d
r

∫
a curve joining
p
0
to
(
a,b
)
F
·
d
r
)
=
1
h
(
∫
the line joining
(
a,b
)
to
(
a
+
h,b
)
F
·
d
r
)
=
1
h
∫
h
0
(
f
(
a
+
t, b
)
, g
(
a
+
t, b
))
·
(1
,
0)
dt
=
1
h
∫
h
0
f
(
a
+
t, b
)
dt
which tends to
d
ds
∫
s
0
f
(
a
+
t, b
)
dt

s
=0
as
h
tends to 0. This derivative is just
f
(
a, b
) by the fundamental
theorem of calculus. Thus,
f
=
∂ϕ
∂x
. Similarly,
g
=
∂ϕ
∂y
and we conclude that
F
=
∇
ϕ
.
4.4
Surface Integrals
Definition 4.4.1.
Let
f
be a function in 3 variables.
S
is a surface parametrized by
r
(
u, v
) for (
u, v
)
in a certain region
R
. We define the surface integral of
f
over
S
to be
∫
S
fdσ
=
∫
R
f
(
r
(
u, v
))
∂
r
∂u
×
∂
r
∂v
dudv.
Remark 4.4.1.
If a piece of metal plate occupies the surface
S
and
f
(
x, y, z
) =the density of the metal
plate at the point (
x, y, z
). Then
∫
S
fdσ
=total mass of the metal plate.
Remark 4.4.2.
The surface integral of a function over a surface is independent of the parametrization
chosen for the surface.
However, we are unable to prove it here for the lack of a
change of variable
formula.
Example 4.4.1.
Let
f
(
x, y, z
) =
x
2
+
y
2
and let
S
be the unit northern hemisphere. Evaluate
∫
S
fdσ
.
42
Remark 4.4.4.
We saw that
n
=
∂
r
∂u
×
∂
r
∂v
∥
∂
r
∂u
×
∂
r
∂v
∥
is indeed an unit normal to
S
, while
∥
∂
r
∂u
×
∂
r
∂v
∥
dudv
is
the surface area element
dσ
of
S
. Therefore, the flux integral of
F
over
S
is indeed the surface integral
of
F
·
n
over
S
.
Thats why such a symbol is adopted.
However, there are always two choices for an
unit normal to the surface. We make the following convention when dealing with closed surfaces (i.e.
a surface that divides the whole space into two parts: the interior and exterior.), we always make the
choice of the unit outward normal.
Remark 4.4.5.
Consider the case when we are studying the sea.
F
(
x, y, z
) is the flow velocity at the
point (
x, y, z
). Let
S
be an imaginary surface in the sea. Then,
∫
S
F
·
n
dσ
is the volume of water flowing
THROUGH
S
per unit time. In case
S
is a closed surface and
R
is the interior of
S
,

∫
S
F
·
n
dσ
is the
volume of water accumulated in
R
per unit time.