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6. We have said that U = U(T,V) and H-H(T,P). However, it is possible to write...
Let H be a complex Hilbert space. 6. (a) Let φ, ψ E H \ {0} . Define the linear operator T on H by Using the Cauchy-Schwarz inequality, show that llll = Hell ll [4 marks] (b) A bounded linear operator A is said to have rank one if there exists v e H [0 such that for any u E H we have Au cu, where cu E C is a constant depending on u. (i) Show that...
6. Suppose that, instead of boundary conditions Eqs. (2) and (3), we have u(x, o, t) -f^(r), u(r, b, t)() 0<x<a, 0<t (2') u(0,y, t)-gi(v), u(a,y,t)-89(v) 0 <y<b, o<t (3) Show that the steady-state solution involves the potential equation, and indicate how to solve it. 6. Suppose that, instead of boundary conditions Eqs. (2) and (3), we have u(x, o, t) -f^(r), u(r, b, t)() 0
6. Assume that ( U U ), ( V V ) and (W, w) are three normed vector spaces over R. Show that if A: U V and B: V W are bounded, linear operators, then C = BoA is a bounded, linear operator. Show that C| < |A|B| and find an example where we have strict inequality (it is possible to find simple, finite dimensional examples).
Show that for the vectors Tu and Ty, we have the formula u, V U,V Show that for the vectors Tu and Ty, we have the formula u, V U,V
We have seen the microcanonical ensemble, Ω(U,V,N), the canonical ensemble, Z(T,V,N), and the grand canonical ensemble, Z(T,V,μ). Why do we not create a hyper-grand canonical ensemble, Ξ(T, P, μ)?
and the references it needs H-13. Equation 16.5 gives P for the van der Waals equation as a function of V and T. Show that P expressed as a function of V, T, and n is nRT n2a Now evaluate (aP/aV), from Equation 16.5 and (aP/aV), from Equation 1 above and show that (see Problem H-12) a P H-12. Prove that and that a P T,n where Y = Y(P, T, n) is an extensive variable. We were unable to...
4. The enthalpy H may be written as a function of temperature T and pressure P. If we have a system whose composition remains constant and using Maxwell's equations and the total differential, we can write dH as avdP where Cp is the heat capacity at constant pressure and the subscript of P on the partial derivative represents the partial of volume with respect to temperature holding pressure connstant. Find the change in enthalpy (A) for an ideal gas undergoing...
5. In class we saw that the function r(u, v) = (sin u, (2 + cos u) cos v, (2 + cos u) sin v), 0<u<27, 050521 parametrizes a torus T, which is depicted below. (a) Calculate ||ru x rull. (b) Show that T is smooth. (c) Find the equation of the tangent plane to T at (0,). (d) Find the surface area of T (e) Earlier in the semester, we observed that a torus can be built out of...
Consider and impulse response h(t)= p(t ,T ), where p(t,T) is the pulse function u(t) - u(t - T) Find the output for the following inputs to the system via convolution: a) p(t,T) b) u(t) c) r(t) = 0, t < 0 a t, 0 < t < T 0, t > T
The internal energy, U, is the total energy of a system. For any isolated system, the internal energy is constant. U is a state function, meaning that any path used in calculating AU ill result in the same answer. For any pure substance or fixed mixture of substances, the internal energy, U, can be determined from any two of the variablesP, V, and T. It is often most convenient to choose V and T as the variables. It is helpful,...