5120, Homework 1
1. Do the following exercises for these functions:
a) f = x2 + y 2
b) f = xy
∂2f
∂2f
- find ∂f
∂x , ∂x∂y , ∂y∂x , ∇f
- Determine whether there are any critical points. Which if any are local
maxima?
- (computing) Visualize the surface and the level curves given by the equation
z = f (x, y)
in Matlab. Explore different functions for plotting: mesh, meshz, meshc,
surface and see what they do. Type
help mesh
or similar to see the available options. Here is a sample code to help you:
[X,Y] = meshgrid(-5:.2:5, -6:.2:6);
z=sin(X).*cos(Y);
meshc(X,Y,z)
xlabel(’x’)
ylabel(’y’)
2. Determine whether the following vector fields are gradient fields. If
so, find φ such that F = ∇φ:
a) (x + y, x − y)
b) (x2 y, y 2 x)
3. The cross-sectional area of the small intestine varies periodically in
space and time due to peristaltic motion of the gut muscles. Suppose that
at position x the area can be described by
a
A(x, t) = [2 + cos(x − vt)],
2
where v is a constant.
a) Write an equation of balance for c(x, t), the concentration of digested
material at location x.
b) Suppose there is a constant flux of material throughout the intestine from
the stomach (that is, J(x, t) = 1) and that material is absorbed from the
gut into the bloodstream at a rate proportional to its concentration . Give
the appropriate balance equation.
c) Show that even if J(x, t) = 0 and σ(x, t) = 0, the concentration c(x, t)
appears to change
1