Lecture51 : Applications of Divergence theorem [Section 51.2]
51.2.4
Special cases of Green's Identity :
(1)
Let in . Then, as we have
Thus, if , (in which case the scalar field is called harmonic ), we have
The integral is the average of the rate of change of along the normal on .
Thus, for a harmonic function on , average of its rate of change on is zero.
This is called the Laplace theorem .
in 
. Then, as 
we have 
, (in which case the scalar field 
is called harmonic ), we have 
along the normal on 
.
Thus, for a harmonic function on 
, average of its rate of change on 
is zero.
This is called the Laplace theorem . 
in (75). Then, 
or 
on 
Then, 
is harmonic, i.e., 
, we have 
