Natural Science Forum / Physics / Research / October 2007

You are right that heat (or rather energy) is conserved, but only
globally. In each individual volume element, the amount of heat changes
at the rate given by the total heat flux through its boundary. You can
think of each volume having a little pouch where heat is stored.
Sometimes heat is added to it, sometimes heat is taken away from it. In
either case, it is this pouch that acts as the source (or sink) for the
heat flow vector field. So, unless dq/dt = 0, your h is not "source
free".

BTW, I have to chide Feynman for his use of the term "heat". I looked
over the section containing the above equation and he uses the word
"heat" in place of what most people would call energy (or at least
internal energy). Thermodynamics teaches us that heat is always
associated with a process, where heat is transfered between systems. The
first law of thermodynamics teaches us that what's being transfered is
just energy. But energy can be transfered in other ways as well,
mechanical or chemical.

In static situations, there is no mechanical energy transfer. So, all
energy flow has to be in the form of heat flow, that's where the h
vector field comes in. Then, "dq/dt" should represent the rate of heat
flow into the volume. However, there is no "heat density" q of which it
is a time derivative. While, in the static case, q may be identified
with the total energy per volume, as soon as the situation is no longer
static, the total energy can change by other means than heat transfer
and "dq/dt" is no longer a derivative of any quantity.  Instead of q we
should have the total energy density and specify that other terms enter
the left hand side of the above equation besides div h, representing
mechanical and other kinds of energy flow, aka work.

Hope this helps,

Igor

It may help your understanding if you rewrite the continuity equation
somewhat: generally speaking, a continuity equation equates the total
differential of a quantity with the sum of the local production and
loss rates, i.e.

Dq(r,t)/Dt = p(r,t)+l(r,t)

(the capital D shall signify the total differential; p(r,t) is the
local production rate (for instance associated with the absorption of
radiation), l(r,t) is the local loss rate (for instance through
inelastic collisions resulting in radiation)). Now Dq(r,t)/Dt is the
sum of the partial differential dq(r,t)/dt (commonly, the small greek
delta is used instead of d here) and the divergence of the flux of
q(r,t) (here h(r,t)), so we have

dq(r,t)/dt +div(h(r,t)) = p(r,t)+l(r,t)

which simply means that a net local production either leads to an
explicity locally time dependent quantity q(r,t) or a net heat flux
(or both).

Now you see that your assumption div(h(r,t))=-dq(r,t)/dt means that
p(r,t)+l(r,t)=0, i.e. there is no net source of heat here. In other
words, the change of the heat density in the volume element is solely
due to heat leaving or entering it to/from other volume elements. If
that is overall zero as well, we have obviously div(h(r,t))=dq(r,t)/
dt=0, i.e. the heat density is constant with time.

Thomas