By completing the following steps, prove Theorem 1. Let (h(s), k(s)) be a parame
ID: 1720096 • Letter: B
Question
By completing the following steps, prove Theorem 1. Let (h(s), k(s)) be a parametrization of the side condition curve in Theorem 1 by an arclength parameter s. For each point (x, y) in the plane, show that there are unique numbers sigma (x, y) and r(x, y), such that x = h(sigma (x, y))+ ar(x, y) and y = k(sigma(x, y)) + br(x, y). (Draw a picture.) Using the functions sigma (x, y) and r(x, y) of part (a), show that with the change of variables and with v(s, t) = u(x, y), the PDE au_x + bu_y + cu = f(x, y) becomes v_t + cv = F(s, t), where F(s, t) f(h(s) + at, k(s) + bt). Show that v(s, t) = e^-ct(integral_0^1 e^cr F(s, r) dr + U(s), where U(s) = u(s, 0) = is the C^-1 function which specifies the values of u on the side-condition curve. Thus, the unique solution of the problem in Theorem 1 is the C^1 function u(x, y) = v(sigma(x, y)) r(x, y) (Note that the Jacobian x_s y_t - y_s x_t = h'(s)b a 0 (Why?), so that sigma and tau are C^-1 by the Inverse Function Theorem which is covered in most advanced calculus books, e.g., [Taylor and Mann].)Explanation / Answer
When we multiply a , b and c we get:
a.b.c=abc
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