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Thisexpressioncanbewritteninthedoublesumform,whereas
thesummationfirstfollowsl,andthenk:
3
3
3
A
ij
=
ΣΣ
abT
ik
jlkl
=
Σ
(
abT
ik
j
1
k
1
+
abT
ik
j
2
k
2
+
abT
ik
j
3
k
3
)
k
=
1
l
=
1
k
=
1
=
abT
i
1
j
111
+
abT
i
1
j
212
+
abT
i
1
j
313
+
abT
i
2
j
121
+
abT
i
2
j
222
+
abT
i
2
j
323
+
abT
i
3
j
131
+
abT
i
3
j
232
+
abT
i
3
j
333
.
Byreplacingtheindicesiandjwithanynumbers,e.g.i=1,j=3,
weobtaintheexpressionfor
A:
13
A
13
=
abT
1
k
3
lkl
=
abT
1
k
31
k
1
+
abT
1
k
32
k
2
+
abT
1
k
33
k
3
=
abT
113111
+
abT
123121
+
abT
133131
+
abT
113212
+
abT
123222
+
abT
133232
+
abT
113313
+
abT
123323
+
abT
133333.
1040Vectorsintherectangularcoordinate
system
Letabeanyvectorinaplaneinarectangularcoordinatesystem
Oxyzand
a
i
betheprojectionsofthatvectorontheaxesOx,Oy
andOz(Fig.1.5).
If
e
i
areversorsofaxes(orunitvectorswithsensescoincident
withthesensesoftheaxes)andletusassumethattheterminal
ą
ą
pointsofvectors
OA
and
OB
(thesearepositionvectors)have
ą
coordinates
x
i
and
y
i
,respectively,thenvector
a
=
AB
iswritten
inthefollowingform(Fig.1.5):
a
=
(
x
1
B
−
x
1
A
)
e
1
+
(
x
2
B
−
x
2
A
)
e
2
+
(
x
3
B
−
x
3
A
)
e
3
or
a
=
(
x
i
B
−
x
i
A
)
e
i
=
a
ii
e.
Thenumbers
a
i
=
(
x
i
B
−
x
i
A
)
arevectorcoordinates.Coordinates
canbepositive,negativeorequaltozero.Inthethree-dimensional
space,thegivenvectorawillbewrittenasfollows(seeFig.1.5):
aa
=
aaa
1
2
3
=
aaa
1
2
3
=
aaa
x
y
,]
z
={}
[
|
a
a
1
2
]
|
.
(,
,)[,
,][,
|
[
a
3
|
J
CHAPTER1
|
FUNDAMENTALSOFVECTORANDTENSORCALCULUS
———
———
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——
17
