  We extend Jarzynski's work relation and the second law of thermodynamics to a
heat conducting system which is operated by an external agent. These extensions
contain a new non equilibrium contribution expressed as the violation of the
(linear) response relation caused by the operation. We find that a natural
extension of the minimum work principle involves information about the
time-reversed operation, and is far from straightforward. Our work relation may
be tested experimentally especially when the temperature gradient is small.
