classical electrodynamics - fudan...
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Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
§ 1.2 ���©©©�!FÝ
E��Æ ÔnX ��� Mï� 1
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
§ 1.2 ���©©©�!FÝ
FÝ´�¥þ
∇T =∂T
∂xex +
∂T
∂yey +
∂T
∂zez
E��Æ ÔnX ��� Mï� 1
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
§ 1.2 ���©©©�!FÝ
FÝ´�¥þ
∇T =∂T
∂xex +
∂T
∂yey +
∂T
∂zez
��©
dT =∂T
∂xdx +
∂T
∂ydy +
∂T
∂zdz
E��Æ ÔnX ��� Mï� 1
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
§ 1.2 ���©©©�!FÝ
FÝ´�¥þ
∇T =∂T
∂xex +
∂T
∂yey +
∂T
∂zez
��©
dT =∂T
∂xdx +
∂T
∂ydy +
∂T
∂zdz d~l = exdx + eydy + ezdz
E��Æ ÔnX ��� Mï� 1
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
§ 1.2 ���©©©�!FÝ
FÝ´�¥þ
∇T =∂T
∂xex +
∂T
∂yey +
∂T
∂zez
��©
dT =∂T
∂xdx +
∂T
∂ydy +
∂T
∂zdz d~l = exdx + eydy + ezdz
= (∇T ) · (d~l) = |∇T | |d~l| cos θ
E��Æ ÔnX ��� Mï� 1
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
§ 1.2 ���©©©�!FÝ
FÝ´�¥þ
∇T =∂T
∂xex +
∂T
∂yey +
∂T
∂zez
��©
dT =∂T
∂xdx +
∂T
∂ydy +
∂T
∂zdz d~l = exdx + eydy + ezdz
= (∇T ) · (d~l) = |∇T | |d~l| cos θ
Aۿµ
E��Æ ÔnX ��� Mï� 1
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
§ 1.2 ���©©©�!FÝ
FÝ´�¥þ
∇T =∂T
∂xex +
∂T
∂yey +
∂T
∂zez
��©
dT =∂T
∂xdx +
∂T
∂ydy +
∂T
∂zdz d~l = exdx + eydy + ezdz
= (∇T ) · (d~l) = |∇T | |d~l| cos θ
Aۿµ
FÝ ∇T �����¼ê T ���CzÇ£���ê¤��§Ù��=�¼ê T ���CzÇ£���ê¤"
E��Æ ÔnX ��� Mï� 1
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
�!�Î ∇
E��Æ ÔnX ��� Mï� 2
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
�!�Î ∇FÝ�¤
E��Æ ÔnX ��� Mï� 2
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
�!�Î ∇FÝ�¤
∇T =∂T
∂xex +
∂T
∂yey +
∂T
∂zez =
(ex
∂
∂x+ey
∂
∂y+ez
∂
∂z
)T
E��Æ ÔnX ��� Mï� 2
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
�!�Î ∇FÝ�¤
∇T =∂T
∂xex +
∂T
∂yey +
∂T
∂zez =
(ex
∂
∂x+ey
∂
∂y+ez
∂
∂z
)T
del ¥þ�Î ∇ £Q´¥þq´�Τ
E��Æ ÔnX ��� Mï� 2
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
�!�Î ∇FÝ�¤
∇T =∂T
∂xex +
∂T
∂yey +
∂T
∂zez =
(ex
∂
∂x+ey
∂
∂y+ez
∂
∂z
)T
del ¥þ�Î ∇ £Q´¥þq´�Τ
∇ = ex∂
∂x+ey
∂
∂y+ez
∂
∂z
E��Æ ÔnX ��� Mï� 2
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
a'µ
~A a =⇒ ∇T FÝ (gradient)
E��Æ ÔnX ��� Mï� 3
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
a'µ
~A a =⇒ ∇T FÝ (gradient)
~A · ~B =⇒ ∇ · ~v ÑÝ (divergence)
E��Æ ÔnX ��� Mï� 3
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
a'µ
~A a =⇒ ∇T FÝ (gradient)
~A · ~B =⇒ ∇ · ~v ÑÝ (divergence)
~A × ~B =⇒ ∇× ~v ^Ý (curl)
E��Æ ÔnX ��� Mï� 3
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
a'µ
~A a =⇒ ∇T FÝ (gradient)
~A · ~B =⇒ ∇ · ~v ÑÝ (divergence)
~A × ~B =⇒ ∇× ~v ^Ý (curl)
n!ÑÝ
E��Æ ÔnX ��� Mï� 3
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
a'µ
~A a =⇒ ∇T FÝ (gradient)
~A · ~B =⇒ ∇ · ~v ÑÝ (divergence)
~A × ~B =⇒ ∇× ~v ^Ý (curl)
n!ÑÝ
∇ · ~v =(ex
∂
∂x+ey
∂
∂y+ez
∂
∂z
)· (vx ex + vy ey + vz ez)
=∂vx
∂x+
∂vy
∂y+
∂vz
∂z
E��Æ ÔnX ��� Mï� 3
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
o!^Ý
E��Æ ÔnX ��� Mï� 4
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
o!^Ý
∇× ~v =(ex
∂
∂x+ey
∂
∂y+ez
∂
∂z
)× (vx ex + vy ey + vz ez)
E��Æ ÔnX ��� Mï� 4
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
o!^Ý
∇× ~v =(ex
∂
∂x+ey
∂
∂y+ez
∂
∂z
)× (vx ex + vy ey + vz ez)
=
∣∣∣∣∣∣∣ex ey ez∂∂x
∂∂y
∂∂z
vx vy vz
∣∣∣∣∣∣∣
E��Æ ÔnX ��� Mï� 4
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
o!^Ý
∇× ~v =(ex
∂
∂x+ey
∂
∂y+ez
∂
∂z
)× (vx ex + vy ey + vz ez)
=
∣∣∣∣∣∣∣ex ey ez∂∂x
∂∂y
∂∂z
vx vy vz
∣∣∣∣∣∣∣= ex
(∂vz
∂y− ∂vy
∂z
)+ ey
(∂vx
∂z− ∂vz
∂x
)+ ez
(∂vy
∂x− ∂vx
∂y
)
E��Æ ÔnX ��� Mï� 4
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
o!^Ý
∇× ~v =(ex
∂
∂x+ey
∂
∂y+ez
∂
∂z
)× (vx ex + vy ey + vz ez)
=
∣∣∣∣∣∣∣ex ey ez∂∂x
∂∂y
∂∂z
vx vy vz
∣∣∣∣∣∣∣= ex
(∂vz
∂y− ∂vy
∂z
)+ ey
(∂vx
∂z− ∂vz
∂x
)+ ez
(∂vy
∂x− ∂vx
∂y
)Ê!~K
E��Æ ÔnX ��� Mï� 4
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
o!^Ý
∇× ~v =(ex
∂
∂x+ey
∂
∂y+ez
∂
∂z
)× (vx ex + vy ey + vz ez)
=
∣∣∣∣∣∣∣ex ey ez∂∂x
∂∂y
∂∂z
vx vy vz
∣∣∣∣∣∣∣= ex
(∂vz
∂y− ∂vy
∂z
)+ ey
(∂vx
∂z− ∂vz
∂x
)+ ez
(∂vy
∂x− ∂vx
∂y
)Ê!~K
∇r =∂√
x2 + y2 + z2
∂xex +
∂√
x2 + y2 + z2
∂yey +
∂√
x2 + y2 + z2
∂zez
E��Æ ÔnX ��� Mï� 4
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
o!^Ý
∇× ~v =(ex
∂
∂x+ey
∂
∂y+ez
∂
∂z
)× (vx ex + vy ey + vz ez)
=
∣∣∣∣∣∣∣ex ey ez∂∂x
∂∂y
∂∂z
vx vy vz
∣∣∣∣∣∣∣= ex
(∂vz
∂y− ∂vy
∂z
)+ ey
(∂vx
∂z− ∂vz
∂x
)+ ez
(∂vy
∂x− ∂vx
∂y
)Ê!~K
∇r =∂√
x2 + y2 + z2
∂xex +
∂√
x2 + y2 + z2
∂yey +
∂√
x2 + y2 + z2
∂zez
∇r =x
rex +
y
rey +
z
rez
E��Æ ÔnX ��� Mï� 4
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
o!^Ý
∇× ~v =(ex
∂
∂x+ey
∂
∂y+ez
∂
∂z
)× (vx ex + vy ey + vz ez)
=
∣∣∣∣∣∣∣ex ey ez∂∂x
∂∂y
∂∂z
vx vy vz
∣∣∣∣∣∣∣= ex
(∂vz
∂y− ∂vy
∂z
)+ ey
(∂vx
∂z− ∂vz
∂x
)+ ez
(∂vy
∂x− ∂vx
∂y
)Ê!~K
∇r =∂√
x2 + y2 + z2
∂xex +
∂√
x2 + y2 + z2
∂yey +
∂√
x2 + y2 + z2
∂zez
∇r =x
rex +
y
rey +
z
rez=
~r
r= er
E��Æ ÔnX ��� Mï� 4
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇r =~r
r= er,
E��Æ ÔnX ��� Mï� 5
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇r =~r
r= er, ∇f(u) =
df
du∇u
E��Æ ÔnX ��� Mï� 5
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇r =~r
r= er, ∇f(u) =
df
du∇u
∇ · ~A(u) = (∇u) · d ~A(u)du
,
E��Æ ÔnX ��� Mï� 5
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇r =~r
r= er, ∇f(u) =
df
du∇u
∇ · ~A(u) = (∇u) · d ~A(u)du
, ∇× ~A(u) = (∇u) × d ~A(u)du
E��Æ ÔnX ��� Mï� 5
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇r =~r
r= er, ∇f(u) =
df
du∇u
∇ · ~A(u) = (∇u) · d ~A(u)du
, ∇× ~A(u) = (∇u) × d ~A(u)du
∇r2 = 2r∇r = 2~r
E��Æ ÔnX ��� Mï� 5
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇r =~r
r= er, ∇f(u) =
df
du∇u
∇ · ~A(u) = (∇u) · d ~A(u)du
, ∇× ~A(u) = (∇u) × d ~A(u)du
∇r2 = 2r∇r = 2~r
∇ 1r
= − 1r2∇r = − 1
r2er = −
~r
r3
E��Æ ÔnX ��� Mï� 5
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇r =~r
r= er, ∇f(u) =
df
du∇u
∇ · ~A(u) = (∇u) · d ~A(u)du
, ∇× ~A(u) = (∇u) × d ~A(u)du
∇r2 = 2r∇r = 2~r
∇ 1r
= − 1r2∇r = − 1
r2er = −
~r
r3
∇ · ~r =(ex
∂
∂x+ey
∂
∂y+ez
∂
∂z
)· (x ex + y ey + z ez) = 3
E��Æ ÔnX ��� Mï� 5
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇r =~r
r= er, ∇f(u) =
df
du∇u
∇ · ~A(u) = (∇u) · d ~A(u)du
, ∇× ~A(u) = (∇u) × d ~A(u)du
∇r2 = 2r∇r = 2~r
∇ 1r
= − 1r2∇r = − 1
r2er = −
~r
r3
∇ · ~r =(ex
∂
∂x+ey
∂
∂y+ez
∂
∂z
)· (x ex + y ey + z ez) = 3
∇× ~r =
∣∣∣∣∣∣∣ex ey ez∂∂x
∂∂y
∂∂z
x y z
∣∣∣∣∣∣∣ = 0
E��Æ ÔnX ��� Mï� 5
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
8!¦È�FÝ!ÑÝ!^Ý
E��Æ ÔnX ��� Mï� 6
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
8!¦È�FÝ!ÑÝ!^Ý
∇ Q´¥þq´�5�Î
E��Æ ÔnX ��� Mï� 6
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
8!¦È�FÝ!ÑÝ!^Ý
∇ Q´¥þq´�5�Î
©�ǵ
E��Æ ÔnX ��� Mï� 6
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
8!¦È�FÝ!ÑÝ!^Ý
∇ Q´¥þq´�5�Î
©�ǵ
∇ (f + g) = ∇ f +∇ g
∇ · ( ~A + ~B) = ∇ · ~A +∇ · ~B
∇× ( ~A + ~B) = ∇× ~A +∇× ~B
E��Æ ÔnX ��� Mï� 6
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
8!¦È�FÝ!ÑÝ!^Ý
∇ Q´¥þq´�5�Î
©�ǵ
∇ (f + g) = ∇ f +∇ g
∇ · ( ~A + ~B) = ∇ · ~A +∇ · ~B
∇× ( ~A + ~B) = ∇× ~A +∇× ~B
XJ k ´~ê
E��Æ ÔnX ��� Mï� 6
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
8!¦È�FÝ!ÑÝ!^Ý
∇ Q´¥þq´�5�Î
©�ǵ
∇ (f + g) = ∇ f +∇ g
∇ · ( ~A + ~B) = ∇ · ~A +∇ · ~B
∇× ( ~A + ~B) = ∇× ~A +∇× ~B
XJ k ´~ê∇ (kf) = k∇ f
∇ · (k ~A) = k∇ · ~A
∇× (k ~A) = k∇× ~AE��Æ ÔnX ��� Mï� 6
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
¦È/ªµfg, ~A · ~B, f ~A, ~A × ~B
E��Æ ÔnX ��� Mï� 7
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
¦È/ªµfg, ~A · ~B, f ~A, ~A × ~B
∇(fg)
E��Æ ÔnX ��� Mï� 7
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
¦È/ªµfg, ~A · ~B, f ~A, ~A × ~B
∇(fg)k|^ ∇ ��Σ¦�¤A5§¦È�¦�©¤ü�¶
1. ∇ é f �^£¦�¤§g À�~þ§P� ∇f
2. ∇ é g �^£¦�¤§f À�~þ§P� ∇g
E��Æ ÔnX ��� Mï� 7
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
¦È/ªµfg, ~A · ~B, f ~A, ~A × ~B
∇(fg) = ∇f(fg) +∇g(fg)k|^ ∇ ��Σ¦�¤A5§¦È�¦�©¤ü�¶
1. ∇ é f �^£¦�¤§g À�~þ§P� ∇f
2. ∇ é g �^£¦�¤§f À�~þ§P� ∇g
E��Æ ÔnX ��� Mï� 7
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
¦È/ªµfg, ~A · ~B, f ~A, ~A × ~B
∇(fg) = ∇f(fg) +∇g(fg)k|^ ∇ ��Σ¦�¤A5§¦È�¦�©¤ü�¶
1. ∇ é f �^£¦�¤§g À�~þ§P� ∇f
2. ∇ é g �^£¦�¤§f À�~þ§P� ∇g
2|^ ∇ �¥þ$�A5¶ ~C(fg) = g ~Cf = f ~Cg
òÉ ∇ �^�¼ê£� ∇ �m>§ØÉ ∇ �^�¼ê£� ∇ ��>
E��Æ ÔnX ��� Mï� 7
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
¦È/ªµfg, ~A · ~B, f ~A, ~A × ~B
∇(fg) = ∇f(fg) +∇g(fg)k|^ ∇ ��Σ¦�¤A5§¦È�¦�©¤ü�¶
1. ∇ é f �^£¦�¤§g À�~þ§P� ∇f
2. ∇ é g �^£¦�¤§f À�~þ§P� ∇g
= g∇ff + f∇gg2|^ ∇ �¥þ$�A5¶ ~C(fg) = g ~Cf = f ~Cg
òÉ ∇ �^�¼ê£� ∇ �m>§ØÉ ∇ �^�¼ê£� ∇ ��>
E��Æ ÔnX ��� Mï� 7
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
¦È/ªµfg, ~A · ~B, f ~A, ~A × ~B
∇(fg) = ∇f(fg) +∇g(fg)k|^ ∇ ��Σ¦�¤A5§¦È�¦�©¤ü�¶
1. ∇ é f �^£¦�¤§g À�~þ§P� ∇f
2. ∇ é g �^£¦�¤§f À�~þ§P� ∇g
= g∇ff + f∇gg2|^ ∇ �¥þ$�A5¶ ~C(fg) = g ~Cf = f ~Cg
òÉ ∇ �^�¼ê£� ∇ �m>§ØÉ ∇ �^�¼ê£� ∇ ��>
∇(fg) = g∇f + f∇g
E��Æ ÔnX ��� Mï� 7
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
¦È/ªµfg, ~A · ~B, f ~A, ~A × ~B
∇(fg) = ∇f(fg) +∇g(fg)k|^ ∇ ��Σ¦�¤A5§¦È�¦�©¤ü�¶
1. ∇ é f �^£¦�¤§g À�~þ§P� ∇f
2. ∇ é g �^£¦�¤§f À�~þ§P� ∇g
= g∇ff + f∇gg2|^ ∇ �¥þ$�A5¶ ~C(fg) = g ~Cf = f ~Cg
òÉ ∇ �^�¼ê£� ∇ �m>§ØÉ ∇ �^�¼ê£� ∇ ��>
∇(fg) = g∇f + f∇g
∇ · (f ~A)
E��Æ ÔnX ��� Mï� 7
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
¦È/ªµfg, ~A · ~B, f ~A, ~A × ~B
∇(fg) = ∇f(fg) +∇g(fg)k|^ ∇ ��Σ¦�¤A5§¦È�¦�©¤ü�¶
1. ∇ é f �^£¦�¤§g À�~þ§P� ∇f
2. ∇ é g �^£¦�¤§f À�~þ§P� ∇g
= g∇ff + f∇gg2|^ ∇ �¥þ$�A5¶ ~C(fg) = g ~Cf = f ~Cg
òÉ ∇ �^�¼ê£� ∇ �m>§ØÉ ∇ �^�¼ê£� ∇ ��>
∇(fg) = g∇f + f∇g
∇ · (f ~A)k|^ ∇ �¦�A5§¦È�¦�©¤ü�¶
1. ∇ é f ¦�§ ~A À�~þ§P� ∇f
2. ∇ é ~A ¦�§f À�~þ§P� ∇A
E��Æ ÔnX ��� Mï� 7
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
¦È/ªµfg, ~A · ~B, f ~A, ~A × ~B
∇(fg) = ∇f(fg) +∇g(fg)k|^ ∇ ��Σ¦�¤A5§¦È�¦�©¤ü�¶
1. ∇ é f �^£¦�¤§g À�~þ§P� ∇f
2. ∇ é g �^£¦�¤§f À�~þ§P� ∇g
= g∇ff + f∇gg2|^ ∇ �¥þ$�A5¶ ~C(fg) = g ~Cf = f ~Cg
òÉ ∇ �^�¼ê£� ∇ �m>§ØÉ ∇ �^�¼ê£� ∇ ��>
∇(fg) = g∇f + f∇g
∇ · (f ~A) = ∇f · (f ~A) +∇A · (f ~A)k|^ ∇ �¦�A5§¦È�¦�©¤ü�¶
1. ∇ é f ¦�§ ~A À�~þ§P� ∇f
2. ∇ é ~A ¦�§f À�~þ§P� ∇A
E��Æ ÔnX ��� Mï� 7
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
¦È/ªµfg, ~A · ~B, f ~A, ~A × ~B
∇(fg) = ∇f(fg) +∇g(fg)k|^ ∇ ��Σ¦�¤A5§¦È�¦�©¤ü�¶
1. ∇ é f �^£¦�¤§g À�~þ§P� ∇f
2. ∇ é g �^£¦�¤§f À�~þ§P� ∇g
= g∇ff + f∇gg2|^ ∇ �¥þ$�A5¶ ~C(fg) = g ~Cf = f ~Cg
òÉ ∇ �^�¼ê£� ∇ �m>§ØÉ ∇ �^�¼ê£� ∇ ��>
∇(fg) = g∇f + f∇g
∇ · (f ~A) = ∇f · (f ~A) +∇A · (f ~A)k|^ ∇ �¦�A5§¦È�¦�©¤ü�¶
1. ∇ é f ¦�§ ~A À�~þ§P� ∇f
2. ∇ é ~A ¦�§f À�~þ§P� ∇A
2|^ ∇ �¥þ$�A5¶~C · (f ~A) = ~A · ( ~Cf) = f( ~C · ~A)
òÉ ∇ �^�¼ê£� ∇ �m>§ØÉ ∇ �^�¼ê£� ∇ ��>
E��Æ ÔnX ��� Mï� 7
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
¦È/ªµfg, ~A · ~B, f ~A, ~A × ~B
∇(fg) = ∇f(fg) +∇g(fg)k|^ ∇ ��Σ¦�¤A5§¦È�¦�©¤ü�¶
1. ∇ é f �^£¦�¤§g À�~þ§P� ∇f
2. ∇ é g �^£¦�¤§f À�~þ§P� ∇g
= g∇ff + f∇gg2|^ ∇ �¥þ$�A5¶ ~C(fg) = g ~Cf = f ~Cg
òÉ ∇ �^�¼ê£� ∇ �m>§ØÉ ∇ �^�¼ê£� ∇ ��>
∇(fg) = g∇f + f∇g
∇ · (f ~A) = ∇f · (f ~A) +∇A · (f ~A)k|^ ∇ �¦�A5§¦È�¦�©¤ü�¶
1. ∇ é f ¦�§ ~A À�~þ§P� ∇f
2. ∇ é ~A ¦�§f À�~þ§P� ∇A
= ~A · (∇ff) + f(∇A · ~A) 2|^ ∇ �¥þ$�A5¶~C · (f ~A) = ~A · ( ~Cf) = f( ~C · ~A)
òÉ ∇ �^�¼ê£� ∇ �m>§ØÉ ∇ �^�¼ê£� ∇ ��>
E��Æ ÔnX ��� Mï� 7
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
¦È/ªµfg, ~A · ~B, f ~A, ~A × ~B
∇(fg) = ∇f(fg) +∇g(fg)k|^ ∇ ��Σ¦�¤A5§¦È�¦�©¤ü�¶
1. ∇ é f �^£¦�¤§g À�~þ§P� ∇f
2. ∇ é g �^£¦�¤§f À�~þ§P� ∇g
= g∇ff + f∇gg2|^ ∇ �¥þ$�A5¶ ~C(fg) = g ~Cf = f ~Cg
òÉ ∇ �^�¼ê£� ∇ �m>§ØÉ ∇ �^�¼ê£� ∇ ��>
∇(fg) = g∇f + f∇g
∇ · (f ~A) = ∇f · (f ~A) +∇A · (f ~A)k|^ ∇ �¦�A5§¦È�¦�©¤ü�¶
1. ∇ é f ¦�§ ~A À�~þ§P� ∇f
2. ∇ é ~A ¦�§f À�~þ§P� ∇A
= ~A · (∇ff) + f(∇A · ~A) 2|^ ∇ �¥þ$�A5¶~C · (f ~A) = ~A · ( ~Cf) = f( ~C · ~A)
òÉ ∇ �^�¼ê£� ∇ �m>§ØÉ ∇ �^�¼ê£� ∇ ��>∇ · (f ~A) = ~A ·∇f + f∇ · ~A
E��Æ ÔnX ��� Mï� 7
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇× (f ~A)
E��Æ ÔnX ��� Mï� 8
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇× (f ~A)k|^ ∇ �¦�A5§¦�©¤ü�¶1. ∇ é f ¦�§ ~A À�~þ§= ∇f
2. ∇ é ~A ¦�§f À�~þ§= ∇A
E��Æ ÔnX ��� Mï� 8
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇× (f ~A) = ∇f × (f ~A) +∇A × (f ~A)k|^ ∇ �¦�A5§¦�©¤ü�¶1. ∇ é f ¦�§ ~A À�~þ§= ∇f
2. ∇ é ~A ¦�§f À�~þ§= ∇A
E��Æ ÔnX ��� Mï� 8
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇× (f ~A) = ∇f × (f ~A) +∇A × (f ~A)k|^ ∇ �¦�A5§¦�©¤ü�¶1. ∇ é f ¦�§ ~A À�~þ§= ∇f
2. ∇ é ~A ¦�§f À�~þ§= ∇A
2|^ ∇ �¥þ$�A5¶~C × (f ~A) = − ~A × ( ~Cf)
= f( ~C × ~A)òÉ ∇ �^�¼ê£� ∇ �m>§ØÉ ∇ �^�¼ê£� ∇ ��>
E��Æ ÔnX ��� Mï� 8
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇× (f ~A) = ∇f × (f ~A) +∇A × (f ~A)k|^ ∇ �¦�A5§¦�©¤ü�¶1. ∇ é f ¦�§ ~A À�~þ§= ∇f
2. ∇ é ~A ¦�§f À�~þ§= ∇A
= − ~A × (∇ff) + f(∇A × ~A)
2|^ ∇ �¥þ$�A5¶~C × (f ~A) = − ~A × ( ~Cf)
= f( ~C × ~A)òÉ ∇ �^�¼ê£� ∇ �m>§ØÉ ∇ �^�¼ê£� ∇ ��>
E��Æ ÔnX ��� Mï� 8
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇× (f ~A) = ∇f × (f ~A) +∇A × (f ~A)k|^ ∇ �¦�A5§¦�©¤ü�¶1. ∇ é f ¦�§ ~A À�~þ§= ∇f
2. ∇ é ~A ¦�§f À�~þ§= ∇A
= − ~A × (∇ff) + f(∇A × ~A)
2|^ ∇ �¥þ$�A5¶~C × (f ~A) = − ~A × ( ~Cf)
= f( ~C × ~A)òÉ ∇ �^�¼ê£� ∇ �m>§ØÉ ∇ �^�¼ê£� ∇ ��>
∇× (f ~A) = − ~A ×∇f + f∇× ~A
E��Æ ÔnX ��� Mï� 8
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇ · ( ~A × ~B)
E��Æ ÔnX ��� Mï� 9
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇ · ( ~A × ~B)
k|^ ∇ �¦�A5§¦�©¤ü�¶1. ∇ é ~A ¦�§ ~B À�~þ§P� ∇A
2. ∇ é ~B ¦�§ ~A À�~þ§P� ∇B
E��Æ ÔnX ��� Mï� 9
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇ · ( ~A × ~B) = ∇A · ( ~A × ~B) +∇B · ( ~A × ~B)
k|^ ∇ �¦�A5§¦�©¤ü�¶1. ∇ é ~A ¦�§ ~B À�~þ§P� ∇A
2. ∇ é ~B ¦�§ ~A À�~þ§P� ∇B
E��Æ ÔnX ��� Mï� 9
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇ · ( ~A × ~B) = ∇A · ( ~A × ~B) +∇B · ( ~A × ~B)
k|^ ∇ �¦�A5§¦�©¤ü�¶1. ∇ é ~A ¦�§ ~B À�~þ§P� ∇A
2. ∇ é ~B ¦�§ ~A À�~þ§P� ∇B
2|^ ∇ �¥þ$�A5¶~C · ( ~A × ~B) = ~B · ( ~C × ~A) = − ~A · ( ~C × ~B)
òÉ ∇ �^�¼ê£� ∇ �m>§ØÉ ∇ �^�¼ê£� ∇ ��>
E��Æ ÔnX ��� Mï� 9
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇ · ( ~A × ~B) = ∇A · ( ~A × ~B) +∇B · ( ~A × ~B)
k|^ ∇ �¦�A5§¦�©¤ü�¶1. ∇ é ~A ¦�§ ~B À�~þ§P� ∇A
2. ∇ é ~B ¦�§ ~A À�~þ§P� ∇B
= ~B · (∇A × ~A)− ~A · (∇B × ~B)
2|^ ∇ �¥þ$�A5¶~C · ( ~A × ~B) = ~B · ( ~C × ~A) = − ~A · ( ~C × ~B)
òÉ ∇ �^�¼ê£� ∇ �m>§ØÉ ∇ �^�¼ê£� ∇ ��>
E��Æ ÔnX ��� Mï� 9
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇ · ( ~A × ~B) = ∇A · ( ~A × ~B) +∇B · ( ~A × ~B)
k|^ ∇ �¦�A5§¦�©¤ü�¶1. ∇ é ~A ¦�§ ~B À�~þ§P� ∇A
2. ∇ é ~B ¦�§ ~A À�~þ§P� ∇B
= ~B · (∇A × ~A)− ~A · (∇B × ~B)
2|^ ∇ �¥þ$�A5¶~C · ( ~A × ~B) = ~B · ( ~C × ~A) = − ~A · ( ~C × ~B)
òÉ ∇ �^�¼ê£� ∇ �m>§ØÉ ∇ �^�¼ê£� ∇ ��>
∇ · ( ~A × ~B) = ~B · (∇× ~A)− ~A · (∇× ~B)
E��Æ ÔnX ��� Mï� 9
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇× ( ~A × ~B)
E��Æ ÔnX ��� Mï� 10
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇× ( ~A × ~B)
k|^ ∇ �¦�A5§¦�©¤ü�¶1. ∇ é ~A ¦�§ ~B À�~þ§P� ∇A
2. ∇ é ~B ¦�§ ~A À�~þ§P� ∇B
E��Æ ÔnX ��� Mï� 10
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇× ( ~A × ~B) = ∇A × ( ~A × ~B) +∇B × ( ~A × ~B)
k|^ ∇ �¦�A5§¦�©¤ü�¶1. ∇ é ~A ¦�§ ~B À�~þ§P� ∇A
2. ∇ é ~B ¦�§ ~A À�~þ§P� ∇B
E��Æ ÔnX ��� Mï� 10
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇× ( ~A × ~B) = ∇A × ( ~A × ~B) +∇B × ( ~A × ~B)
k|^ ∇ �¦�A5§¦�©¤ü�¶1. ∇ é ~A ¦�§ ~B À�~þ§P� ∇A
2. ∇ é ~B ¦�§ ~A À�~þ§P� ∇B
2|^ ∇ �¥þ$�A5¶
~C × ( ~A × ~B) = ( ~B · ~C) ~A− ~B( ~C · ~A)
= ~A( ~C · ~B)− ( ~A · ~C) ~B
òÉ ∇ �^�¼ê£� ∇ �m>§ØÉ ∇ �^�¼ê£� ∇ ��>
E��Æ ÔnX ��� Mï� 10
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇× ( ~A × ~B) = ∇A × ( ~A × ~B) +∇B × ( ~A × ~B)
k|^ ∇ �¦�A5§¦�©¤ü�¶1. ∇ é ~A ¦�§ ~B À�~þ§P� ∇A
2. ∇ é ~B ¦�§ ~A À�~þ§P� ∇B
= ( ~B ·∇A) ~A− ~B(∇A · ~A) + ~A(∇B · ~B)− ( ~A ·∇B) ~B
2|^ ∇ �¥þ$�A5¶
~C × ( ~A × ~B) = ( ~B · ~C) ~A− ~B( ~C · ~A)
= ~A( ~C · ~B)− ( ~A · ~C) ~B
òÉ ∇ �^�¼ê£� ∇ �m>§ØÉ ∇ �^�¼ê£� ∇ ��>
E��Æ ÔnX ��� Mï� 10
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇× ( ~A × ~B) = ∇A × ( ~A × ~B) +∇B × ( ~A × ~B)
k|^ ∇ �¦�A5§¦�©¤ü�¶1. ∇ é ~A ¦�§ ~B À�~þ§P� ∇A
2. ∇ é ~B ¦�§ ~A À�~þ§P� ∇B
= ( ~B ·∇A) ~A− ~B(∇A · ~A) + ~A(∇B · ~B)− ( ~A ·∇B) ~B
2|^ ∇ �¥þ$�A5¶
~C × ( ~A × ~B) = ( ~B · ~C) ~A− ~B( ~C · ~A)
= ~A( ~C · ~B)− ( ~A · ~C) ~B
òÉ ∇ �^�¼ê£� ∇ �m>§ØÉ ∇ �^�¼ê£� ∇ ��>
∇× ( ~A × ~B) = ( ~B ·∇) ~A− ~B(∇ · ~A)− ( ~A ·∇) ~B + ~A(∇ · ~B)
E��Æ ÔnX ��� Mï� 10
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
Ù¥
( ~B ·∇) =[(Bx ex + By ey + Bz ez) ·
(ex
∂
∂x+ey
∂
∂y+ez
∂
∂z
)]
E��Æ ÔnX ��� Mï� 11
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
Ù¥
( ~B ·∇) =[(Bx ex + By ey + Bz ez) ·
(ex
∂
∂x+ey
∂
∂y+ez
∂
∂z
)]= Bx
∂
∂x+ By
∂
∂y+ Bz
∂
∂z
E��Æ ÔnX ��� Mï� 11
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
Ù¥
( ~B ·∇) =[(Bx ex + By ey + Bz ez) ·
(ex
∂
∂x+ey
∂
∂y+ez
∂
∂z
)]= Bx
∂
∂x+ By
∂
∂y+ Bz
∂
∂z=⇒ Iþ�Î
E��Æ ÔnX ��� Mï� 11
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
Ù¥
( ~B ·∇) =[(Bx ex + By ey + Bz ez) ·
(ex
∂
∂x+ey
∂
∂y+ez
∂
∂z
)]= Bx
∂
∂x+ By
∂
∂y+ Bz
∂
∂z=⇒ Iþ�Î
l
( ~B ·∇) ~A =(
Bx∂
∂x+ By
∂
∂y+ Bz
∂
∂z
)(Ax ex + Ay ey + Az ez)
E��Æ ÔnX ��� Mï� 11
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
Ù¥
( ~B ·∇) =[(Bx ex + By ey + Bz ez) ·
(ex
∂
∂x+ey
∂
∂y+ez
∂
∂z
)]= Bx
∂
∂x+ By
∂
∂y+ Bz
∂
∂z=⇒ Iþ�Î
l
( ~B ·∇) ~A =(
Bx∂
∂x+ By
∂
∂y+ Bz
∂
∂z
)(Ax ex + Ay ey + Az ez)
= Bx∂Ax
∂xex + By
∂Ax
∂yex + Bz
∂Ax
∂zex
+ Bx∂Ay
∂xey + By
∂Ay
∂yey + Bz
∂Ay
∂zey
+ Bx∂Az
∂xez + By
∂Az
∂yez + Bz
∂Az
∂zez
E��Æ ÔnX ��� Mï� 11
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇( ~A · ~B)
E��Æ ÔnX ��� Mï� 12
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇( ~A · ~B)
k|^ ∇ �¦�A5§¦�©¤ü�¶1. ∇ é ~A ¦�§ ~B À�~þ§P� ∇A
2. ∇ é ~B ¦�§ ~A À�~þ§P� ∇B
E��Æ ÔnX ��� Mï� 12
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇( ~A · ~B) = ∇A( ~A · ~B) +∇B( ~A · ~B)
k|^ ∇ �¦�A5§¦�©¤ü�¶1. ∇ é ~A ¦�§ ~B À�~þ§P� ∇A
2. ∇ é ~B ¦�§ ~A À�~þ§P� ∇B
E��Æ ÔnX ��� Mï� 12
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇( ~A · ~B) = ∇A( ~A · ~B) +∇B( ~A · ~B)
k|^ ∇ �¦�A5§¦�©¤ü�¶1. ∇ é ~A ¦�§ ~B À�~þ§P� ∇A
2. ∇ é ~B ¦�§ ~A À�~þ§P� ∇B
2|^ ∇ �¥þ$�A5¶
~B × ( ~C × ~A) = ~C( ~A · ~B)− ( ~B · ~C) ~A
⇓~C( ~A · ~B) = ~B × ( ~C × ~A) + ( ~B · ~C) ~A
~C( ~A · ~B) = ~A × ( ~C × ~B) + ( ~A · ~C) ~B
òÉ ∇ �^�¼ê£� ∇ �m>§ØÉ ∇ �^�¼ê£� ∇ ��>
E��Æ ÔnX ��� Mï� 12
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇( ~A · ~B) = ∇A( ~A · ~B) +∇B( ~A · ~B)
k|^ ∇ �¦�A5§¦�©¤ü�¶1. ∇ é ~A ¦�§ ~B À�~þ§P� ∇A
2. ∇ é ~B ¦�§ ~A À�~þ§P� ∇B
= ~B × (∇A × ~A) + ( ~B ·∇A) ~A + ~A × (∇B × ~B) + ( ~A ·∇B) ~B
2|^ ∇ �¥þ$�A5¶
~B × ( ~C × ~A) = ~C( ~A · ~B)− ( ~B · ~C) ~A
⇓~C( ~A · ~B) = ~B × ( ~C × ~A) + ( ~B · ~C) ~A
~C( ~A · ~B) = ~A × ( ~C × ~B) + ( ~A · ~C) ~B
òÉ ∇ �^�¼ê£� ∇ �m>§ØÉ ∇ �^�¼ê£� ∇ ��>
E��Æ ÔnX ��� Mï� 12
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇( ~A · ~B) = ∇A( ~A · ~B) +∇B( ~A · ~B)
k|^ ∇ �¦�A5§¦�©¤ü�¶1. ∇ é ~A ¦�§ ~B À�~þ§P� ∇A
2. ∇ é ~B ¦�§ ~A À�~þ§P� ∇B
= ~B × (∇A × ~A) + ( ~B ·∇A) ~A + ~A × (∇B × ~B) + ( ~A ·∇B) ~B
2|^ ∇ �¥þ$�A5¶
~B × ( ~C × ~A) = ~C( ~A · ~B)− ( ~B · ~C) ~A
⇓~C( ~A · ~B) = ~B × ( ~C × ~A) + ( ~B · ~C) ~A
~C( ~A · ~B) = ~A × ( ~C × ~B) + ( ~A · ~C) ~B
òÉ ∇ �^�¼ê£� ∇ �m>§ØÉ ∇ �^�¼ê£� ∇ ��>
∇( ~A · ~B) = ~B × (∇× ~A) + ( ~B ·∇) ~A + ~A × (∇× ~B) + ( ~A ·∇) ~B
E��Æ ÔnX ��� Mï� 12
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
Ô!��Î�^
E��Æ ÔnX ��� Mï� 13
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
Ô!��Î�^�U��¹µ
E��Æ ÔnX ��� Mï� 13
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
Ô!��Î�^�U��¹µ
FÝ�ÑÝ ∇ · (∇f)
E��Æ ÔnX ��� Mï� 13
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
Ô!��Î�^�U��¹µ
FÝ�ÑÝ ∇ · (∇f)
FÝ�^Ý ∇× (∇f)
E��Æ ÔnX ��� Mï� 13
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
Ô!��Î�^�U��¹µ
FÝ�ÑÝ ∇ · (∇f)
FÝ�^Ý ∇× (∇f)
}å©uIþ¼ê
E��Æ ÔnX ��� Mï� 13
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
Ô!��Î�^�U��¹µ
FÝ�ÑÝ ∇ · (∇f)
FÝ�^Ý ∇× (∇f)
}å©uIþ¼ê
ÑÝ�FÝ ∇(∇ · ~A)
E��Æ ÔnX ��� Mï� 13
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
Ô!��Î�^�U��¹µ
FÝ�ÑÝ ∇ · (∇f)
FÝ�^Ý ∇× (∇f)
}å©uIþ¼ê
ÑÝ�FÝ ∇(∇ · ~A)
^Ý�ÑÝ ∇ · (∇× ~A)
E��Æ ÔnX ��� Mï� 13
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
Ô!��Î�^�U��¹µ
FÝ�ÑÝ ∇ · (∇f)
FÝ�^Ý ∇× (∇f)
}å©uIþ¼ê
ÑÝ�FÝ ∇(∇ · ~A)
^Ý�ÑÝ ∇ · (∇× ~A)
^Ý�^Ý ∇× (∇× ~A)
E��Æ ÔnX ��� Mï� 13
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
Ô!��Î�^�U��¹µ
FÝ�ÑÝ ∇ · (∇f)
FÝ�^Ý ∇× (∇f)
}å©uIþ¼ê
ÑÝ�FÝ ∇(∇ · ~A)
^Ý�ÑÝ ∇ · (∇× ~A)
^Ý�^Ý ∇× (∇× ~A)
å©u¥þ¼ê
E��Æ ÔnX ��� Mï� 13
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
Ô!��Î�^�U��¹µ
FÝ�ÑÝ ∇ · (∇f)
FÝ�^Ý ∇× (∇f)
}å©uIþ¼ê
ÑÝ�FÝ ∇(∇ · ~A)
^Ý�ÑÝ ∇ · (∇× ~A)
^Ý�^Ý ∇× (∇× ~A)
å©u¥þ¼ê
∇ · (∇f) =(ex
∂
∂x+ey
∂
∂y+ez
∂
∂z
)·(ex
∂f
∂x+ey
∂f
∂y+ez
∂f
∂z
)=
∂2f
∂x2+
∂2f
∂y2+
∂2f
∂z2≡ ∇2f Iþ¼ê f � Laplacian, �´�Iþ
5¿ ex, ey, ez þ�~¥þE��Æ ÔnX ��� Mï� 13
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
Iþ�Î Laplacianµ∇2
∇2 = ∇ ·∇ =∂
∂x2+
∂
∂y2+
∂
∂z2
E��Æ ÔnX ��� Mï� 14
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
Iþ�Î Laplacianµ∇2
∇2 = ∇ ·∇ =∂
∂x2+
∂
∂y2+
∂
∂z2
∇2f =∂2f
∂x2+
∂2f
∂y2+
∂2f
∂z2
E��Æ ÔnX ��� Mï� 14
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
Iþ�Î Laplacianµ∇2
∇2 = ∇ ·∇ =∂
∂x2+
∂
∂y2+
∂
∂z2
∇2f =∂2f
∂x2+
∂2f
∂y2+
∂2f
∂z2Iþ�LaplacianE�Iþ
E��Æ ÔnX ��� Mï� 14
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
Iþ�Î Laplacianµ∇2
∇2 = ∇ ·∇ =∂
∂x2+
∂
∂y2+
∂
∂z2
∇2f =∂2f
∂x2+
∂2f
∂y2+
∂2f
∂z2Iþ�LaplacianE�Iþ
∇2 ~A =(
∂2
∂x2+
∂2
∂y2+
∂2
∂z2
)(Ax ex + Ay ey + Az ez)
E��Æ ÔnX ��� Mï� 14
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
Iþ�Î Laplacianµ∇2
∇2 = ∇ ·∇ =∂
∂x2+
∂
∂y2+
∂
∂z2
∇2f =∂2f
∂x2+
∂2f
∂y2+
∂2f
∂z2Iþ�LaplacianE�Iþ
∇2 ~A =(
∂2
∂x2+
∂2
∂y2+
∂2
∂z2
)(Ax ex + Ay ey + Az ez)
=(
∂2Ax
∂x2+
∂2Ax
∂y2+
∂2Ax
∂z2
)ex +
(∂2Ay
∂x2+
∂2Ay
∂y2+
∂2Ay
∂z2
)ey
+(
∂2Az
∂x2+
∂2Az
∂y2+
∂2Az
∂z2
)ez
E��Æ ÔnX ��� Mï� 14
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
Iþ�Î Laplacianµ∇2
∇2 = ∇ ·∇ =∂
∂x2+
∂
∂y2+
∂
∂z2
∇2f =∂2f
∂x2+
∂2f
∂y2+
∂2f
∂z2Iþ�LaplacianE�Iþ
∇2 ~A =(
∂2
∂x2+
∂2
∂y2+
∂2
∂z2
)(Ax ex + Ay ey + Az ez)
=(
∂2Ax
∂x2+
∂2Ax
∂y2+
∂2Ax
∂z2
)ex +
(∂2Ay
∂x2+
∂2Ay
∂y2+
∂2Ay
∂z2
)ey
+(
∂2Az
∂x2+
∂2Az
∂y2+
∂2Az
∂z2
)ez ¥þ�LaplacianE�¥þ
E��Æ ÔnX ��� Mï� 14
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
Iþ�Î Laplacianµ∇2
∇2 = ∇ ·∇ =∂
∂x2+
∂
∂y2+
∂
∂z2
∇2f =∂2f
∂x2+
∂2f
∂y2+
∂2f
∂z2Iþ�LaplacianE�Iþ
∇2 ~A =(
∂2
∂x2+
∂2
∂y2+
∂2
∂z2
)(Ax ex + Ay ey + Az ez)
=(
∂2Ax
∂x2+
∂2Ax
∂y2+
∂2Ax
∂z2
)ex +
(∂2Ay
∂x2+
∂2Ay
∂y2+
∂2Ay
∂z2
)ey
+(
∂2Az
∂x2+
∂2Az
∂y2+
∂2Az
∂z2
)ez ¥þ�LaplacianE�¥þ
∇2 ~A = (∇2Ax) ex + (∇2Ay) ey + (∇2Az) ez
E��Æ ÔnX ��� Mï� 14
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
Iþ�Î Laplacianµ∇2
∇2 = ∇ ·∇ =∂
∂x2+
∂
∂y2+
∂
∂z2
∇2f =∂2f
∂x2+
∂2f
∂y2+
∂2f
∂z2Iþ�LaplacianE�Iþ
∇2 ~A =(
∂2
∂x2+
∂2
∂y2+
∂2
∂z2
)(Ax ex + Ay ey + Az ez)
=(
∂2Ax
∂x2+
∂2Ax
∂y2+
∂2Ax
∂z2
)ex +
(∂2Ay
∂x2+
∂2Ay
∂y2+
∂2Ay
∂z2
)ey
+(
∂2Az
∂x2+
∂2Az
∂y2+
∂2Az
∂z2
)ez ¥þ�LaplacianE�¥þ
∇2 ~A = (∇2Ax) ex + (∇2Ay) ey + (∇2Az) ez ex, ey, ez �~¥þ"
E��Æ ÔnX ��� Mï� 14
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
FÝ�^Ýð�"µ
∇× (∇f) = (∇×∇)f = 0
E��Æ ÔnX ��� Mï� 15
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
FÝ�^Ýð�"µ
∇× (∇f) = (∇×∇)f = 0 Ál©þ/ª\±y²
E��Æ ÔnX ��� Mï� 15
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
FÝ�^Ýð�"µ
∇× (∇f) = (∇×∇)f = 0 Ál©þ/ª\±y²
5¿µ
( ~Cg) × ( ~Cf) = ( ~C × ~C)gf = 0,
E��Æ ÔnX ��� Mï� 15
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
FÝ�^Ýð�"µ
∇× (∇f) = (∇×∇)f = 0 Ál©þ/ª\±y²
5¿µ
( ~Cg) × ( ~Cf) = ( ~C × ~C)gf = 0, ~C �¥þ
E��Æ ÔnX ��� Mï� 15
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
FÝ�^Ýð�"µ
∇× (∇f) = (∇×∇)f = 0 Ál©þ/ª\±y²
5¿µ
( ~Cg) × ( ~Cf) = ( ~C × ~C)gf = 0, ~C �¥þ
(∇g) × (∇f) = (∇g g) × (∇f f) = (∇g ×∇f) gf 6= 0
E��Æ ÔnX ��� Mï� 15
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
FÝ�^Ýð�"µ
∇× (∇f) = (∇×∇)f = 0 Ál©þ/ª\±y²
5¿µ
( ~Cg) × ( ~Cf) = ( ~C × ~C)gf = 0, ~C �¥þ
(∇g) × (∇f) = (∇g g) × (∇f f) = (∇g ×∇f) gf 6= 0
ÑÝ�FÝØ~^µ∇(∇ · ~A) 6= ∇2 ~A
E��Æ ÔnX ��� Mï� 15
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
FÝ�^Ýð�"µ
∇× (∇f) = (∇×∇)f = 0 Ál©þ/ª\±y²
5¿µ
( ~Cg) × ( ~Cf) = ( ~C × ~C)gf = 0, ~C �¥þ
(∇g) × (∇f) = (∇g g) × (∇f f) = (∇g ×∇f) gf 6= 0
ÑÝ�FÝØ~^µ∇(∇ · ~A) 6= ∇2 ~A
^Ý�ÑÝð�"µ
∇ · (∇× ~A) = (∇×∇) · ~A = 0
E��Æ ÔnX ��� Mï� 15
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
FÝ�^Ýð�"µ
∇× (∇f) = (∇×∇)f = 0 Ál©þ/ª\±y²
5¿µ
( ~Cg) × ( ~Cf) = ( ~C × ~C)gf = 0, ~C �¥þ
(∇g) × (∇f) = (∇g g) × (∇f f) = (∇g ×∇f) gf 6= 0
ÑÝ�FÝØ~^µ∇(∇ · ~A) 6= ∇2 ~A
^Ý�ÑÝð�"µ
∇ · (∇× ~A) = (∇×∇) · ~A = 0
|^µ ~B · ( ~C × ~A) = ~A · ( ~B × ~C) = ( ~B × ~C) · ~A
E��Æ ÔnX ��� Mï� 15
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
^Ý�^Ý~^u½Â¥þ� Laplacianµ
∇× (∇× ~A) = ∇(∇ · ~A)− (∇ ·∇) ~A = ∇(∇ · ~A)−∇2 ~A
E��Æ ÔnX ��� Mï� 16
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
^Ý�^Ý~^u½Â¥þ� Laplacianµ
∇× (∇× ~A) = ∇(∇ · ~A)− (∇ ·∇) ~A = ∇(∇ · ~A)−∇2 ~A
|^µ ~B × ( ~C × ~A) = ~C( ~B · ~A)− ( ~B · ~C) ~A
E��Æ ÔnX ��� Mï� 16
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
^Ý�^Ý~^u½Â¥þ� Laplacianµ
∇× (∇× ~A) = ∇(∇ · ~A)− (∇ ·∇) ~A = ∇(∇ · ~A)−∇2 ~A
|^µ ~B × ( ~C × ~A) = ~C( ~B · ~A)− ( ~B · ~C) ~A
��¥þ� Laplacian¶
∇2 ~A = ∇(∇ · ~A)−∇× (∇× ~A)
E��Æ ÔnX ��� Mï� 16
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
^Ý�^Ý~^u½Â¥þ� Laplacianµ
∇× (∇× ~A) = ∇(∇ · ~A)− (∇ ·∇) ~A = ∇(∇ · ~A)−∇2 ~A
|^µ ~B × ( ~C × ~A) = ~C( ~B · ~A)− ( ~B · ~C) ~A
��¥þ� Laplacian¶
∇2 ~A = ∇(∇ · ~A)−∇× (∇× ~A)
∇2 ~A = (∇2Ax) ex + (∇2Ay) ey + (∇2Az) ez =é���I¤á"
E��Æ ÔnX ��� Mï� 16
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
^Ý�^Ý~^u½Â¥þ� Laplacianµ
∇× (∇× ~A) = ∇(∇ · ~A)− (∇ ·∇) ~A = ∇(∇ · ~A)−∇2 ~A
|^µ ~B × ( ~C × ~A) = ~C( ~B · ~A)− ( ~B · ~C) ~A
��¥þ� Laplacian¶
∇2 ~A = ∇(∇ · ~A)−∇× (∇× ~A)
∇2 ~A = (∇2Ax) ex + (∇2Ay) ey + (∇2Az) ez =é���I¤á"
XJ
~R = ~r − ~r ′
~r = x ex + y ey + z ez,
~r ′ = x′ ex + y′ ey + z′ ez,
∇ = ei∂
∂xi, ∇′ = ei
∂
∂x′i
=⇒ ∇′ [g( ~R)] = −∇ [g( ~R)]
E��Æ ÔnX ��� Mï� 16
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
l!~K
E��Æ ÔnX ��� Mï� 17
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
l!~K
±e ~A �?¿¥þ§~a �~¥þ"
( ~A ·∇)~r = (Ax∂
∂x+ Ay
∂
∂y+ Az
∂
∂z)(x ex + y ey + z ez) = ~A
E��Æ ÔnX ��� Mï� 17
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
l!~K
±e ~A �?¿¥þ§~a �~¥þ"
( ~A ·∇)~r = (Ax∂
∂x+ Ay
∂
∂y+ Az
∂
∂z)(x ex + y ey + z ez) = ~A
( ~A ·∇)r = (Ax∂
∂x+ Ay
∂
∂y+ Az
∂
∂z)√
x2 + y2 + z2
= (Axx + Ayy + Azz)/r = ~A ·~r
r
E��Æ ÔnX ��� Mï� 17
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
l!~K
±e ~A �?¿¥þ§~a �~¥þ"
( ~A ·∇)~r = (Ax∂
∂x+ Ay
∂
∂y+ Az
∂
∂z)(x ex + y ey + z ez) = ~A
( ~A ·∇)r = (Ax∂
∂x+ Ay
∂
∂y+ Az
∂
∂z)√
x2 + y2 + z2
= (Axx + Ayy + Azz)/r = ~A ·~r
r= ~A · (∇r)
E��Æ ÔnX ��� Mï� 17
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
l!~K
±e ~A �?¿¥þ§~a �~¥þ"
( ~A ·∇)~r = (Ax∂
∂x+ Ay
∂
∂y+ Az
∂
∂z)(x ex + y ey + z ez) = ~A
( ~A ·∇)r = (Ax∂
∂x+ Ay
∂
∂y+ Az
∂
∂z)√
x2 + y2 + z2
= (Axx + Ayy + Azz)/r = ~A ·~r
r= ~A · (∇r)
∇(~a · ~r)
E��Æ ÔnX ��� Mï� 17
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
l!~K
±e ~A �?¿¥þ§~a �~¥þ"
( ~A ·∇)~r = (Ax∂
∂x+ Ay
∂
∂y+ Az
∂
∂z)(x ex + y ey + z ez) = ~A
( ~A ·∇)r = (Ax∂
∂x+ Ay
∂
∂y+ Az
∂
∂z)√
x2 + y2 + z2
= (Axx + Ayy + Azz)/r = ~A ·~r
r= ~A · (∇r)
∇(~a · ~r) = ∇(axx + ayy + azz)
E��Æ ÔnX ��� Mï� 17
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
l!~K
±e ~A �?¿¥þ§~a �~¥þ"
( ~A ·∇)~r = (Ax∂
∂x+ Ay
∂
∂y+ Az
∂
∂z)(x ex + y ey + z ez) = ~A
( ~A ·∇)r = (Ax∂
∂x+ Ay
∂
∂y+ Az
∂
∂z)√
x2 + y2 + z2
= (Axx + Ayy + Azz)/r = ~A ·~r
r= ~A · (∇r)
∇(~a · ~r) = ∇(axx + ayy + azz) = ax ex + ay ey + az ez
E��Æ ÔnX ��� Mï� 17
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
l!~K
±e ~A �?¿¥þ§~a �~¥þ"
( ~A ·∇)~r = (Ax∂
∂x+ Ay
∂
∂y+ Az
∂
∂z)(x ex + y ey + z ez) = ~A
( ~A ·∇)r = (Ax∂
∂x+ Ay
∂
∂y+ Az
∂
∂z)√
x2 + y2 + z2
= (Axx + Ayy + Azz)/r = ~A ·~r
r= ~A · (∇r)
∇(~a · ~r) = ∇(axx + ayy + azz) = ax ex + ay ey + az ez = ~a
E��Æ ÔnX ��� Mï� 17
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
l!~K
±e ~A �?¿¥þ§~a �~¥þ"
( ~A ·∇)~r = (Ax∂
∂x+ Ay
∂
∂y+ Az
∂
∂z)(x ex + y ey + z ez) = ~A
( ~A ·∇)r = (Ax∂
∂x+ Ay
∂
∂y+ Az
∂
∂z)√
x2 + y2 + z2
= (Axx + Ayy + Azz)/r = ~A ·~r
r= ~A · (∇r)
∇(~a · ~r) = ∇(axx + ayy + azz) = ax ex + ay ey + az ez = ~a
∇ · (~a × ~r)
E��Æ ÔnX ��� Mï� 17
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
l!~K
±e ~A �?¿¥þ§~a �~¥þ"
( ~A ·∇)~r = (Ax∂
∂x+ Ay
∂
∂y+ Az
∂
∂z)(x ex + y ey + z ez) = ~A
( ~A ·∇)r = (Ax∂
∂x+ Ay
∂
∂y+ Az
∂
∂z)√
x2 + y2 + z2
= (Axx + Ayy + Azz)/r = ~A ·~r
r= ~A · (∇r)
∇(~a · ~r) = ∇(axx + ayy + azz) = ax ex + ay ey + az ez = ~a
∇ · (~a × ~r) = ∇r · (~a × ~r)|^ ~b · (~a × ~c) = −~a · (~b × ~c)
E��Æ ÔnX ��� Mï� 17
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
l!~K
±e ~A �?¿¥þ§~a �~¥þ"
( ~A ·∇)~r = (Ax∂
∂x+ Ay
∂
∂y+ Az
∂
∂z)(x ex + y ey + z ez) = ~A
( ~A ·∇)r = (Ax∂
∂x+ Ay
∂
∂y+ Az
∂
∂z)√
x2 + y2 + z2
= (Axx + Ayy + Azz)/r = ~A ·~r
r= ~A · (∇r)
∇(~a · ~r) = ∇(axx + ayy + azz) = ax ex + ay ey + az ez = ~a
∇ · (~a × ~r) = ∇r · (~a × ~r) = −~a · (∇× ~r)|^ ~b · (~a × ~c) = −~a · (~b × ~c)
E��Æ ÔnX ��� Mï� 17
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
l!~K
±e ~A �?¿¥þ§~a �~¥þ"
( ~A ·∇)~r = (Ax∂
∂x+ Ay
∂
∂y+ Az
∂
∂z)(x ex + y ey + z ez) = ~A
( ~A ·∇)r = (Ax∂
∂x+ Ay
∂
∂y+ Az
∂
∂z)√
x2 + y2 + z2
= (Axx + Ayy + Azz)/r = ~A ·~r
r= ~A · (∇r)
∇(~a · ~r) = ∇(axx + ayy + azz) = ax ex + ay ey + az ez = ~a
∇ · (~a × ~r) = ∇r · (~a × ~r) = −~a · (∇× ~r) = 0
E��Æ ÔnX ��� Mï� 17
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
l!~K
±e ~A �?¿¥þ§~a �~¥þ"
( ~A ·∇)~r = (Ax∂
∂x+ Ay
∂
∂y+ Az
∂
∂z)(x ex + y ey + z ez) = ~A
( ~A ·∇)r = (Ax∂
∂x+ Ay
∂
∂y+ Az
∂
∂z)√
x2 + y2 + z2
= (Axx + Ayy + Azz)/r = ~A ·~r
r= ~A · (∇r)
∇(~a · ~r) = ∇(axx + ayy + azz) = ax ex + ay ey + az ez = ~a
∇ · (~a × ~r) = ∇r · (~a × ~r) = −~a · (∇× ~r) = 0
∇× (~a × ~r)
E��Æ ÔnX ��� Mï� 17
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
l!~K
±e ~A �?¿¥þ§~a �~¥þ"
( ~A ·∇)~r = (Ax∂
∂x+ Ay
∂
∂y+ Az
∂
∂z)(x ex + y ey + z ez) = ~A
( ~A ·∇)r = (Ax∂
∂x+ Ay
∂
∂y+ Az
∂
∂z)√
x2 + y2 + z2
= (Axx + Ayy + Azz)/r = ~A ·~r
r= ~A · (∇r)
∇(~a · ~r) = ∇(axx + ayy + azz) = ax ex + ay ey + az ez = ~a
∇ · (~a × ~r) = ∇r · (~a × ~r) = −~a · (∇× ~r) = 0
∇× (~a × ~r) = ∇r × (~a × ~r)
E��Æ ÔnX ��� Mï� 17
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
l!~K
±e ~A �?¿¥þ§~a �~¥þ"
( ~A ·∇)~r = (Ax∂
∂x+ Ay
∂
∂y+ Az
∂
∂z)(x ex + y ey + z ez) = ~A
( ~A ·∇)r = (Ax∂
∂x+ Ay
∂
∂y+ Az
∂
∂z)√
x2 + y2 + z2
= (Axx + Ayy + Azz)/r = ~A ·~r
r= ~A · (∇r)
∇(~a · ~r) = ∇(axx + ayy + azz) = ax ex + ay ey + az ez = ~a
∇ · (~a × ~r) = ∇r · (~a × ~r) = −~a · (∇× ~r) = 0
∇× (~a × ~r) = ∇r × (~a × ~r)|^ ~b × (~a × ~c) = ~a(~b · ~c)− (~a · ~b)~c
E��Æ ÔnX ��� Mï� 17
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
l!~K
±e ~A �?¿¥þ§~a �~¥þ"
( ~A ·∇)~r = (Ax∂
∂x+ Ay
∂
∂y+ Az
∂
∂z)(x ex + y ey + z ez) = ~A
( ~A ·∇)r = (Ax∂
∂x+ Ay
∂
∂y+ Az
∂
∂z)√
x2 + y2 + z2
= (Axx + Ayy + Azz)/r = ~A ·~r
r= ~A · (∇r)
∇(~a · ~r) = ∇(axx + ayy + azz) = ax ex + ay ey + az ez = ~a
∇ · (~a × ~r) = ∇r · (~a × ~r) = −~a · (∇× ~r) = 0
∇× (~a × ~r) = ∇r × (~a × ~r) = ~a(∇ · ~r)− (~a ·∇)~r|^ ~b × (~a × ~c) = ~a(~b · ~c)− (~a · ~b)~c
E��Æ ÔnX ��� Mï� 17
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
l!~K
±e ~A �?¿¥þ§~a �~¥þ"
( ~A ·∇)~r = (Ax∂
∂x+ Ay
∂
∂y+ Az
∂
∂z)(x ex + y ey + z ez) = ~A
( ~A ·∇)r = (Ax∂
∂x+ Ay
∂
∂y+ Az
∂
∂z)√
x2 + y2 + z2
= (Axx + Ayy + Azz)/r = ~A ·~r
r= ~A · (∇r)
∇(~a · ~r) = ∇(axx + ayy + azz) = ax ex + ay ey + az ez = ~a
∇ · (~a × ~r) = ∇r · (~a × ~r) = −~a · (∇× ~r) = 0
∇× (~a × ~r) = ∇r × (~a × ~r) = ~a(∇ · ~r)− (~a ·∇)~r|^ ∇ · ~r = 3 Ú (~a ·∇)~r = ~a
E��Æ ÔnX ��� Mï� 17
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
l!~K
±e ~A �?¿¥þ§~a �~¥þ"
( ~A ·∇)~r = (Ax∂
∂x+ Ay
∂
∂y+ Az
∂
∂z)(x ex + y ey + z ez) = ~A
( ~A ·∇)r = (Ax∂
∂x+ Ay
∂
∂y+ Az
∂
∂z)√
x2 + y2 + z2
= (Axx + Ayy + Azz)/r = ~A ·~r
r= ~A · (∇r)
∇(~a · ~r) = ∇(axx + ayy + azz) = ax ex + ay ey + az ez = ~a
∇ · (~a × ~r) = ∇r · (~a × ~r) = −~a · (∇× ~r) = 0
∇× (~a × ~r) = ∇r × (~a × ~r) = ~a(∇ · ~r)− (~a ·∇)~r = 2~a|^ ∇ · ~r = 3 Ú (~a ·∇)~r = ~a
E��Æ ÔnX ��� Mï� 17
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇ · [(~a · ~r)~r]
E��Æ ÔnX ��� Mï� 18
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇ · [(~a · ~r)~r]|^ ∇ · (f ~A) = (∇f) · ~A + f∇ · ~A
E��Æ ÔnX ��� Mï� 18
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇ · [(~a · ~r)~r] = [∇(~a · ~r)] · ~r + (~a · ~r)∇ · ~r
|^ ∇ · (f ~A) = (∇f) · ~A + f∇ · ~A
E��Æ ÔnX ��� Mï� 18
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇ · [(~a · ~r)~r] = [∇(~a · ~r)] · ~r + (~a · ~r)∇ · ~r
|^ ∇(~a · ~r) = ~a 9 ∇ · ~r = 3
E��Æ ÔnX ��� Mï� 18
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇ · [(~a · ~r)~r] = [∇(~a · ~r)] · ~r + (~a · ~r)∇ · ~r
= ~a · ~r + 3~a · ~r = 4~a · ~r
|^ ∇(~a · ~r) = ~a 9 ∇ · ~r = 3
E��Æ ÔnX ��� Mï� 18
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇ · [(~a · ~r)~r] = [∇(~a · ~r)] · ~r + (~a · ~r)∇ · ~r
= ~a · ~r + 3~a · ~r = 4~a · ~r
E��Æ ÔnX ��� Mï� 18
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇ · [(~a · ~r)~r] = [∇(~a · ~r)] · ~r + (~a · ~r)∇ · ~r
= ~a · ~r + 3~a · ~r = 4~a · ~r
∇× [(~a · ~r)~r]|^ ∇× (f ~A) = (∇f) × ~A + f∇× ~A
E��Æ ÔnX ��� Mï� 18
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇ · [(~a · ~r)~r] = [∇(~a · ~r)] · ~r + (~a · ~r)∇ · ~r
= ~a · ~r + 3~a · ~r = 4~a · ~r
∇× [(~a · ~r)~r] = [∇(~a · ~r)] × ~r + (~a · ~r)∇× ~r
|^ ∇× (f ~A) = (∇f) × ~A + f∇× ~A
E��Æ ÔnX ��� Mï� 18
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇ · [(~a · ~r)~r] = [∇(~a · ~r)] · ~r + (~a · ~r)∇ · ~r
= ~a · ~r + 3~a · ~r = 4~a · ~r
∇× [(~a · ~r)~r] = [∇(~a · ~r)] × ~r + (~a · ~r)∇× ~r
|^ ∇(~a · ~r) = ~a 9 ∇× ~r = 0
E��Æ ÔnX ��� Mï� 18
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇ · [(~a · ~r)~r] = [∇(~a · ~r)] · ~r + (~a · ~r)∇ · ~r
= ~a · ~r + 3~a · ~r = 4~a · ~r
∇× [(~a · ~r)~r] = [∇(~a · ~r)] × ~r + (~a · ~r)∇× ~r
= ~a × ~r
|^ ∇(~a · ~r) = ~a 9 ∇× ~r = 0
E��Æ ÔnX ��� Mï� 18
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇ · [(~a · ~r)~r] = [∇(~a · ~r)] · ~r + (~a · ~r)∇ · ~r
= ~a · ~r + 3~a · ~r = 4~a · ~r
∇× [(~a · ~r)~r] = [∇(~a · ~r)] × ~r + (~a · ~r)∇× ~r
= ~a × ~r
∇ · [~r × ~A(r)]
E��Æ ÔnX ��� Mï� 18
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇ · [(~a · ~r)~r] = [∇(~a · ~r)] · ~r + (~a · ~r)∇ · ~r
= ~a · ~r + 3~a · ~r = 4~a · ~r
∇× [(~a · ~r)~r] = [∇(~a · ~r)] × ~r + (~a · ~r)∇× ~r
= ~a × ~r
∇ · [~r × ~A(r)] = ∇r · [~r × ~A(r)] +∇A · [~r × ~A(r)]
E��Æ ÔnX ��� Mï� 18
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇ · [(~a · ~r)~r] = [∇(~a · ~r)] · ~r + (~a · ~r)∇ · ~r
= ~a · ~r + 3~a · ~r = 4~a · ~r
∇× [(~a · ~r)~r] = [∇(~a · ~r)] × ~r + (~a · ~r)∇× ~r
= ~a × ~r
∇ · [~r × ~A(r)] = ∇r · [~r × ~A(r)] +∇A · [~r × ~A(r)]
|^ ~a · (~b × ~c) = (~a × ~b) · ~c = −~b · (~a × ~c)
E��Æ ÔnX ��� Mï� 18
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇ · [(~a · ~r)~r] = [∇(~a · ~r)] · ~r + (~a · ~r)∇ · ~r
= ~a · ~r + 3~a · ~r = 4~a · ~r
∇× [(~a · ~r)~r] = [∇(~a · ~r)] × ~r + (~a · ~r)∇× ~r
= ~a × ~r
∇ · [~r × ~A(r)] = ∇r · [~r × ~A(r)] +∇A · [~r × ~A(r)]
= (∇× ~r) · ~A(r)− ~r · [∇× ~A(r)]
|^ ~a · (~b × ~c) = (~a × ~b) · ~c = −~b · (~a × ~c)
E��Æ ÔnX ��� Mï� 18
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇ · [(~a · ~r)~r] = [∇(~a · ~r)] · ~r + (~a · ~r)∇ · ~r
= ~a · ~r + 3~a · ~r = 4~a · ~r
∇× [(~a · ~r)~r] = [∇(~a · ~r)] × ~r + (~a · ~r)∇× ~r
= ~a × ~r
∇ · [~r × ~A(r)] = ∇r · [~r × ~A(r)] +∇A · [~r × ~A(r)]
= (∇× ~r) · ~A(r)− ~r · [∇× ~A(r)]
|^ ∇× ~r = 0 9 ∇× ~A(r) = − ~A′(r) ×∇r
E��Æ ÔnX ��� Mï� 18
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇ · [(~a · ~r)~r] = [∇(~a · ~r)] · ~r + (~a · ~r)∇ · ~r
= ~a · ~r + 3~a · ~r = 4~a · ~r
∇× [(~a · ~r)~r] = [∇(~a · ~r)] × ~r + (~a · ~r)∇× ~r
= ~a × ~r
∇ · [~r × ~A(r)] = ∇r · [~r × ~A(r)] +∇A · [~r × ~A(r)]
= (∇× ~r) · ~A(r)− ~r · [∇× ~A(r)]
= ~r · [ ~A′(r) ×∇r]|^ ∇× ~r = 0 9 ∇× ~A(r) = − ~A′(r) ×∇r
E��Æ ÔnX ��� Mï� 18
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇ · [(~a · ~r)~r] = [∇(~a · ~r)] · ~r + (~a · ~r)∇ · ~r
= ~a · ~r + 3~a · ~r = 4~a · ~r
∇× [(~a · ~r)~r] = [∇(~a · ~r)] × ~r + (~a · ~r)∇× ~r
= ~a × ~r
∇ · [~r × ~A(r)] = ∇r · [~r × ~A(r)] +∇A · [~r × ~A(r)]
= (∇× ~r) · ~A(r)− ~r · [∇× ~A(r)]
= ~r · [ ~A′(r) ×∇r]|^ ∇r = ~r/r
E��Æ ÔnX ��� Mï� 18
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇ · [(~a · ~r)~r] = [∇(~a · ~r)] · ~r + (~a · ~r)∇ · ~r
= ~a · ~r + 3~a · ~r = 4~a · ~r
∇× [(~a · ~r)~r] = [∇(~a · ~r)] × ~r + (~a · ~r)∇× ~r
= ~a × ~r
∇ · [~r × ~A(r)] = ∇r · [~r × ~A(r)] +∇A · [~r × ~A(r)]
= (∇× ~r) · ~A(r)− ~r · [∇× ~A(r)]
= ~r · [ ~A′(r) ×∇r] = ~r · [ ~A′(r) × ~r/r]|^ ∇r = ~r/r
E��Æ ÔnX ��� Mï� 18
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇ · [(~a · ~r)~r] = [∇(~a · ~r)] · ~r + (~a · ~r)∇ · ~r
= ~a · ~r + 3~a · ~r = 4~a · ~r
∇× [(~a · ~r)~r] = [∇(~a · ~r)] × ~r + (~a · ~r)∇× ~r
= ~a × ~r
∇ · [~r × ~A(r)] = ∇r · [~r × ~A(r)] +∇A · [~r × ~A(r)]
= (∇× ~r) · ~A(r)− ~r · [∇× ~A(r)]
= ~r · [ ~A′(r) ×∇r] = ~r · [ ~A′(r) × ~r/r]
|^ [ ~A′(r) × ~r] ⊥ ~r
E��Æ ÔnX ��� Mï� 18
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇ · [(~a · ~r)~r] = [∇(~a · ~r)] · ~r + (~a · ~r)∇ · ~r
= ~a · ~r + 3~a · ~r = 4~a · ~r
∇× [(~a · ~r)~r] = [∇(~a · ~r)] × ~r + (~a · ~r)∇× ~r
= ~a × ~r
∇ · [~r × ~A(r)] = ∇r · [~r × ~A(r)] +∇A · [~r × ~A(r)]
= (∇× ~r) · ~A(r)− ~r · [∇× ~A(r)]
= ~r · [ ~A′(r) ×∇r] = ~r · [ ~A′(r) × ~r/r] = 0
|^ [ ~A′(r) × ~r] ⊥ ~r
E��Æ ÔnX ��� Mï� 18
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇ · [φ(r)(~a × ~r)]
E��Æ ÔnX ��� Mï� 19
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇ · [φ(r)(~a × ~r)]|^ ∇ · (f ~A) = (∇f) · ~A + f∇ · ~A
E��Æ ÔnX ��� Mï� 19
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇ · [φ(r)(~a × ~r)] = [∇φ(r)] · (~a × ~r) + φ(r)∇ · (~a × ~r)|^ ∇ · (f ~A) = (∇f) · ~A + f∇ · ~A
E��Æ ÔnX ��� Mï� 19
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇ · [φ(r)(~a × ~r)] = [∇φ(r)] · (~a × ~r) + φ(r)∇ · (~a × ~r)
|^ ∇f(u) = f ′(u)∇u Ú ∇ · (~a × ~r) = 0
E��Æ ÔnX ��� Mï� 19
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇ · [φ(r)(~a × ~r)] = [∇φ(r)] · (~a × ~r) + φ(r)∇ · (~a × ~r)
= [φ′(r)∇r] · (~a × ~r)
|^ ∇f(u) = f ′(u)∇u Ú ∇ · (~a × ~r) = 0
E��Æ ÔnX ��� Mï� 19
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇ · [φ(r)(~a × ~r)] = [∇φ(r)] · (~a × ~r) + φ(r)∇ · (~a × ~r)
= [φ′(r)∇r] · (~a × ~r)
|^ ∇r =~r
r
E��Æ ÔnX ��� Mï� 19
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇ · [φ(r)(~a × ~r)] = [∇φ(r)] · (~a × ~r) + φ(r)∇ · (~a × ~r)
= [φ′(r)∇r] · (~a × ~r) =φ′(r)
r~r · (~a × ~r)
|^ ∇r =~r
r
E��Æ ÔnX ��� Mï� 19
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇ · [φ(r)(~a × ~r)] = [∇φ(r)] · (~a × ~r) + φ(r)∇ · (~a × ~r)
= [φ′(r)∇r] · (~a × ~r) =φ′(r)
r~r · (~a × ~r)
|^ ~a × ~r ⊥ ~r
E��Æ ÔnX ��� Mï� 19
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇ · [φ(r)(~a × ~r)] = [∇φ(r)] · (~a × ~r) + φ(r)∇ · (~a × ~r)
= [φ′(r)∇r] · (~a × ~r) =φ′(r)
r~r · (~a × ~r) = 0
|^ ~a × ~r ⊥ ~r
E��Æ ÔnX ��� Mï� 19
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇ · [φ(r)(~a × ~r)] = [∇φ(r)] · (~a × ~r) + φ(r)∇ · (~a × ~r)
= [φ′(r)∇r] · (~a × ~r) =φ′(r)
r~r · (~a × ~r) = 0
∇× [ ~E0 sin(~k · ~r − ωt)]
E��Æ ÔnX ��� Mï� 19
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇ · [φ(r)(~a × ~r)] = [∇φ(r)] · (~a × ~r) + φ(r)∇ · (~a × ~r)
= [φ′(r)∇r] · (~a × ~r) =φ′(r)
r~r · (~a × ~r) = 0
∇× [ ~E0 sin(~k · ~r − ωt)]
|^ ∇× ( ~Af) = (∇× ~A)f − ~A × (∇f)
E��Æ ÔnX ��� Mï� 19
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇ · [φ(r)(~a × ~r)] = [∇φ(r)] · (~a × ~r) + φ(r)∇ · (~a × ~r)
= [φ′(r)∇r] · (~a × ~r) =φ′(r)
r~r · (~a × ~r) = 0
∇× [ ~E0 sin(~k · ~r − ωt)] = (∇× ~E0)︸ ︷︷ ︸0
sin(~k · ~r − ωt)− ~E0 × [∇ sin(~k · ~r − ωt)]
|^ ∇× ( ~Af) = (∇× ~A)f − ~A × (∇f)
E��Æ ÔnX ��� Mï� 19
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇ · [φ(r)(~a × ~r)] = [∇φ(r)] · (~a × ~r) + φ(r)∇ · (~a × ~r)
= [φ′(r)∇r] · (~a × ~r) =φ′(r)
r~r · (~a × ~r) = 0
∇× [ ~E0 sin(~k · ~r − ωt)] = (∇× ~E0)︸ ︷︷ ︸0
sin(~k · ~r − ωt)− ~E0 × [∇ sin(~k · ~r − ωt)]
|^ ∇f(u) =df
du∇u
E��Æ ÔnX ��� Mï� 19
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇ · [φ(r)(~a × ~r)] = [∇φ(r)] · (~a × ~r) + φ(r)∇ · (~a × ~r)
= [φ′(r)∇r] · (~a × ~r) =φ′(r)
r~r · (~a × ~r) = 0
∇× [ ~E0 sin(~k · ~r − ωt)] = (∇× ~E0)︸ ︷︷ ︸0
sin(~k · ~r − ωt)− ~E0 × [∇ sin(~k · ~r − ωt)]
= − ~E0 × [cos(~k · ~r − ωt)]∇(~k · ~r)
|^ ∇f(u) =df
du∇u
E��Æ ÔnX ��� Mï� 19
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇ · [φ(r)(~a × ~r)] = [∇φ(r)] · (~a × ~r) + φ(r)∇ · (~a × ~r)
= [φ′(r)∇r] · (~a × ~r) =φ′(r)
r~r · (~a × ~r) = 0
∇× [ ~E0 sin(~k · ~r − ωt)] = (∇× ~E0)︸ ︷︷ ︸0
sin(~k · ~r − ωt)− ~E0 × [∇ sin(~k · ~r − ωt)]
= − ~E0 × [cos(~k · ~r − ωt)]∇(~k · ~r)
|^ ∇(~k · ~r) = ~k
E��Æ ÔnX ��� Mï� 19
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇ · [φ(r)(~a × ~r)] = [∇φ(r)] · (~a × ~r) + φ(r)∇ · (~a × ~r)
= [φ′(r)∇r] · (~a × ~r) =φ′(r)
r~r · (~a × ~r) = 0
∇× [ ~E0 sin(~k · ~r − ωt)] = (∇× ~E0)︸ ︷︷ ︸0
sin(~k · ~r − ωt)− ~E0 × [∇ sin(~k · ~r − ωt)]
= − ~E0 × [cos(~k · ~r − ωt)]∇(~k · ~r)
= ~k × ~E0 cos(~k · ~r − ωt)
|^ ∇(~k · ~r) = ~k
E��Æ ÔnX ��� Mï� 19
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇(
~a ·~r
r3
)
E��Æ ÔnX ��� Mï� 20
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇(
~a ·~r
r3
)|^� ~a �~¥þ�µ∇(~a · ~A) = (~a ·∇) ~A + ~a × (∇× ~A)
E��Æ ÔnX ��� Mï� 20
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇(
~a ·~r
r3
)= (~a ·∇)
(~r
r3
)+ ~a ×
(∇×
~r
r3
)|^� ~a �~¥þ�µ∇(~a · ~A) = (~a ·∇) ~A + ~a × (∇× ~A)
E��Æ ÔnX ��� Mï� 20
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇(
~a ·~r
r3
)= (~a ·∇)
(~r
r3
)+ ~a ×
(∇×
~r
r3
)
|^ ∇×~r
rn= 0
E��Æ ÔnX ��� Mï� 20
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇(
~a ·~r
r3
)= (~a ·∇)
(~r
r3
)+ ~a ×
(∇×
~r
r3
)= (~a ·∇)
(~r
r3
)|^ ∇×
~r
rn= 0
E��Æ ÔnX ��� Mï� 20
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇(
~a ·~r
r3
)= (~a ·∇)
(~r
r3
)+ ~a ×
(∇×
~r
r3
)= (~a ·∇)
(~r
r3
)Iþ�Î (~a ·∇) ©O�^u ~r � 1
r3
E��Æ ÔnX ��� Mï� 20
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇(
~a ·~r
r3
)= (~a ·∇)
(~r
r3
)+ ~a ×
(∇×
~r
r3
)= (~a ·∇)
(~r
r3
)=
1r3
[ (~a ·∇)~r ] + ~r
[(~a ·∇)
1r3
]Iþ�Î (~a ·∇) ©O�^u ~r � 1
r3
E��Æ ÔnX ��� Mï� 20
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇(
~a ·~r
r3
)= (~a ·∇)
(~r
r3
)+ ~a ×
(∇×
~r
r3
)= (~a ·∇)
(~r
r3
)=
1r3
[ (~a ·∇)~r ] + ~r
[(~a ·∇)
1r3
]
|^ (~a ·∇)~r = ~a Ú (~a ·∇)f(u) = f ′(u)[(a ·∇)u]
E��Æ ÔnX ��� Mï� 20
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇(
~a ·~r
r3
)= (~a ·∇)
(~r
r3
)+ ~a ×
(∇×
~r
r3
)= (~a ·∇)
(~r
r3
)=
1r3
[ (~a ·∇)~r ] + ~r
[(~a ·∇)
1r3
]=
~a
r3+ ~r
(− 3
r4
)[(~a ·∇)r]
|^ (~a ·∇)~r = ~a Ú (~a ·∇)f(u) = f ′(u)[(a ·∇)u]
E��Æ ÔnX ��� Mï� 20
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇(
~a ·~r
r3
)= (~a ·∇)
(~r
r3
)+ ~a ×
(∇×
~r
r3
)= (~a ·∇)
(~r
r3
)=
1r3
[ (~a ·∇)~r ] + ~r
[(~a ·∇)
1r3
]=
~a
r3+ ~r
(− 3
r4
)[(~a ·∇)r]
|^ (~a ·∇)r =(~a · ~r)
r
E��Æ ÔnX ��� Mï� 20
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇(
~a ·~r
r3
)= (~a ·∇)
(~r
r3
)+ ~a ×
(∇×
~r
r3
)= (~a ·∇)
(~r
r3
)=
1r3
[ (~a ·∇)~r ] + ~r
[(~a ·∇)
1r3
]=
~a
r3+ ~r
(− 3
r4
)[(~a ·∇)r] =
~a
r3−
(3~r
r4
)(~a · ~r)
r
|^ (~a ·∇)r =(~a · ~r)
r
E��Æ ÔnX ��� Mï� 20
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇(
~a ·~r
r3
)= (~a ·∇)
(~r
r3
)+ ~a ×
(∇×
~r
r3
)= (~a ·∇)
(~r
r3
)=
1r3
[ (~a ·∇)~r ] + ~r
[(~a ·∇)
1r3
]=
~a
r3+ ~r
(− 3
r4
)[(~a ·∇)r] =
~a
r3−
(3~r
r4
)(~a · ~r)
r
=~a
r3− 3(~a · ~r)~r
r5
E��Æ ÔnX ��� Mï� 20
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇(
~a ·~r
r3
)= (~a ·∇)
(~r
r3
)+ ~a ×
(∇×
~r
r3
)= (~a ·∇)
(~r
r3
)=
1r3
[ (~a ·∇)~r ] + ~r
[(~a ·∇)
1r3
]=
~a
r3+ ~r
(− 3
r4
)[(~a ·∇)r] =
~a
r3−
(3~r
r4
)(~a · ~r)
r
=~a
r3− 3(~a · ~r)~r
r5
∇×(
~a ×~r
r3
)
E��Æ ÔnX ��� Mï� 20
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇(
~a ·~r
r3
)= (~a ·∇)
(~r
r3
)+ ~a ×
(∇×
~r
r3
)= (~a ·∇)
(~r
r3
)=
1r3
[ (~a ·∇)~r ] + ~r
[(~a ·∇)
1r3
]=
~a
r3+ ~r
(− 3
r4
)[(~a ·∇)r] =
~a
r3−
(3~r
r4
)(~a · ~r)
r
=~a
r3− 3(~a · ~r)~r
r5
∇×(
~a ×~r
r3
)|^� ~a �~¥þ�µ∇× (~a × ~A) = ~a(∇ · ~A)− (~a ·∇) ~A
E��Æ ÔnX ��� Mï� 20
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇(
~a ·~r
r3
)= (~a ·∇)
(~r
r3
)+ ~a ×
(∇×
~r
r3
)= (~a ·∇)
(~r
r3
)=
1r3
[ (~a ·∇)~r ] + ~r
[(~a ·∇)
1r3
]=
~a
r3+ ~r
(− 3
r4
)[(~a ·∇)r] =
~a
r3−
(3~r
r4
)(~a · ~r)
r
=~a
r3− 3(~a · ~r)~r
r5
∇×(
~a ×~r
r3
)= ~a
(∇ ·
~r
r3
)− (~a ·∇)
(~r
r3
)|^� ~a �~¥þ�µ∇× (~a × ~A) = ~a(∇ · ~A)− (~a ·∇) ~A
E��Æ ÔnX ��� Mï� 20
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇(
~a ·~r
r3
)= (~a ·∇)
(~r
r3
)+ ~a ×
(∇×
~r
r3
)= (~a ·∇)
(~r
r3
)=
1r3
[ (~a ·∇)~r ] + ~r
[(~a ·∇)
1r3
]=
~a
r3+ ~r
(− 3
r4
)[(~a ·∇)r] =
~a
r3−
(3~r
r4
)(~a · ~r)
r
=~a
r3− 3(~a · ~r)~r
r5
∇×(
~a ×~r
r3
)= ~a
(∇ ·
~r
r3
)− (~a ·∇)
(~r
r3
)|^ ∇ ·
~r
r3=∇ · ~r
r3+
(∇ 1
r3
)· ~r =
3r3
+(− 3
r4
~r
r
)· ~r = 0
E��Æ ÔnX ��� Mï� 20
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇(
~a ·~r
r3
)= (~a ·∇)
(~r
r3
)+ ~a ×
(∇×
~r
r3
)= (~a ·∇)
(~r
r3
)=
1r3
[ (~a ·∇)~r ] + ~r
[(~a ·∇)
1r3
]=
~a
r3+ ~r
(− 3
r4
)[(~a ·∇)r] =
~a
r3−
(3~r
r4
)(~a · ~r)
r
=~a
r3− 3(~a · ~r)~r
r5
∇×(
~a ×~r
r3
)= ~a
(∇ ·
~r
r3
)− (~a ·∇)
(~r
r3
)= −(~a ·∇)
(~r
r3
)|^ ∇ ·
~r
r3=∇ · ~r
r3+
(∇ 1
r3
)· ~r =
3r3
+(− 3
r4
~r
r
)· ~r = 0
E��Æ ÔnX ��� Mï� 20
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇(
~a ·~r
r3
)= (~a ·∇)
(~r
r3
)+ ~a ×
(∇×
~r
r3
)= (~a ·∇)
(~r
r3
)=
1r3
[ (~a ·∇)~r ] + ~r
[(~a ·∇)
1r3
]=
~a
r3+ ~r
(− 3
r4
)[(~a ·∇)r] =
~a
r3−
(3~r
r4
)(~a · ~r)
r
=~a
r3− 3(~a · ~r)~r
r5
∇×(
~a ×~r
r3
)= ~a
(∇ ·
~r
r3
)− (~a ·∇)
(~r
r3
)= −(~a ·∇)
(~r
r3
)|^ (~a ·∇)
(~r
r3
)=
~a
r3− 3(~a · ~r)~r
r5£þ¡ùÚÜ©¤
E��Æ ÔnX ��� Mï� 20
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇(
~a ·~r
r3
)= (~a ·∇)
(~r
r3
)+ ~a ×
(∇×
~r
r3
)= (~a ·∇)
(~r
r3
)=
1r3
[ (~a ·∇)~r ] + ~r
[(~a ·∇)
1r3
]=
~a
r3+ ~r
(− 3
r4
)[(~a ·∇)r] =
~a
r3−
(3~r
r4
)(~a · ~r)
r
=~a
r3− 3(~a · ~r)~r
r5
∇×(
~a ×~r
r3
)= ~a
(∇ ·
~r
r3
)− (~a ·∇)
(~r
r3
)= −(~a ·∇)
(~r
r3
)|^ (~a ·∇)
(~r
r3
)=
~a
r3− 3(~a · ~r)~r
r5£þ¡ùÚÜ©¤
= −~a
r3+
3(~a · ~r)~rr5
E��Æ ÔnX ��� Mï� 20
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
∇(
~a ·~r
r3
)= (~a ·∇)
(~r
r3
)+ ~a ×
(∇×
~r
r3
)= (~a ·∇)
(~r
r3
)=
1r3
[ (~a ·∇)~r ] + ~r
[(~a ·∇)
1r3
]=
~a
r3+ ~r
(− 3
r4
)[(~a ·∇)r] =
~a
r3−
(3~r
r4
)(~a · ~r)
r
=~a
r3− 3(~a · ~r)~r
r5
∇×(
~a ×~r
r3
)= ~a
(∇ ·
~r
r3
)− (~a ·∇)
(~r
r3
)= −(~a ·∇)
(~r
r3
)= −
~a
r3+
3(~a · ~r)~rr5
E��Æ ÔnX ��� Mï� 20
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
( ~A ×∇)× ~r
E��Æ ÔnX ��� Mï� 21
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
( ~A ×∇)× ~r =[ ~A × ∇ � k ©þ︷ ︸︸ ︷
εijk(Ai∂j) ek
]︸ ︷︷ ︸éEeI¦Ú
×(xl el)
E��Æ ÔnX ��� Mï� 21
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
( ~A ×∇)× ~r =[ ~A × ∇ � k ©þ︷ ︸︸ ︷
εijk(Ai∂j) ek
]︸ ︷︷ ︸éEeI¦Ú
×(xl el)
= εijk(Ai∂jxl)( ek × el)
E��Æ ÔnX ��� Mï� 21
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
( ~A ×∇)× ~r =[ ~A × ∇ � k ©þ︷ ︸︸ ︷
εijk(Ai∂j) ek
]︸ ︷︷ ︸éEeI¦Ú
×(xl el)
= εijk(Ai∂jxl)( ek × el) |^
{∂jxl = δjl
ek × el = εklm em
E��Æ ÔnX ��� Mï� 21
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
( ~A ×∇)× ~r =[ ~A × ∇ � k ©þ︷ ︸︸ ︷
εijk(Ai∂j) ek
]︸ ︷︷ ︸éEeI¦Ú
×(xl el)
= εijk(Ai∂jxl)( ek × el) |^
{∂jxl = δjl
ek × el = εklm em
= εijk(Aiδjl)(εklm em)
E��Æ ÔnX ��� Mï� 21
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
( ~A ×∇)× ~r =[ ~A × ∇ � k ©þ︷ ︸︸ ︷
εijk(Ai∂j) ek
]︸ ︷︷ ︸éEeI¦Ú
×(xl el)
= εijk(Ai∂jxl)( ek × el) |^
{∂jxl = δjl
ek × el = εklm em
= εijk(Aiδjl)(εklm em) k δjl �§
{é l k¦Ú
�3 l = j �
E��Æ ÔnX ��� Mï� 21
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
( ~A ×∇)× ~r =[ ~A × ∇ � k ©þ︷ ︸︸ ︷
εijk(Ai∂j) ek
]︸ ︷︷ ︸éEeI¦Ú
×(xl el)
= εijk(Ai∂jxl)( ek × el) |^
{∂jxl = δjl
ek × el = εklm em
= εijk(Aiδjl)(εklm em) k δjl �§
{é l k¦Ú
�3 l = j �
= εijkAiεkjm em
E��Æ ÔnX ��� Mï� 21
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
( ~A ×∇)× ~r =[ ~A × ∇ � k ©þ︷ ︸︸ ︷
εijk(Ai∂j) ek
]︸ ︷︷ ︸éEeI¦Ú
×(xl el)
= εijk(Ai∂jxl)( ek × el) |^
{∂jxl = δjl
ek × el = εklm em
= εijk(Aiδjl)(εklm em) k δjl �§
{é l k¦Ú
�3 l = j �
= εijkAiεkjm em ëY|^ εijk = −εikj
E��Æ ÔnX ��� Mï� 21
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
( ~A ×∇)× ~r =[ ~A × ∇ � k ©þ︷ ︸︸ ︷
εijk(Ai∂j) ek
]︸ ︷︷ ︸éEeI¦Ú
×(xl el)
= εijk(Ai∂jxl)( ek × el) |^
{∂jxl = δjl
ek × el = εklm em
= εijk(Aiδjl)(εklm em) k δjl �§
{é l k¦Ú
�3 l = j �
= εijkAiεkjm em ëY|^ εijk = −εikj
= −[εijkεmjk]Ai em
E��Æ ÔnX ��� Mï� 21
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
( ~A ×∇)× ~r =[ ~A × ∇ � k ©þ︷ ︸︸ ︷
εijk(Ai∂j) ek
]︸ ︷︷ ︸éEeI¦Ú
×(xl el)
= εijk(Ai∂jxl)( ek × el) |^
{∂jxl = δjl
ek × el = εklm em
= εijk(Aiδjl)(εklm em) k δjl �§
{é l k¦Ú
�3 l = j �
= εijkAiεkjm em ëY|^ εijk = −εikj
= −[εijkεmjk]Ai em [ ] S�Levi-Civita ÎÒ�ü¦Ú
E��Æ ÔnX ��� Mï� 21
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
( ~A ×∇)× ~r =[ ~A × ∇ � k ©þ︷ ︸︸ ︷
εijk(Ai∂j) ek
]︸ ︷︷ ︸éEeI¦Ú
×(xl el)
= εijk(Ai∂jxl)( ek × el) |^
{∂jxl = δjl
ek × el = εklm em
= εijk(Aiδjl)(εklm em) k δjl �§
{é l k¦Ú
�3 l = j �
= εijkAiεkjm em ëY|^ εijk = −εikj
= −[εijkεmjk]Ai em [ ] S�Levi-Civita ÎÒ�ü¦Ú
= −2δimAi em ÷v εijkεmjk = 2δim
E��Æ ÔnX ��� Mï� 21
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
( ~A ×∇)× ~r =[ ~A × ∇ � k ©þ︷ ︸︸ ︷
εijk(Ai∂j) ek
]︸ ︷︷ ︸éEeI¦Ú
×(xl el)
= εijk(Ai∂jxl)( ek × el) |^
{∂jxl = δjl
ek × el = εklm em
= εijk(Aiδjl)(εklm em) k δjl �§
{é l k¦Ú
�3 l = j �
= εijkAiεkjm em ëY|^ εijk = −εikj
= −[εijkεmjk]Ai em [ ] S�Levi-Civita ÎÒ�ü¦Ú
= −2δimAi em ÷v εijkεmjk = 2δim
= −2Ai ei = −2 ~A
E��Æ ÔnX ��� Mï� 21
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
( ~A ×∇)× ~r =[ ~A × ∇ � k ©þ︷ ︸︸ ︷
εijk(Ai∂j) ek
]︸ ︷︷ ︸éEeI¦Ú
×(xl el)
= εijk(Ai∂jxl)( ek × el) |^
{∂jxl = δjl
ek × el = εklm em
= εijk(Aiδjl)(εklm em) k δjl �§
{é l k¦Ú
�3 l = j �
= εijkAiεkjm em ëY|^ εijk = −εikj
= −[εijkεmjk]Ai em [ ] S�Levi-Civita ÎÒ�ü¦Ú
= −2δimAi em ÷v εijkεmjk = 2δim
= −2Ai ei = −2 ~A
( ~A ×∇) × ~r = −2 ~A
E��Æ ÔnX ��� Mï� 21
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
( ~A ×∇)× ~r =[ ~A × ∇ � k ©þ︷ ︸︸ ︷
εijk(Ai∂j) ek
]︸ ︷︷ ︸éEeI¦Ú
×(xl el)
= εijk(Ai∂jxl)( ek × el) |^
{∂jxl = δjl
ek × el = εklm em
= εijk(Aiδjl)(εklm em) k δjl �§
{é l k¦Ú
�3 l = j �
= εijkAiεkjm em ëY|^ εijk = −εikj
= −[εijkεmjk]Ai em [ ] S�Levi-Civita ÎÒ�ü¦Ú
= −2δimAi em ÷v εijkεmjk = 2δim
= −2Ai ei = −2 ~A
( ~A ×∇) × ~r = −2 ~A ~A �±�?¿¥þ
E��Æ ÔnX ��� Mï� 21
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
( ~A ×∇) · ~r
E��Æ ÔnX ��� Mï� 22
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
( ~A ×∇) · ~r =[ ~A × ∇ � k ©þ︷ ︸︸ ︷
εijk(Ai∂j) ek
]︸ ︷︷ ︸éEeI¦Ú
·(xl el)
E��Æ ÔnX ��� Mï� 22
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
( ~A ×∇) · ~r =[ ~A × ∇ � k ©þ︷ ︸︸ ︷
εijk(Ai∂j) ek
]︸ ︷︷ ︸éEeI¦Ú
·(xl el)
= εijk(Ai∂jxl)( ek · el)
E��Æ ÔnX ��� Mï� 22
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
( ~A ×∇) · ~r =[ ~A × ∇ � k ©þ︷ ︸︸ ︷
εijk(Ai∂j) ek
]︸ ︷︷ ︸éEeI¦Ú
·(xl el)
= εijk(Ai∂jxl)( ek · el) |^
{∂jxl = δjl
ek · el = δkl
E��Æ ÔnX ��� Mï� 22
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
( ~A ×∇) · ~r =[ ~A × ∇ � k ©þ︷ ︸︸ ︷
εijk(Ai∂j) ek
]︸ ︷︷ ︸éEeI¦Ú
·(xl el)
= εijk(Ai∂jxl)( ek · el) |^
{∂jxl = δjl
ek · el = δkl
= εijk(Aiδjl)δkl
E��Æ ÔnX ��� Mï� 22
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
( ~A ×∇) · ~r =[ ~A × ∇ � k ©þ︷ ︸︸ ︷
εijk(Ai∂j) ek
]︸ ︷︷ ︸éEeI¦Ú
·(xl el)
= εijk(Ai∂jxl)( ek · el) |^
{∂jxl = δjl
ek · el = δkl
= εijk(Aiδjl)δkl = εillAi = 0
( ~A ×∇) · ~r = 0
E��Æ ÔnX ��� Mï� 22
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
( ~A ×∇) · ~r =[ ~A × ∇ � k ©þ︷ ︸︸ ︷
εijk(Ai∂j) ek
]︸ ︷︷ ︸éEeI¦Ú
·(xl el)
= εijk(Ai∂jxl)( ek · el) |^
{∂jxl = δjl
ek · el = δkl
= εijk(Aiδjl)δkl = εillAi = 0
( ~A ×∇) · ~r = 0 ~A �±�?¿¥þ
E��Æ ÔnX ��� Mï� 22
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
( ~A ×∇) · ~r =[ ~A × ∇ � k ©þ︷ ︸︸ ︷
εijk(Ai∂j) ek
]︸ ︷︷ ︸éEeI¦Ú
·(xl el)
= εijk(Ai∂jxl)( ek · el) |^
{∂jxl = δjl
ek · el = δkl
= εijk(Aiδjl)δkl = εillAi = 0
( ~A ×∇) · ~r = 0 ~A �±�?¿¥þ
,y
E��Æ ÔnX ��� Mï� 22
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
( ~A ×∇) · ~r =[ ~A × ∇ � k ©þ︷ ︸︸ ︷
εijk(Ai∂j) ek
]︸ ︷︷ ︸éEeI¦Ú
·(xl el)
= εijk(Ai∂jxl)( ek · el) |^
{∂jxl = δjl
ek · el = δkl
= εijk(Aiδjl)δkl = εillAi = 0
( ~A ×∇) · ~r = 0 ~A �±�?¿¥þ
,y ( ~A ×∇) · ~r = ~A · (∇× ~r) = 0
E��Æ ÔnX ��� Mï� 22
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
( ~A ×∇) · ~r =[ ~A × ∇ � k ©þ︷ ︸︸ ︷
εijk(Ai∂j) ek
]︸ ︷︷ ︸éEeI¦Ú
·(xl el)
= εijk(Ai∂jxl)( ek · el) |^
{∂jxl = δjl
ek · el = δkl
= εijk(Aiδjl)δkl = εillAi = 0
( ~A ×∇) · ~r = 0 ~A �±�?¿¥þ
,y ( ~A ×∇) · ~r = ~A · (∇× ~r) = 0
( ~A ×∇) × ~r = ( ~A ×∇r) × ~r
E��Æ ÔnX ��� Mï� 22
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
( ~A ×∇) · ~r =[ ~A × ∇ � k ©þ︷ ︸︸ ︷
εijk(Ai∂j) ek
]︸ ︷︷ ︸éEeI¦Ú
·(xl el)
= εijk(Ai∂jxl)( ek · el) |^
{∂jxl = δjl
ek · el = δkl
= εijk(Aiδjl)δkl = εillAi = 0
( ~A ×∇) · ~r = 0 ~A �±�?¿¥þ
,y ( ~A ×∇) · ~r = ~A · (∇× ~r) = 0
( ~A ×∇) × ~r = ( ~A ×∇r) × ~r |^ (~a × ~b) × ~c = ~b (~a · ~c)− ~a (~b · ~c)
E��Æ ÔnX ��� Mï� 22
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
( ~A ×∇) · ~r =[ ~A × ∇ � k ©þ︷ ︸︸ ︷
εijk(Ai∂j) ek
]︸ ︷︷ ︸éEeI¦Ú
·(xl el)
= εijk(Ai∂jxl)( ek · el) |^
{∂jxl = δjl
ek · el = δkl
= εijk(Aiδjl)δkl = εillAi = 0
( ~A ×∇) · ~r = 0 ~A �±�?¿¥þ
,y ( ~A ×∇) · ~r = ~A · (∇× ~r) = 0
( ~A ×∇) × ~r = ( ~A ×∇r) × ~r |^ (~a × ~b) × ~c = ~b (~a · ~c)− ~a (~b · ~c)
= ∇r( ~A · ~r)− ~A(∇r · ~r)
E��Æ ÔnX ��� Mï� 22
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
( ~A ×∇) · ~r =[ ~A × ∇ � k ©þ︷ ︸︸ ︷
εijk(Ai∂j) ek
]︸ ︷︷ ︸éEeI¦Ú
·(xl el)
= εijk(Ai∂jxl)( ek · el) |^
{∂jxl = δjl
ek · el = δkl
= εijk(Aiδjl)δkl = εillAi = 0
( ~A ×∇) · ~r = 0 ~A �±�?¿¥þ
,y ( ~A ×∇) · ~r = ~A · (∇× ~r) = 0
( ~A ×∇) × ~r = ( ~A ×∇r) × ~r |^ (~a × ~b) × ~c = ~b (~a · ~c)− ~a (~b · ~c)
= ∇r( ~A · ~r)− ~A(∇r · ~r) ∇r( ~A · ~r) ¥ ~A À�~¥þ|^ ∇(~a · ~r) = ~a Ú ∇ · ~r = 3
E��Æ ÔnX ��� Mï� 22
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
( ~A ×∇) · ~r =[ ~A × ∇ � k ©þ︷ ︸︸ ︷
εijk(Ai∂j) ek
]︸ ︷︷ ︸éEeI¦Ú
·(xl el)
= εijk(Ai∂jxl)( ek · el) |^
{∂jxl = δjl
ek · el = δkl
= εijk(Aiδjl)δkl = εillAi = 0
( ~A ×∇) · ~r = 0 ~A �±�?¿¥þ
,y ( ~A ×∇) · ~r = ~A · (∇× ~r) = 0
( ~A ×∇) × ~r = ( ~A ×∇r) × ~r |^ (~a × ~b) × ~c = ~b (~a · ~c)− ~a (~b · ~c)
= ∇r( ~A · ~r)− ~A(∇r · ~r) ∇r( ~A · ~r) ¥ ~A À�~¥þ|^ ∇(~a · ~r) = ~a Ú ∇ · ~r = 3
= −2 ~A
E��Æ ÔnX ��� Mï� 22
Let there be light²;>ÄåÆ�Ø
1�ÙµêÆÄ: § 1.2
( ~A ×∇) · ~r =[ ~A × ∇ � k ©þ︷ ︸︸ ︷
εijk(Ai∂j) ek
]︸ ︷︷ ︸éEeI¦Ú
·(xl el)
= εijk(Ai∂jxl)( ek · el) |^
{∂jxl = δjl
ek · el = δkl
= εijk(Aiδjl)δkl = εillAi = 0
( ~A ×∇) · ~r = 0 ~A �±�?¿¥þ
,y ( ~A ×∇) · ~r = ~A · (∇× ~r) = 0
( ~A ×∇) × ~r = ( ~A ×∇r) × ~r |^ (~a × ~b) × ~c = ~b (~a · ~c)− ~a (~b · ~c)
= ∇r( ~A · ~r)− ~A(∇r · ~r) ∇r( ~A · ~r) ¥ ~A À�~¥þ|^ ∇(~a · ~r) = ~a Ú ∇ · ~r = 3
= −2 ~A ( ~A ×∇) × ~r = −2 ~A
E��Æ ÔnX ��� Mï� 22
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