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MATH1211/LSweek33/TNK/1011(2)
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH1211: Multivariable Calculus
Lecture Summary – Week 33
April 11, 2011
We redrew
Figure 7.13
in
Section 7.1
(p.414) as shown below.
The image of the small rectangular region in
D
with sides given by vectors
¢
s
i
and
¢
t
j
under the
mapping
X
is approximately equal to the parallelogram with sides
¢
s
T
s
and
¢
t
T
t
, the area of
which is given by
k
¢
s
T
s
kk
¢
t
T
t
k
sin
μ
=
¡
k
T
s
kk
T
t
k
sin
μ
¢
¢
s
¢
t
=
k
T
s
£
T
t
k
¢
s
¢
t;
where
μ
is the angle made by the vectors
¢
s
T
s
and
¢
t
T
t
. Thus the total surface area of
S
=
X
(
D
)
is approximately equal to
P
i;j
k
T
s
(
s
i
¡
1
;t
j
¡
1
)
£
T
t
(
s
i
¡
1
;t
j
¡
1
)
k
¢
s
i
¢
t
j
where the
sum is taken over all small rectangles resulted from partitioning
D
. By taking limit of this sum when
all
¢
s
i
;
¢
t
j
!
0
, we obtain that the surface area of
S
is
RR
D
k
T
s
£
T
t
k
ds dt
(see p.414
−
415).
We noted that the surface area formula
RR
D
k
T
s
£
T
t
k
ds dt
is analogous to the arclength formula
R
b
a
k
x
0
(
t
)
k
dt
of a curve parametrized by
x
(
t
)
;a
·
t
·
b
.
We remarked that in equation (7) on p.416, we may write the term
¡
@
(
x; z
)
@
(
s; t
)
as
@
(
z;x
)
@
(
s; t
)
. This is
based on the fact that, for any square matrix, when two columns (or two rows) are interchanged, the
determinant will become negative of the original determinant. Accordingly, we may use the cycle of
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 Spring '11
 wang
 Multivariable Calculus

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