1. G.H. Patel College of Engineering and
Technology
Subject: Field Theory
Name Enrollment no.
Champaneria Dhvanil J. 150113109004
Chauhan Nisarg D. 150113109005
Jadav Prashant 150113109009
Limbani Milan P. 150113109011
3. The Divergence Theorem
In this section, we will learn about:
The Divergence Theorem for simple solid regions,
and its applications in electric fields and fluid flow.
4. INTRODUCTION
• In Section 16.5, we rewrote Green’s
Theorem in a vector version as:
• where C is the positively oriented
boundary curve of the plane region D.
div ( , )
C
D
ds x y dA× =∫ ∫∫F n F
5. INTRODUCTION
• If we were seeking to extend this theorem to
vector fields on, we might make the guess that
• where S is the boundary surface
of the solid region E.
div ( , , )
S E
dS x y z dV× =∫∫ ∫∫∫F n F ……Equation 1
6. DIVERGENCE THEOREM
• It turns out that Equation 1 is true, under appropriate
hypotheses, and is called the Divergence Theorem.
•Notice its similarity to Green’s Theorem
and Stokes’ Theorem in that:
•It relates the integral of a derivative of a function (div F in
this case) over a region to the integral
of the original function F over the boundary of
the region.
7. SIMPLE SOLID REGION
• We state and prove the Divergence Theorem for regions E
that are simultaneously of types 1, 2, and 3.
• We call such regions simple solid regions. For instance,
regions bounded by ellipsoids or rectangular boxes are
simple solid regions.
•The boundary of E is a closed surface.
•That is, the unit normal vector n is directed outward from
E.
8. THE DIVERGENCE THEOREM
• Let:
–E be a simple solid region and let S be
the boundary surface of E, given with positive
(outward) orientation.
–F be a vector field whose component functions
have continuous partial derivatives on an open region
that contains E.
• Then, div
S E
d dV× =∫∫ ∫∫∫F S F
9. • Thus, the Divergence Theorem states that:
–Under the given conditions, the flux of F across the
boundary surface of E is equal to the triple integral of the
divergence of F over E.
10. THE DIVERGENCE THEOREM
• Let F = P i + Q j + R k
–Then,
–Hence,
div
P Q R
x y z
∂ ∂ ∂
= + +
∂ ∂ ∂
F
Proof
div
E
E E E
dV
P Q R
dV dV dV
x y z
∂ ∂ ∂
= + +
∂ ∂ ∂
∫∫∫
∫∫∫ ∫∫∫ ∫∫∫
F
11. • If n is the unit outward normal of S, then the surface
integral on the left side of the Divergence Theorem is:
( )
S S
S
S S S
d d
P Q R dS
P dS Q dS R dS
× = ×
= + + ×
= × + × + ×
∫∫ ∫∫
∫∫
∫∫ ∫∫ ∫∫
F S F n S
i j k n
i n j n k n
12. • So, to prove the theorem, it suffices to prove these equations:
S E
S E
S E
P
P dS dV
x
Q
Q dS dV
y
R
R dS dV
z
∂
× =
∂
∂
× =
∂
∂
× =
∂
∫∫ ∫∫∫
∫∫ ∫∫∫
∫∫ ∫∫∫
i n
j n
k n
• To prove Equation 4, we use the fact that E is a type 1 region:
where D is the projection of E onto the xy-plane.
( ) ( ) ( ) ( ){ }1 2, , , , , ,
E
x y z x y D u x y z u x y
=
∈ ≤ ≤
13. • By Equation 6 ,we have:
( )( )
( )2
1
,
,
, ,
u x y
u x y
E D
R R
dV x y z dz dA
z z
∂ ∂
= ∂ ∂
∫∫∫ ∫∫ ∫
•Thus, by the Fundamental Theorem of Calculus,
( )( ) ( )( )2 1, , , , , ,
E
D
R
dV
z
R x y u x y R x y u x y dA
∂
∂
= −
∫∫∫
∫∫
14. • The boundary surface S consists of three pieces:
–Bottom surface S1
–Top surface S2
–Possibly a vertical
surface S3, which lies
above the boundary
curve of D
(It might happen that
S3 doesn’t appear,
as in the case of
15. • Notice that, on S3, we have k ∙ n = 0,
because k is vertical and n is horizontal.
–Thus,
3
3
0 0
S
S
R dS
dS
×
= =
∫∫
∫∫
k n
• Thus, regardless of whether there
is a vertical surface, we can write:
1 2S S S
R dS R dS R dS× = × + ×∫∫ ∫∫ ∫∫k n k n k n
16. • The equation of S2 is z = u2(x, y), (x, y) D, and the outward
normal n points upward.
–So, from Equation 10
(with F replaced by
R k), we have:
( )( )
2
2, , ,
S
D
R dS
R x y u x y dA
× =∫∫
∫∫
k n
•On S1, we have z = u1(x, y).
•However, here, n points downward.
• So, we multiply
by –1:
( )( )
1
1, , ,
S
D
R dS
R x y u x y dA
× =
−
∫∫
∫∫
k n
17. • Therefore, Equation 6 gives:
( )( ) ( )( )2 1, , , , , ,
S
D
R dS
R x y u x y R x y u x y dA
×
= −
∫∫
∫∫
k n
•Comparison with Equation 5 shows that:
•Equations 2 and 3 are proved in a similar manner using the
expressions for E as a type 2 or type 3 region, respectively.
S E
R
R dS dV
z
∂
× =
∂∫∫ ∫∫∫k n
18. • Find the flux of the vector field
F(x, y, z) = z i + y j + x k
over the unit sphere
x2
+ y2
+ z2
= 1
–First, we compute the divergence of F:
Example 1
( ) ( ) ( )div 1z y x
x y z
∂ ∂ ∂
= + + =
∂ ∂ ∂
F
19. • The unit sphere S is the boundary of the unit ball B given by:
x2
+ y2
+ z2
≤ 1
–So, the Divergence Theorem gives the flux
as:
( ) ( )
34
3
div 1
4
1
3
S B B
F dS dV dV
V B
π
π
× = =
= = =
∫∫ ∫∫∫ ∫∫∫F
20. UNIONS OF SIMPLE SOLID REGIONS
• The Divergence Theorem can also be proved for regions
that are finite unions of simple solid regions.
•For example, let’s consider the region E that lies between the closed surfaces S1 and
S2, where S1 lies inside S2.
•Let n1 and n2 be outward normal
of S1 and S2.
21. • Then, the boundary surface of E is:
S = S1 S2
Its normal n is given
by:
n = –n1 on S1
n = n2 on S2
22. • Applying the Divergence Theorem to S, we get:
( )
1 2
1 2
1 2
div
E S
S
S S
S S
dV d
dS
dS dS
d d
= ×
= ×
= × − + ×
= − × + ×
∫∫∫ ∫∫
∫∫
∫∫ ∫∫
∫∫ ∫∫
F F S
F n
F n F n
F S F S