Lab Project V
The first two problems below ask for Maple output. Attach a printout
of this output to your solutions to the six problems below.
The Laplacian and the Heat Equation
Given a multivariate scalar function f(x, y,
z)
(the number of input variables here is not important, as long as there
are more than 1), we have already encountered the gradient as a type of
(vector) derivative for f:
On the other hand, we have also learned that one way to define a derivative
of a vector function F(x, y, z) = (P(x,
y,
z),
Q(x,
y, z), R(x,
y,
z)) is with the
divergence:
It follows that a type of second derivative of the scalar function f
can be obtained by computing the divergence of the gradient of f;
this quantity is called the Laplacian of f:
1. Compute the Laplacian of these functions: (a) f(x,
y,
z)
= x3 - 2xy + y2;
(b) f(x, y, z) = 1/(x2
+ y2 + z2). Check your work with
Maple.
2. An important class of functions are those whose Laplacians
vanish; these are called harmonic functions. Show that 
= 0 for the following functions: (a) f(x,
y,
z)
= x2 - y2;
(b) f(x,
y, z) = sin x cosh y.
[This is not a typo: cosh y is the hyperbolic cosine function, defined
as cosh y = (1/2)(ey + e-y).]
We know that heat flows in the direction of decreasing temperature.
(This is why radiators are so popular as home heating devices.) Physical
experiment shows that the rate of this flow is in fact proportional to
the rate of change of the temperature across the space in which the heat
flows. Therefore, if T = f(x,
y,
z;
t)
is the (scalar) function that gives the temperature
T at any point
(x, y, z) within some solid body at time
t,
then the velocity (vector) v of the flow will satisfy the relation
for some positive constant k called the thermal conductivity
of the body. If the body is some solid region of space S with
a smooth boundary surface 
,
then the amount of heat leaving S per unit of time is measured by
the flux integral
where n is chosen consistently across the surface
as the outward pointing unit normal vector.
3. Apply the divergence theorem to argue that this can be rewritten
in the form 
.
On the other hand, the total amount of heat H within the solid
S can be computed with the integral
where s is the specific heat of the material
of which the solid is made (the amount of heat per unit mass per unit of
temperature, a property of the material) and r
is the mass density (the amount of mass per unit volume); both are
essentially constant in typical circumstances. Clearly, the dimension
of the quantity srT(x, y,
z)
is that of heat per unit of volume, so integrating this quantity across
S
gives the total heat found within S.
Consequently, the rate at which heat leaves S over time is
(negative because it is a rate of decrease).
4. Deduce that 
.
Because this integral equation holds for any region of space S,
it holds for, say, a sphere with very small radius centered at a point
P
= (x0, y0, z0).
If we choose the radius of the sphere small enough, we can assume that
the integrand is essentially constant over the sphere, equal to its value
at P, so
whence we conclude that
As this is true for any point P, it follows that
universally across space. This equation is called the heat equation,
a central result in thermodynamics.
5. Which of the four functions in #1 and #2 above satisfy the
heat equation?
6. If the temperature function T = f(x,
y,
z;
t)
is a harmonic function of the space variables (x,
y,
z),
explain how we can conclude that temperature is independent of time.
That is, a harmonic function T provides a steady-state solution
to the heat equation.
Due date: Friday, December 14, 2001.