Use the Debye approximation to find the following themodynamic functions of a solid as a function of the absolute temperature T
a) the fee energy F
b) the mean energy
c) the entropy S
Express your answers in terms of the Debye function D(y) =
and the Debye temperature D =
hwmax/k
e) Evaluate the function D(y) in the limit when y >> land
y<<1. Use these results to express the thermodynamic
functions F, and S
in the llimiting cases
when T << D and
T>>D
Use the Debye approximation to find the following themodynamic functions of a solid as a function...
For the following functions, answer if the function is homogeneous in (x , y), and if yes, what degree it is. a. b. c. Does any of these three production functions display constant returns to scale? F(z,y) %3D zy-y을 We were unable to transcribe this imageWe were unable to transcribe this image
a) By direct substitution determine which of the following functions satisfy the wave equation. 1. g(x, t) = Acos(kx − t) where A, k, are positive constants. 2. h(x, t) = Ae where A, k, are positive constants. 3. p(x, t) = Asinh(kx − t) where A, k, are positive constants. 4. q(x, t) = Ae where A, a, are positive constants. 5. An arbitrary function: f(x, t) = f(kx−t) where k and are positive constants. (Hint: Be careful with...
Find the inverse (unilateral) Laplace transforms of the following functions: (a) (b) (c) (d) (e) (f) (g) (h) (i) We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Consider the following nonlinear program: min s.t. - (a) Express the objective function of the above problem in the standard quadratic function form: (b) Find the gradient and the Hessian of f(x). (c) If possible, solve the minimisation problem and give reasons why the solution you found is a global minimum rather than just a local minimum. Otherwise, demonstrate that the problem is unbounded. f (x: y) = (x + 2y)2-2x-y We were unable to transcribe this imageWe were unable...
Use Cauchy Reimann equation to find the function is analytic and differential. a) Express the following in the form of (x+iy) b) c) We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Show that a function , which minimises, among all smooth functions , s.t. on , solves the following equation: in and on We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
The figure below shows a graph of the derivative of a function . Use this graph to answer parts (a) and (b) (a) On what intervals is increasing or decreasing? (b) For what values of does have a local maximum or minimum? (It asks to be specific). Only the values are needed (not ordered pairs). We were unable to transcribe this imageWe were unable to transcribe this imagepe & Bl apr derivative f' of a function f. Use this graph...
Find an equation for each polar graph. Express as a function of t. (click on graphs to enlarge) f (b) Five-petal rose (a) Cardioid (c) Circle N 4 We were unable to transcribe this imageWe were unable to transcribe this image
3. Consider a gas of fermion at a) Express the mean number of particle , and mean energy by polylogarithm function a) For a gas of fermion with density of state , show that the chemical potential is given by b) At finite temperature find the occupation number of the quantum state with energy . Explain qualitatively how this distribution would influence on the specific heat of the system. T7 0 We were unable to transcribe this imageWe were unable...
(a) Find the Fourier transform of the following function (b) Using Fourier transforms, solve the wave equation , -∞<x<∞ t>0 and bounded as ∞ f(r)e We were unable to transcribe this imageu(r, 0)e 4(r.0) =0 , t ur. We were unable to transcribe this image f(r)e u(r, 0)e 4(r.0) =0 , t ur.