If the function u (x, y) is a harmonic conjugate of v (x, y) prove that the curves u (x, y) = st. and v (x, y) = stations. are orthogonal to each other. These curves are called level curves. Now consider the function f (z) = 1 / z
defined throughout the complex plane except the beginning of the axes. Draw them
level curves for the real and imaginary part of this function
and notice that they are two...
Find u(x,y) harmonic in S with given boundary values:
S = {(x,y): 1 < y < 3} , u(x,y) = 5 (if y=1) and = 7
(when y=3)
S = {(x,y): 1 < x2 + y2 < 4},
u(x,y)= 5 (on outer circle) and = 7 (on inner circle)
I have these two problems to solve, and I'm not sure where to
start. Any help would be appreciated. Thanks!
Let u(x, y) be the harmonic function in the unit disk with the
boundary values u(x, y) = x^2 on {x^2 + y^2 = 1}. Find its
Rayleigh–Ritz approximation of the form x^2 +C1*(1−x^2
−y^2).
Consider the following utility functions:
(i) u(x,y) = x2y
(ii) u(x,y) = max{x,y}
(iii) u(x,y) = √x + y
(a) For each example, with prices px = 2 and
py = 4 find the expenditure minimising bundle to achieve
utility level of 10.
(b) Verify, in each case, that if you use the expenditure
minimizing amount as income and face the same prices, then the
expenditure minimizing bundle will maximize your utility.
Let X ∈ L(U, V ) and Y ∈ L(V, W). You may assume that V is
finite-dimensional.
1)Prove that dim(range Y) ≤ min(dim V, dim W). Explain the
corresponding result for matrices in terms of rank
2) If dim(range Y) = dim V, what can you conclude of Y? Give
some explanation
3) If dim(range Y) = dim W, what can you conclude of Y? Give
some explanation