  Applying the well-known Feynman-Kac formula of inhomogeneous case, an
interesting and rigorous mathematical proof of generalized Jarzynski's equality
of inhomogeneous multidimensional diffusion processes is presented, followed by
an extension of the second law of thermodynamics. Then, we explain its physical
meaning and applications, extending Hummer and Szabo's work ({\em Proc. Natl.
Acad. Sci. USA} {\bf 98}(7), 3658--3661 (2001)) and Hatano-Sasa equality of
steady state thermodynamics ({\em Phys. Rev. Lett.} {\bf 86}, 3463--3466
(2001)) to the general multidimensional case.
