ES202
A SUMMARY ON INTEGRAL THEOREMS
DIVERGENCE THEOREM: Let V be a bounded region in R with a boundary S

consisting
of a piecewise smooth surface. Let n be the outward unit normal to S

and F (x,y,z) be a vector field with components having partial derivatives that are
continuous at all points of V and S, then
Region R
 
 
   FdV   F  ndS
V
S
Surface S enclosing region R
STOKES’S THEOREM: Let S be a piecewise smooth orientablesurface with
unit normal n and having as boundary the simple closed curve C. Let F(x,y,z) be a
vector field whose components have continuous first order partial derivatives on S,
then
S (surface)

b



 (   F )  n dS =  F  dr
S

t

n
C (boundary)
C
provided that the orientations of C and S are compatible.
GREEN'S THEOREM: If V is a volume enclosed
by a regular surface S and if


u(x,y,z)
and (x,y,z) are scalar functions and if F  u then divergence theorem on

F will give



   u dV   u  n dS
V
 
  S 
Since   up  u  p  p u , the integral of the right-hand side of the
equation will give the first term of Green's theorem:
2



 [ u   (u)  ()] dV   u  n dS
V
(a)
S
If we interchange u and  in equation (a) we'll have
2



 [ u  ()  (u)] dV   u  n dS
V
(b)
S
Now subtract (b) from (a), we'll obtain the second form of Green's theorem
2
2


 ( u    u) dV   ( u  u)  n dS
V
Special Cases:
S
above
i) If u =  = , then equation (a) will be
2

 [   
2
 
] dV     n dS
V
S

  
 2 
where  = ()  () and note that   n 
, so:
n
2

 [   
2
] dV   
V
S

dS
n


ii) If u= and =1, then   0 and  2  0 and

2
   dV   n dS
V
S
GREEN'S THEOREM IN PLANE: Let R be a closed region in the x-y plane having
its boundary as simple, closed, piecewise smooth curve C. Let M(x,y) and
N(x,y) be continuous functions of x and y with continuous first order partial
derivatives in R. Then
N M
 M(x, y)dx + N(x, y)dy   ( x - y ) dxdy
C
R
where C is traversed in positive direction.
Corollary: The are bounded by a simple closed curve C is given by
A
1
 (xdy - ydx)
2C



Proof of Green's theorem in plane: Let P = M(x, y) e1 + N(x, y) e 2 be a vector function
defined in a closed region S in x-y plane with boundary C. For this special case we get
:
  
N M
 
; if n  e 3 , so Stokes’s theorem,
(  P)  e3 =
x y
yields



 (   F )  n dS =  F  dr
S
C
y
N M

n
 ( x - y ) dxdy   (M, N)  (dx, dy)   (Mdx + Ndy)
S
C



where dr = (dx, dy) = dx e1 + dy e2 .
C

e2

e1
C

dr
x