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a(’?) = ’(a?)
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a￡(kb) = k(a￡b)
a￠(kb) = k(a￠b)
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O
￥
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1 è??
^?B è
a￠(b￡c) ; a￡(b￡c) ; a￠(b￠c) ; a￡(b￠c)
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x 1.2 ?
O
￥
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a￠(b￡c) =
flfl
flfl
flfl
a1 a2 a3
b1 b2 b3
c1 c2 c3
flfl
flfl
flfl = b￠(c￡a) = c￠(a￡b)
=?a￠(c￡b) =?b￠(a￡c) =?c￠(b￡a)
}w
~a; b; c
() a ￠ (b ￡ c) = 0 a; a; cB?
() a￠(a￡c) = 0
1
x 1.3 ?
O
￥
O
O￥Z_
!c￡(a￡b) = f5fa; b
ü
 ODc?
ü
￥
g?<°; ?fa; b
ü

5 V[s31
?Z_c￡(a￡b) = (b￠c)￠a?(a￠c)￠b
(a￡b)￡c = (a￠c)￠b?(b￠c)￠a
}w
~
a￡(ex ￡ey) = (a￠ey)￠ex?(a￠ex)￠ey = ay ￠ex?ax ￠ey
ex ￡(a￡ey) = (ex ￠ey)￠a?(a￠ex)￠ey =?ax ￠ey
￡
ü
O￥
T
} p￡~
c￡(a￡b) = (b￠c)￠a?(a￠c)￠b
}￡
ü~
!
f = c￡(a￡b) = c￡d
#
d = (a2b3?a3b2)ex?(a1b3?a3b1)ey +(a1b2?a2b1)ez
f1 = c2d3?c3d2 = c2(a1b2?a2b1)+c3(a1b3?a3b1)
= a1(b2c2 +b3c3)?b1(a2c2 +a3c3)+(a1b1c1?b1a1c1)
= a1(b￠c)?b1(a￠c)
] ?
f2 = a2(b￠c)?b2(a￠c)
f3 = a3(b￠c)?b3(a￠c)
#N
f = (b￠c)￠a?(a￠c)￠b
x 1.4
O
s3
|
O
s31
?
O
￥ B"bZ_ =a; b
ü
 O<
°cZ_
a￠(b￠c) = (c￠a)￠b+c￡(a￡b) = (c￠a)￠b+(b￡a)￡c
a￠(b￠c) = (a￠b)￠c+b￡(a￡c) = (a￠b)￠c+(c￡a)￡b
}w
~
a = a￠(ex ￠ex) = (a￠ex)￠ex +(ex ￡a)￡ex = ax ￠ex +ex ￡(a￡ex)
2
x 1.5±s
0r
?]B?￥
O
1?r
0 μ±sa
O

×+?b
``3r
0￥vl1r
?
``3r
0￥Z_:_
r = ex ￠ @@x +ey ￠ @@y +ez ￠ @@z
r￠f = @fx@x + @fx@y + @fx@z
r￡f = (@fz@y? @fy@z )￠ex +(@fx@z? @fz@x )￠ey +(@fy@x? @fx@y )￠ez
r’ = @’@x ￠ex + @’@y ￠ey + @’@z ￠ez
x 1.6 r
0
O
a±s+?￥w
a(’?) = ’(a?) )r(’?) = ’(r?)+?(r’)
a￡(kb) = k(a￡b) )r￡(kb) = k(r￡b)+rk￡b
a￠(kb) = k(a￠b) )r￠(kb) = k(r￠b)+rk￠b
a￠(a￡c) = 0 )r￠(r￡c) = 0
a￡(ka) = 0 )r￡(rk) = 0
èB
} p3~
r(f ￠g) ; r￡(f ￡g) ; r￠(f ￡g)
}3~
r(f ￠g) = r(f ￠gc)+r(fc ￠g)
! (gc ￠r)f +(f ￡r)￡gc +(r￠fc)g+(g￡r)￡fc
= (gc ￠r)f +gc ￡(r￡f)+(fc ￠r)g+fc ￡(r￡g)
= (g￠r)f +(f ￠r)g+g￡(r￡f)+f ￡(r￡g)
r￡(f ￡g) = r￡(f ￡gc)+r￡(fc ￡g)
! (r￠gc)f?(r￠f)gc +(r￠g)fc?(r￠fc)g
= (gc ￠r)f?gc (r￠f)+fc (r￠g)?(fc ￠r)g
= (g￠r)f?(f ￠r)g+f (r￠g)?g(r￠f)
3
èB
?
r￠(f ￡g) = r￠(fc ￡g)+r￠(f ￡gc)
!?fc ￠(r￡g)?gc ￠(f ￡r)
=?fc ￠(r￡g)+gc ￠(r￡f)
=?f ￠(r￡g)+g￠(r￡f)
è=
}?
TMD~
(f ￠r)g ; (f ￠r)’
}3~á
ì??
(f ￠r)g???
T
e?9è¨b M
Do
ˉb
(f ￠r)g = (fc ￠r)g = r(fc ￠g)?fc ￡(r￡g)6=f (r￠ g)
(f ￠r)’ = f ￠r’
 è ?/
!k1è
O

(k￠r)r = k
(k￠r)r = r(k￠r)?k￡(r￡r) = r(k￠r) = k
(f ￠r)(’g) = g[(f ￠r)’]+’[(f ￠r)g]
Y
L
?i
O V?
(f ￠r)g = r￠(fg)?(r￠f)g
è ?
}?
TMD~
(f ￡r)￡g ; (f ￡r)￠g ; (f ￡r)’
}3~???
T?z9
á
ì?¨
(f ￡r)￡g = (fc ￡r)￡g = r(fc ￠g)?fc (r￡g)
(f ￡r)￠g = f ￠(r￡g)
(f ￡r)’ = f ￡r’
4
x 1.7ˉf
￥±s
0
rf(u) = ru df(u)du
r￠A(u) = ru￠ dA(u)du
r￡A(u) = ru￡ dA(u)du
(a￠r)A(u) = (a￠r)u dA(u)du
}2
~
r = ru ddu
x 1.8Bt許s
r￠( rr3) = rr￠ ddr( rr3)
= rr￠ ddr(rerr3 ) = (rr￠er) ddr( 1r2)
=? 2r3

w?￥p?
r6=A(r)
}許s~
rr = er ; r1r =?err2
r￠r = 3 ; r￠er = 2r ; r￠( rr3) = 4…–(r)
r￡r = 0 ; r￡er = 0 ; r￡( rr3) = 0
è
1
k¨

T￡
ü
r￠( rrn) = 3?nrn + 4…rn?3–(r)
}￡
ü~
r￠( rrn) = r￠( rr3 ￠ 1rn?3)
= r￠( rr3)￠ 1rn?3 +( rr3)￠r 1rn?3
= 4…rn?3–(r)+( rr3)￠[rr￠ 3?nrn?2 ]
= 4…rn?3–(r)+ 3?nrn
5
x 1.9?US"/￥±s
0
r? = @?@rer + 1r @?@ e + @?@zez
r￠f = 1r @@r(rfr)+ 1r @f @ + @fz@z
r￡f = (1r @fz@? @f @z )er +(@fr@z? @fz@r )e +[1r @@r(rf )? 1r @fr@ ]ez
r2? = 1r @@r(r@?@r )+ 1r2 @
2?
@ 2 +
@2?
@z2
x 1.10?US/許s
rr = er ; rz = ez
r￠er = 1r ; r￡er = 0
r￠e = 0 ; r￡e = 1rez
r￠ez = 0 ; r￡ez = 0
x 1.11 oUS"/￥±s
0
r? = @?@rer + 1r @?@ e + 1rsin @?@`e`
r￠f = 1r2 @@r(r2fr)+ 1rsin @@ (sin f )+ 1rsin @f`@`
r￡f = 1rsin [ @@ (sin f`)? @f @` ]er + 1r[ 1sin @fr@`? @@r(rf`)]e
+ 1r[ @@r(rf )? @fr@ ]e`
r2? = 1r2 @@r(r2@?@r )+ 1r2 sin @@ (sin @?@ )+ 1r2 sin2 @
2?
@`2
x 1.12 oUS/許s
rr = er
r￠er = 2r ; r￡er = 0
r￠e = 1rtan ; r￡e = 1re`
r￠e` = 0 ; r￡e` = 1rtan er? 1re
6
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= p bWò?￥H? <
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B￠dl = 2…rB =?0 I ) B =?0I2…re
#
r￡B = r￡(?0I2…re ) =?0I2… [r￡(1re )]
=?0I2… [(r1r)￡e + 1r(r￡e )]
=?0I2… [rr ddr(1r)￡e + 1r ezr ]
=?0I2… [? 1r2(er ￡e )+ 1r ezr ]
= 0
è?
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??r < a
H
I
L
B￠dl = 2…rB =?0
ZZ
S
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#
r￡B = r￡(?0Ir2…a2e ) =?0I2…a2[r￡(re )]
=?0I2…a2[(rr)￡e +r(r￡e )]
=?0I2…a2[(er ￡e )+rezr ]
=?0I…a2ez
=?0J
?=?f
?
x 2.1USMDD1y
 (Einstein) p??
?f
D bW#USMD
á MM1b ??
x
f bW
=2 5:
°?USMD/￥f

7
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!§"?B?(x1;x2;x3)YVUS"￥?§0"?US1(x01;x02;x03)
5
x0i =
3X
j=1
fiijxj ; i = 1;2;3
? ?fiij = e0i ￠ej1US??￥Z_??? ?￥ ??1USM
D"
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üi?"??·S p p|Pi??·S?-1?·Sb
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T' Ve?
x0i = fiijxj ; i = 1;2;3
USMDD1y
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@
T(1)
 ? ??M￥L?MD?-1??MD
x0ix0i = xixi = const (1)
? bW?
???MDb "
?fiij1B?? ?
~fifi = I
? ?I1?ê ?b
x 2.2f
￥?l
}?l~ ?T
Bt ?
T ??2 58US"/?3n?μ?s
Tl￠￠￠m
í
i OüV?US"§?§0￥MDfiija
@ ?/1"
T0i￠￠￠j = fiil￠￠￠￠￠￠fijm| {z }Tl￠￠￠m
5??
T1n¨f
b
?
,¨f
S
USMD/?M ?é
aè?
T0 = T
?B¨f

O
 ?
a ?aè? <ar
??
T0i = fiijTj
@
@x0i =
@xj
@x0i
@
@xj = fiij
@
@xj
?ú¨f
?¨K￥
^=¨f

8
x 2.3=¨f

?=¨f
 ?f ?f
aè
1 a?8
aoèf
? T0 = ATA0
T0ij = fiilfijmTlm
?=¨f
 V[¨B? ? ?V
U Q-? ?
?f
￥clTijs
jZ_s
T¨/￥iZ_￥Q?rT
?f
￥1? ?B?f
? V[s31 ???s
3
S
Tii1?11
3í?f
Tij = Tji OTii = 01?15è
1
3Q?f
Tij =?Tji1?13
?f
￥???
USMD?M
?
?
O
￥i
O1=¨f
ab =?!T 6= ba ; Tij = aibj ?VrD·SVr
x 2.4 X
= X(Kronecker)?|–ij
9D?|
? X
= X?|–ij
–ij = 1 (i = j)
–ij = 0 (i 6= j)
???–ij = –ji
???M?–0ij = –ij
?9D?–ijvj = vi
??êf
v￠?!I =?!I ￠v = v
x 2.5 à?{ ?
ó(levi{civita)?|"ijk
?|
? à?{ ?
ó?|"ijk
?¨Q?f

"ijk = +1 (ijk = 123;231;312)
"ijk =?1 (ijk = 213;321;132)
"ijk = 0 (ijk = 112;233;￠￠￠)
?Q??"ijk =?"jik
???M?"0ijk = "ijk = "kij = "jki
?
?(a￡b)i = "ijkajbk ; (r￡A)i = "ijk@jAk
9
x 2.6f
￥
?f
M?Aij = Bij?i ?V
U
?f
FEAij +Bij = Cij￡
üCij9
^?f

?S
Df
Me?!C = k?!A,Cij = kAij
?f
￥ie?!C =?!A?!B,Cijlm = AijBlm
``3B?7y?!A?!B 6=?!B?!A?!C￥¨
^?!Aa?!B¨
MF
?2
pDs
￥
p
(?!A?!B)?!C =?!A(?!B?!C )
!A (?!B +?!C ) =?!A?!B +?!A?!C
(?!B +?!C )?!A =?!B?!A +?!C?!A
f
￥

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?
êi¨
Vn?n?2￥
oμn > 2
êi?μil =¨f
￥
êiü
^
``3n¨f
?!A/S(j￠￠￠k)￥
êi?l1
Bi￠￠￠l =
X
j
X
k
Ai￠￠￠j￠￠￠k￠￠￠l ￠–jk = Ai￠￠￠j￠￠￠k￠￠￠l ￠–jk
?
?f
￥
=
êi
``3
?f
?!AD?!B5ieaò |B/SS
êi￥
?1
=¤
?m+n?2¨f
b
``3 ?ê |￥/S
^M
#￥ V[:S?!C =?!A ￠?!B?i ?V
U
Cim = AijBlm–jl = AilBlm
``3 !A
^=¨f
?!B
^B¨f
]" V[¨ ?eEb
?
×
=
?f
5é?ie ?a
Q
êi?!C =?!A,?!B?=¨f
 
×
êi
1S

C = AijBlm–jl–im
f
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?=
?i
O??êi
O￥s3?ê
O
ei￥i
Oeiej
^=¨f
?1?ê
i
Ob
``3e1e2￥ ?V
U"
E12[??1-? (1
, eiejü
^=¨f

￥??b?!
T = Tijeiej
?i
O￥
=
a￠(bc) = (a￠b)c = a￠bc
(ab)￠c = a(b￠c) = ab￠c
ab,cd = (a￠d)(b￠c) = a￠(b￠cd) = (ab￠c)￠d
10
x 2.7i
O￥±s
r￠(fg) = (r￠f)g+(f ￠r)g
r￠?!T = @@x(ex ￠?!T )+ @@y(ey ￠?!T )+ @@z(ez ￠?!T )
r￠(EE) = (r￠E)E +(E￠r)E
r￠(?!IE2) = @@x(ex ￠?!IE2)+ @@y(ey ￠?!IE2)+ @@z(ez ￠?!IE2)
= ex@E
2
@x +ey
@E2
@y +ez
@E2
@z = rE
2
r￠(?!IE2) = (r￠?!I)E2 +(rE2)￠?!I = rE2
I
dS ￠?!T =
ZZZ
dV r￠?!T
I
dS￠(fg) =
ZZZ
dV r￠(fg)
11