(5 marks) Consider a smooth function u(x, y) satisfying: Show that u attains its maximum on the b...
7. (5 marks) Consider a smooth function u(x, y) satisfying: Urx + Uyy + Uzy > 0 in 12. . Show that u atains its maximum on the month ago Show that u attains its maximum on the boundary an.
Problem 3 (12 points): Let D be a bounded domain in R" with smooth boundary. Suppose that K(x, y) is a Green's function for the Neumann . For each x E D, the function y H K(x, y) is a smooth harmonic For each x E D, the normal derivative of the function y K(x, y) . For each z e D, the function y K(x,y)-Г(z-y) is smooth near problem. This means the following: function on D(r satisfies (VyK(x, y).v(b))-arefor...
6. (6 points) Let u = u(,y) be a smooth function in R2 such that au u + ara aya = 0 in a simply connected region R with boundary C. Let it be the unit outward normal to R. Show that ди ds = 0, where ди Vun.
2) Show that a Green's function G(x,y) satisfying the problem a2G = 8(x - y), G (0,y) = 6,(1, y) = 0 does not exist, but a modified Green's function Ĝ(x,y) satisfying a2G 22 = (x - y) -1, G.(0,y)=G.(1,y) = 0 does. How would you use G to solve problem (1) when f satisfies the condition that you found for a solution to exist? Hint: is f(x) = f(u) (8(x - y) - 1) dy?
2. (4 pts) Let f(x,y) =x2+y2.
Mark the locations where f attains its minimum and maximum on the
triangle constraint shown in Figure 1. Clearly indicate “minimum”
or “maximum” at each location.
2 0 X FIGURE 1. Figure for Problem 2. 2. (4 pts) Let f(x, y) = x2 + y². Mark the locations where f attains its minimum and maximum on the triangle constraint shown in Figure 1. Clearly indicate "minimum" or "maximum" at each location.
dy Find the function y(x) satisfying dx = 4x - 9 and y(5) = 0. dy The function y(x) satisfying = 4x - 9 and y(5)= 0 is y(x) = dx
(2 points) Find the maximum and minimum values of the function f(x, y) = 2x2 + 3y2 – 4x – 5 on the domain x2 + y2 < 100. The maximum value of f(x, y) is: List the point(s) where the function attains its maximum as an ordered pair, such as (-6,3), or a list of ordered pairs if there is more than one point, such as (1,3), (-4,7). The minimum value of f(x,y) is: List points where the function...
Consider air flows with velocity of U?=U= 10 m/s over a
semi-finite smooth flat plate with L=97 cm long. Calculate the
followings by assuming ? = 1.568 x 10-5 m2/s and ?=1.177
kg/m3.
Figure 1 : Boundary layer over a flat plate
Consider air flows with velocity of U?=U=10 m/s over a
semi-finite smooth flat plate with L=97 cm long. Calculate the
followings by assuming ? = 1.568 x 10-5 m2/s and ?=1.177
kg/m3.
b) Under some flow and boundary...
07. Show that the function u(x, y) In(5x +y) -5(z +y)2 is concave.
07. Show that the function u(x, y) In(5x +y) -5(z +y)2 is concave.
4 Consider the system represented in state variable form 0 x+ 2 y [1-1x +[0]u B C(sl- A) Show that a transfer function is related to the state equation by H(s) a) D, and find the transfer function for the system above. (5 marks) Sketch the Bode plot. b) (5 marks)
4 Consider the system represented in state variable form 0 x+ 2 y [1-1x +[0]u B C(sl- A) Show that a transfer function is related to the state equation...