MATH 241 (Laskowski) Fall, 2007
MATLAB Problem Set 2
Due Tuesday, October 23
Note: Matlab online help commands gives examples of meshgrid and surf.
1. Enter the function
f(x, y) = cos(x2+ 4y2).
Be sure to make it āarray smartā (or āvectorizedā). i.e., use dots.
a) Graph fover the square D={|x| ā¤ 2,|y| ā¤ 2}. Hint: Use the meshgrid
command to make the set of grid points and then use surf. Label the axes.
b) Plot the level curves (contours) of fin the xy plane in the same square D.
You can use the command contour (works analogously to surf). Label the axes.
Note: You can put the plots of parts a) and b) together on the same page with
the command subplot.
2. Enter the function
u(x, y) = (ā4x3+ 3x2+ 1)(yāy2).
Make sure it is āarray smartā. The function uis the temperature at a point (x, y)
in the unit square Q={0ā¤xā¤1,0ā¤yā¤1}. The heat flux at each point is
defined to be the negative of the gradient vector āāu(x, y) = ā[ux(x, y ), uy(x, y)].
a) Verify by hand that ux(0, y) = 0 for 0 ā¤yā¤1 and that u= 0 on the other
three edges of the square. (*** You do not need to turn this part in.) This means
that the left edge of the square is insulated, and that the temperature is held at
zero on the other three edges.
b) Put a 20 by 20 meshgrid on Q. Graph uover Qusing the command
surf(X,Y,u(X,Y)). Note where the temperature is greatest, and the appearance
of the surface on the edge x= 0.
c) Compute the partial derivatives uxand uy. You can either have MATLAB
do this, or you can compute the partial derivatives by hand and then enter the
functions ux and uy into MATLAB by hand. Then enter the commands
U = u(X,Y);
Ux = ux(X,Y);
Uy = uy(X,Y);
contour(X,Y,U,20)
hold on
quiver(X,Y,-Ux,-Uy)
