without involving differentials of arccos and...
4. ### Tricky differentiation

Hmm, I've transcribed the problem straight from the question sheet. I too had that thought. I'm not sure I follow you with your methodology though. If I try to solve the equation using the standard quotient rule, it gives me zero - is there a reason for that?
5. ### Tricky differentiation

Homework Statement Find \frac{dU_{ave}}{d\beta} where U_{ave}=\sum_{k}\left(\frac{U_{k}exp(-U_{k}\beta)}{exp(-U_{k}\beta)}\right) Homework Equations The Attempt at a Solution My answer is supposed to be -(U_{ave}^{2})+(U_{ave})^{2} However I keep getting zero. I can...
6. ### Formula derivation, algebra

Dang, that will teach me to copy and paste! I'm sorry, Defennder, here's the correct expressions: \frac{V_{2}}{V}=\frac{p-p_{1}}{p_{2}-p_{1}}
7. ### Formula derivation, algebra

I'm trying to derive a formula but can't seem to work the algebra. I need to combine these two: V_{1}p_{1} + V_{2}p_{2} = N V_{1} + V_{2} = V to get this: \frac{V_{1}}{V} = \frac{p-p_{1}}{p_{2}-p_{1}} where p = N/V If anyone could show me the steps that would be a huge help...
8. ### Hermitian Conjugate

And can I also ask why this seems to be a general property of the Hermitian Conjugate? (AB)^{+} = B^{+} A^{+} rather than (AB)^{+} = A^{+} B^{+}
9. ### Hermitian Conjugate

Simple question, and pretty sure I already know the answer - I just wanted confirmation, Considering the Hermitian Conjugate of a matrix, I understand that A^{+} = A where A^{+} = (A^{T})^{*} Explicitly, (A_{nm})^{*} = A_{mn} Would this mean that for a matrix of A, where A is a...
10. ### Separation of Variables / Boundary Conditions

^ Exactly, that's what I'm thinking. I've consulted a textbook and it too has the statement "...the only essential requirement being that k has the same value in both parts of the solution, ie, the part depending on x and the part depending on y". So that would indeed indicate that one can't be...
11. ### Separation of Variables / Boundary Conditions

Ever so sorry, I've missed something off in the transcription of the question: part (a) should read Find the general solution to the equation d²X(x) / dx² = 0 Now i'm even more confused!!
12. ### Separation of Variables / Boundary Conditions

Hmm, i'm not sure I understand. If I ignore the boundary condition when specifying a general solution for X, I would tend to write it as Ae^kx + Be^kx as that's the general solution for an ODE of d²X / dx² - k²X = 0 where k>0. Likewise, the gen. sol. for Y would be what I currently have for X...
13. ### Separation of Variables / Boundary Conditions

Homework Statement The edges of a square sheet of thermally conducting material are at x=0, x=L, y= -L/2 and y=L/2 The temperature of these edges are controlled to be: T = T0 at x = 0 and x = L T = T0 + T1sin(pi*x/L) at y = -L/2 and y = L/2 where T0 and T1 are constants...
14. ### Moment of Inertia / Mass Element proof

Homework Statement A thin square plate of side a has one corner at the origin and two sides along the positive x and y axes. If the density of the plate is given by p(x,y) = xy show that its mass is M=(1/4)a^4 If the distance of the mass element dM = pdS from the origin is r the moment of...