•
One is for scalar Felds (scalar functions):
±±
S
f
(
x, y, z
)
dS
=
R
f
(
r
(
u, v
))
||
r
u
×
r
v
||
dA.
•
The other is for vector Felds:
S
F
(
x, y, z
)
•
d
S
=
R
F
(
r
(
u, v
))
•
(
r
u
×
r
v
)
dA.
Remark:
1. The formulas are similar to the formulas of line integral: for scalar functions, we
multiply the magnitude; for the vector Felds, we use dot product.
2. Given a function (Feld), we can easily determine which formula to use.
3. Usually, two diﬃculties to apply these formulas: Fnd out the expression of the
function, and Fnd out the parametric equations of the surface.
Solution
The solution is quite obvious.
Since we need to apply the two formulas for surface
integrals listed above, so the solution steps are:
•
Find
f
(
r
(
u, v
))
,
F
(
r
(
u, v
))
and
r
u
×
r
v
.
•
Substitute them into the formula, and estimate the double integral on the right
hand side (sketch the region
R
, and change the double integral to iterated integral).
±rom the conditions given in the question, we know
f
(
x, y, z
)=
x
+
y
+
z
and
F
(
x, y, z
x
2
i
+
y
2
j
+
z
2
k
,
so substitute the parametric equation of the surface
r
(
u, v
)=(2
u
+
v
)
i
+(
u
−
2
v
)
j
u
+3
v
)
k
into the two functions, that is, we substitute
x
=2
u
+
v,
y
=
u
−
2
z
=
u
v
and get:
f
(
r
(
u, v
))
=
f
(2
u
+
v, u
−
2
v
)
=(
2
u
+
v
)+(
u
−
2
v
u
v
)
=4
u
+2
v.
F
(
r
(
u, v
))
=
F
(2
u
+
−
2
v
)
2
u
+
v
)
2
i
u
−
2
v
)
2
j
u
v
)
2
k
.
199