Physics 325
Lecture 3
Integration of Vectors
Let
be a vector function of
t
:
A
G
( )
A
A t
=
G
G
.
Then
(3.1)
( )
( )
( )
( )
2
2
2
2
1
1
1
1
ˆ
ˆ
ˆ
t
t
t
t
x
y
z
t
t
t
t
A t dt
i
A
t dt
j
A
t dt
k
A
t dt
=
+
+
∫
∫
∫
∫
G
The vector integral is done by performing three separate ordinary integrations.
Similarly,
if the function depends on
x,y,z
, the integration over volume becomes
(
)
(
)
(
)
(
)
ˆ
ˆ
ˆ
,
,
,
,
,
,
,
,
x
y
z
V
V
V
V
A x y z dV
i
A
x y z dV
j
A
x y z dV
k
A
x y z dV
=
+
+
∫
∫
∫
∫
G
(3.2)
where
2
2
2
1
1
1
x
y
z
V
x
y
z
dV
dx
dy
dz
=
∫
∫
∫
∫
Similarly, the symbol
denotes a double integral over a surface
S
.
S
∫
Surface Integrals:
Let
da
be an element of area of a surface
S
, and
be the unit normal vector to
S
at
da
:
ˆ
n
ˆ
da
da n
=
G
da
The vector
is associated with the element of surface area
da
, and points in a direction
that is outward from and normal to the surface.
“Outward” is well defined for a closed
surface.
For an open surface, we will adopt the right-hand rule to define the “outward”
direction.
da
G
If
, then
.
We can now define the integral over a surface of the
projection of a vector function
ˆ
ˆ
n
i
=
ˆ
ˆ
x
da
ida
idydz
=
=
G
(
)
,
,
A x y z
G
onto the normal of the surface as
(
)
ˆ
x
x
y
y
z
z
S
S
S
A da
A nda
A da
A da
A da
=
=
+
+
∫
∫
∫
G
G
G
i
i
(3.3)

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