Give at least 2 applications of the interconversion of the different states of matter in real life/ every day life.

Suppose that A,, is a covector field, and consider the object Fμv=O₁ Av - O₂ A₁. (a) Show explicitly that F is a tensor, that is, show that it transforms appropriately under a coordinate transformation.

Problem 2) Consider the following Maxwell Boltzmann distribution of molecular speeds: P(v) = 4( m 27kBT. mp² v²e 2kgT To calculate average values for say f(v) (function of v) one just integrates f(v) with P(v)dv from zero to infinity = P(v)f(v)dv, where signifies average of f(v). Of course, the distribution should be normalized: P(v)dv=1, (is a requirement for any probability distribution). a) Check the last equation. b) Calculate the average of v. c) Calculate the average of v². d) Calculate from c) the RMS value of the speed. e) Calculate the most probable value of v. f) Square the results of b, d and e and rank them from smallest to the largest value.

Starting from the Maxwell distribution of speeds, 3/2 v² cxp m mv² f (v) = 47N 2nkgT 2kBT find (a) I s(v) dv, (b) vf (v) dv, and (c) | Ef (v) dv, 0. where E is the kinetic energy of the gas.