classical electrodynamics - fudan...

Let there be light²;>ÄåÆ�Ø

1�ÙµêÆÄ: § 1.2

§ 1.2 ��©©©�!FÝ

E��Æ ÔnX �� Mï� 1


1�ÙµêÆÄ: § 1.2

§ 1.2 ��©©©�!FÝ

FÝ´�¥þ

∇T =∂T

∂xex +

∂T

∂yey +

∂T

∂zez



1�ÙµêÆÄ: § 1.2

§ 1.2 ��©©©�!FÝ

FÝ´�¥þ

∇T =∂T

∂xex +

∂T

∂yey +

∂T

∂zez

��©

dT =∂T

∂xdx +

∂T

∂ydy +

∂T

∂zdz



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



1�ÙµêÆÄ: § 1.2

§ 1.2 ��©©©�!FÝ

FÝ´�¥þ

∇T =∂T

∂xex +

∂T

∂yey +

∂T

∂zez

��©

dT =∂T

∂xdx +

∂T

∂ydy +

∂T


= (∇T ) · (d~l) = |∇T | |d~l| cos θ



1�ÙµêÆÄ: § 1.2

§ 1.2 ��©©©�!FÝ

FÝ´�¥þ

∇T =∂T

∂xex +

∂T

∂yey +

∂T

∂zez

��©

dT =∂T

∂xdx +

∂T

∂ydy +

∂T


= (∇T ) · (d~l) = |∇T | |d~l| cos θ

AÛ¿Âµ



1�ÙµêÆÄ: § 1.2

§ 1.2 ��©©©�!FÝ

FÝ´�¥þ

∇T =∂T

∂xex +

∂T

∂yey +

∂T

∂zez

��©

dT =∂T

∂xdx +

∂T

∂ydy +

∂T


= (∇T ) · (d~l) = |∇T | |d~l| cos θ

AÛ¿Âµ

FÝ ∇T ��¼ê T ��CzÇ£��ê¤��§Ù��=�¼ê T ��CzÇ£��ê¤"



1�ÙµêÆÄ: § 1.2

�!�Î ∇



1�ÙµêÆÄ: § 1.2

�!�Î ∇FÝ�¤



1�ÙµêÆÄ: § 1.2

�!�Î ∇FÝ�¤

∇T =∂T

∂xex +

∂T

∂yey +

∂T

∂zez =

(ex

∂

∂x+ey

∂

∂y+ez

∂

∂z

)T



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´�Î¤



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´�Î¤

∇ = ex∂

∂x+ey

∂

∂y+ez

∂

∂z



1�ÙµêÆÄ: § 1.2

a'µ

~A a =⇒ ∇T FÝ (gradient)



1�ÙµêÆÄ: § 1.2

a'µ


~A · ~B =⇒ ∇ · ~v ÑÝ (divergence)



1�ÙµêÆÄ: § 1.2

a'µ



~A × ~B =⇒ ∇× ~v ^Ý (curl)



1�ÙµêÆÄ: § 1.2

a'µ



~A × ~B =⇒ ∇× ~v ^Ý (curl)

n!ÑÝ



1�ÙµêÆÄ: § 1.2

a'µ



~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



1�ÙµêÆÄ: § 1.2

o!^Ý



1�ÙµêÆÄ: § 1.2

o!^Ý

∇× ~v =(ex

∂

∂x+ey

∂

∂y+ez

∂

∂z

)× (vx ex + vy ey + vz ez)



1�ÙµêÆÄ: § 1.2

o!^Ý

∇× ~v =(ex

∂

∂x+ey

∂

∂y+ez

∂

∂z


=

∣∣∣∣∣∣∣ex ey ez∂∂x

∂∂y

∂∂z

vx vy vz

∣∣∣∣∣∣∣



1�ÙµêÆÄ: § 1.2

o!^Ý

∇× ~v =(ex

∂

∂x+ey

∂

∂y+ez

∂

∂z


=

∣∣∣∣∣∣∣ex ey ez∂∂x

∂∂y

∂∂z

vx vy vz

∣∣∣∣∣∣∣= ex

(∂vz

∂y− ∂vy

∂z

)+ ey

(∂vx

∂z− ∂vz

∂x

)+ ez

(∂vy

∂x− ∂vx

∂y

)



1�ÙµêÆÄ: § 1.2

o!^Ý

∇× ~v =(ex

∂

∂x+ey

∂

∂y+ez

∂

∂z


=

∣∣∣∣∣∣∣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



1�ÙµêÆÄ: § 1.2

o!^Ý

∇× ~v =(ex

∂

∂x+ey

∂

∂y+ez

∂

∂z


=

∣∣∣∣∣∣∣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



1�ÙµêÆÄ: § 1.2

o!^Ý

∇× ~v =(ex

∂

∂x+ey

∂

∂y+ez

∂

∂z


=

∣∣∣∣∣∣∣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



1�ÙµêÆÄ: § 1.2

o!^Ý

∇× ~v =(ex

∂

∂x+ey

∂

∂y+ez

∂

∂z


=

∣∣∣∣∣∣∣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



1�ÙµêÆÄ: § 1.2

∇r =~r

r= er,



1�ÙµêÆÄ: § 1.2

∇r =~r

r= er, ∇f(u) =

df

du∇u



1�ÙµêÆÄ: § 1.2

∇r =~r

r= er, ∇f(u) =

df

du∇u

∇ · ~A(u) = (∇u) · d ~A(u)du

,



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



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



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



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



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



1�ÙµêÆÄ: § 1.2

8!¦È�FÝ!ÑÝ!^Ý



1�ÙµêÆÄ: § 1.2


∇ Q´¥þq´�5�Î



1�ÙµêÆÄ: § 1.2


∇ Q´¥þq´�5�Î

©�Çµ



1�ÙµêÆÄ: § 1.2


∇ Q´¥þq´�5�Î

©�Çµ

∇ (f + g) = ∇ f +∇ g

∇ · ( ~A + ~B) = ∇ · ~A +∇ · ~B

∇× ( ~A + ~B) = ∇× ~A +∇× ~B



1�ÙµêÆÄ: § 1.2


∇ Q´¥þq´�5�Î

©�Çµ

∇ (f + g) = ∇ f +∇ g

∇ · ( ~A + ~B) = ∇ · ~A +∇ · ~B

∇× ( ~A + ~B) = ∇× ~A +∇× ~B

XJ k ´~ê



1�ÙµêÆÄ: § 1.2


∇ Q´¥þq´�5�Î

©�Çµ

∇ (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


1�ÙµêÆÄ: § 1.2

¦È/ªµfg, ~A · ~B, f ~A, ~A × ~B



1�ÙµêÆÄ: § 1.2

¦È/ªµfg, ~A · ~B, f ~A, ~A × ~B

∇(fg)



1�ÙµêÆÄ: § 1.2

¦È/ªµfg, ~A · ~B, f ~A, ~A × ~B

∇(fg)k|^ ∇ ��Î£¦�¤A5§¦È�¦�©¤ü�¶

1. ∇ é f �^£¦�¤§g À�~þ§P� ∇f

2. ∇ é g �^£¦�¤§f À�~þ§P� ∇g



1�ÙµêÆÄ: § 1.2

¦È/ªµfg, ~A · ~B, f ~A, ~A × ~B

∇(fg) = ∇f(fg) +∇g(fg)k|^ ∇ ��Î£¦�¤A5§¦È�¦�©¤ü�¶

1. ∇ é f �^£¦�¤§g À�~þ§P� ∇f

2. ∇ é g �^£¦�¤§f À�~þ§P� ∇g



1�ÙµêÆÄ: § 1.2

¦È/ªµfg, ~A · ~B, f ~A, ~A × ~B

∇(fg) = ∇f(fg) +∇g(fg)k|^ ∇ ��Î£¦�¤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£� ∇ ��>



1�ÙµêÆÄ: § 1.2

¦È/ªµfg, ~A · ~B, f ~A, ~A × ~B

∇(fg) = ∇f(fg) +∇g(fg)k|^ ∇ ��Î£¦�¤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£� ∇ ��>



1�ÙµêÆÄ: § 1.2

¦È/ªµfg, ~A · ~B, f ~A, ~A × ~B

∇(fg) = ∇f(fg) +∇g(fg)k|^ ∇ ��Î£¦�¤A5§¦È�¦�©¤ü�¶

1. ∇ é f �^£¦�¤§g À�~þ§P� ∇f

2. ∇ é g �^£¦�¤§f À�~þ§P� ∇g


òÉ ∇ �^�¼ê£� ∇ �m>§ØÉ ∇ �^�¼ê£� ∇ ��>

∇(fg) = g∇f + f∇g



1�ÙµêÆÄ: § 1.2

¦È/ªµfg, ~A · ~B, f ~A, ~A × ~B

∇(fg) = ∇f(fg) +∇g(fg)k|^ ∇ ��Î£¦�¤A5§¦È�¦�©¤ü�¶

1. ∇ é f �^£¦�¤§g À�~þ§P� ∇f

2. ∇ é g �^£¦�¤§f À�~þ§P� ∇g


òÉ ∇ �^�¼ê£� ∇ �m>§ØÉ ∇ �^�¼ê£� ∇ ��>

∇(fg) = g∇f + f∇g

∇ · (f ~A)



1�ÙµêÆÄ: § 1.2

¦È/ªµfg, ~A · ~B, f ~A, ~A × ~B

∇(fg) = ∇f(fg) +∇g(fg)k|^ ∇ ��Î£¦�¤A5§¦È�¦�©¤ü�¶

1. ∇ é f �^£¦�¤§g À�~þ§P� ∇f

2. ∇ é g �^£¦�¤§f À�~þ§P� ∇g


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



1�ÙµêÆÄ: § 1.2

¦È/ªµfg, ~A · ~B, f ~A, ~A × ~B

∇(fg) = ∇f(fg) +∇g(fg)k|^ ∇ ��Î£¦�¤A5§¦È�¦�©¤ü�¶

1. ∇ é f �^£¦�¤§g À�~þ§P� ∇f

2. ∇ é g �^£¦�¤§f À�~þ§P� ∇g


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



1�ÙµêÆÄ: § 1.2

¦È/ªµfg, ~A · ~B, f ~A, ~A × ~B

∇(fg) = ∇f(fg) +∇g(fg)k|^ ∇ ��Î£¦�¤A5§¦È�¦�©¤ü�¶

1. ∇ é f �^£¦�¤§g À�~þ§P� ∇f

2. ∇ é g �^£¦�¤§f À�~þ§P� ∇g


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£� ∇ ��>



1�ÙµêÆÄ: § 1.2

¦È/ªµfg, ~A · ~B, f ~A, ~A × ~B

∇(fg) = ∇f(fg) +∇g(fg)k|^ ∇ ��Î£¦�¤A5§¦È�¦�©¤ü�¶

1. ∇ é f �^£¦�¤§g À�~þ§P� ∇f

2. ∇ é g �^£¦�¤§f À�~þ§P� ∇g


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£� ∇ ��>



1�ÙµêÆÄ: § 1.2

¦È/ªµfg, ~A · ~B, f ~A, ~A × ~B

∇(fg) = ∇f(fg) +∇g(fg)k|^ ∇ ��Î£¦�¤A5§¦È�¦�©¤ü�¶

1. ∇ é f �^£¦�¤§g À�~þ§P� ∇f

2. ∇ é g �^£¦�¤§f À�~þ§P� ∇g


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



1�ÙµêÆÄ: § 1.2

∇× (f ~A)



1�ÙµêÆÄ: § 1.2

∇× (f ~A)k|^ ∇ �¦�A5§¦�©¤ü�¶1. ∇ é f ¦�§ ~A À�~þ§= ∇f

2. ∇ é ~A ¦�§f À�~þ§= ∇A



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



1�ÙµêÆÄ: § 1.2


2. ∇ é ~A ¦�§f À�~þ§= ∇A

2|^ ∇ �¥þ$�A5¶~C × (f ~A) = − ~A × ( ~Cf)

= f( ~C × ~A)òÉ ∇ �^�¼ê£� ∇ �m>§ØÉ ∇ �^�¼ê£� ∇ ��>



1�ÙµêÆÄ: § 1.2


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£� ∇ ��>



1�ÙµêÆÄ: § 1.2


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



1�ÙµêÆÄ: § 1.2

∇ · ( ~A × ~B)



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



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



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£� ∇ ��>



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£� ∇ ��>



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)



1�ÙµêÆÄ: § 1.2

∇× ( ~A × ~B)

E��Æ ÔnX �� Mï� 10


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


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


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


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


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


1�ÙµêÆÄ: § 1.2

Ù¥

( ~B ·∇) =[(Bx ex + By ey + Bz ez) ·

(ex

∂

∂x+ey

∂

∂y+ez

∂

∂z

)]

E��Æ ÔnX �� Mï� 11


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


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


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


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


= 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


1�ÙµêÆÄ: § 1.2

∇( ~A · ~B)

E��Æ ÔnX �� Mï� 12


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


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


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


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


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


1�ÙµêÆÄ: § 1.2

Ô!��Î�^

E��Æ ÔnX �� Mï� 13


1�ÙµêÆÄ: § 1.2

Ô!��Î�^�U��¹µ

E��Æ ÔnX �� Mï� 13


1�ÙµêÆÄ: § 1.2

Ô!��Î�^�U��¹µ

FÝ�ÑÝ ∇ · (∇f)

E��Æ ÔnX �� Mï� 13


1�ÙµêÆÄ: § 1.2

Ô!��Î�^�U��¹µ

FÝ�ÑÝ ∇ · (∇f)

FÝ�^Ý ∇× (∇f)

E��Æ ÔnX �� Mï� 13


1�ÙµêÆÄ: § 1.2

Ô!��Î�^�U��¹µ

FÝ�ÑÝ ∇ · (∇f)

FÝ�^Ý ∇× (∇f)

}å©uIþ¼ê

E��Æ ÔnX �� Mï� 13


1�ÙµêÆÄ: § 1.2

Ô!��Î�^�U��¹µ

FÝ�ÑÝ ∇ · (∇f)

FÝ�^Ý ∇× (∇f)

}å©uIþ¼ê

ÑÝ�FÝ ∇(∇ · ~A)

E��Æ ÔnX �� Mï� 13


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


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


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


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


1�ÙµêÆÄ: § 1.2

Iþ�Î Laplacianµ∇2

∇2 = ∇ ·∇ =∂

∂x2+

∂

∂y2+

∂

∂z2

E��Æ ÔnX �� Mï� 14


1�ÙµêÆÄ: § 1.2


∇2 = ∇ ·∇ =∂

∂x2+

∂

∂y2+

∂

∂z2

∇2f =∂2f

∂x2+

∂2f

∂y2+

∂2f

∂z2

E��Æ ÔnX �� Mï� 14


1�ÙµêÆÄ: § 1.2


∇2 = ∇ ·∇ =∂

∂x2+

∂

∂y2+

∂

∂z2

∇2f =∂2f

∂x2+

∂2f

∂y2+

∂2f

∂z2Iþ�LaplacianE�Iþ

E��Æ ÔnX �� Mï� 14


1�ÙµêÆÄ: § 1.2


∇2 = ∇ ·∇ =∂

∂x2+

∂

∂y2+

∂

∂z2

∇2f =∂2f

∂x2+

∂2f

∂y2+

∂2f


∇2 ~A =(

∂2

∂x2+

∂2

∂y2+

∂2

∂z2


E��Æ ÔnX �� Mï� 14


1�ÙµêÆÄ: § 1.2


∇2 = ∇ ·∇ =∂

∂x2+

∂

∂y2+

∂

∂z2

∇2f =∂2f

∂x2+

∂2f

∂y2+

∂2f


∇2 ~A =(

∂2

∂x2+

∂2

∂y2+

∂2

∂z2


=(

∂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


1�ÙµêÆÄ: § 1.2


∇2 = ∇ ·∇ =∂

∂x2+

∂

∂y2+

∂

∂z2

∇2f =∂2f

∂x2+

∂2f

∂y2+

∂2f


∇2 ~A =(

∂2

∂x2+

∂2

∂y2+

∂2

∂z2


=(

∂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


1�ÙµêÆÄ: § 1.2


∇2 = ∇ ·∇ =∂

∂x2+

∂

∂y2+

∂

∂z2

∇2f =∂2f

∂x2+

∂2f

∂y2+

∂2f


∇2 ~A =(

∂2

∂x2+

∂2

∂y2+

∂2

∂z2


=(

∂2Ax

∂x2+

∂2Ax

∂y2+

∂2Ax

∂z2

)ex +

(∂2Ay

∂x2+

∂2Ay

∂y2+

∂2Ay

∂z2

)ey

+(

∂2Az

∂x2+

∂2Az

∂y2+

∂2Az

∂z2


∇2 ~A = (∇2Ax) ex + (∇2Ay) ey + (∇2Az) ez

E��Æ ÔnX �� Mï� 14


1�ÙµêÆÄ: § 1.2


∇2 = ∇ ·∇ =∂

∂x2+

∂

∂y2+

∂

∂z2

∇2f =∂2f

∂x2+

∂2f

∂y2+

∂2f


∇2 ~A =(

∂2

∂x2+

∂2

∂y2+

∂2

∂z2


=(

∂2Ax

∂x2+

∂2Ax

∂y2+

∂2Ax

∂z2

)ex +

(∂2Ay

∂x2+

∂2Ay

∂y2+

∂2Ay

∂z2

)ey

+(

∂2Az

∂x2+

∂2Az

∂y2+

∂2Az

∂z2


∇2 ~A = (∇2Ax) ex + (∇2Ay) ey + (∇2Az) ez ex, ey, ez �~¥þ"

E��Æ ÔnX �� Mï� 14


1�ÙµêÆÄ: § 1.2

FÝ�^Ýð�"µ

∇× (∇f) = (∇×∇)f = 0

E��Æ ÔnX �� Mï� 15


1�ÙµêÆÄ: § 1.2

FÝ�^Ýð�"µ

∇× (∇f) = (∇×∇)f = 0 Ál©þ/ª\±y²

E��Æ ÔnX �� Mï� 15


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


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


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


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


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


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


1�ÙµêÆÄ: § 1.2

^Ý�^Ý~^u½Â¥þ� Laplacianµ

∇× (∇× ~A) = ∇(∇ · ~A)− (∇ ·∇) ~A = ∇(∇ · ~A)−∇2 ~A

E��Æ ÔnX �� Mï� 16


1�ÙµêÆÄ: § 1.2


∇× (∇× ~A) = ∇(∇ · ~A)− (∇ ·∇) ~A = ∇(∇ · ~A)−∇2 ~A

|^µ ~B × ( ~C × ~A) = ~C( ~B · ~A)− ( ~B · ~C) ~A

E��Æ ÔnX �� Mï� 16


1�ÙµêÆÄ: § 1.2


∇× (∇× ~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


1�ÙµêÆÄ: § 1.2


∇× (∇× ~A) = ∇(∇ · ~A)− (∇ ·∇) ~A = ∇(∇ · ~A)−∇2 ~A

|^µ ~B × ( ~C × ~A) = ~C( ~B · ~A)− ( ~B · ~C) ~A


∇2 ~A = ∇(∇ · ~A)−∇× (∇× ~A)

∇2 ~A = (∇2Ax) ex + (∇2Ay) ey + (∇2Az) ez =é��I¤á"

E��Æ ÔnX �� Mï� 16


1�ÙµêÆÄ: § 1.2


∇× (∇× ~A) = ∇(∇ · ~A)− (∇ ·∇) ~A = ∇(∇ · ~A)−∇2 ~A

|^µ ~B × ( ~C × ~A) = ~C( ~B · ~A)− ( ~B · ~C) ~A


∇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


1�ÙµêÆÄ: § 1.2

l!~K

E��Æ ÔnX �� Mï� 17


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


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


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


r= ~A · (∇r)

E��Æ ÔnX �� Mï� 17


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


r= ~A · (∇r)

∇(~a · ~r)

E��Æ ÔnX �� Mï� 17


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


r= ~A · (∇r)

∇(~a · ~r) = ∇(axx + ayy + azz)

E��Æ ÔnX �� Mï� 17


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


r= ~A · (∇r)

∇(~a · ~r) = ∇(axx + ayy + azz) = ax ex + ay ey + az ez

E��Æ ÔnX �� Mï� 17


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


r= ~A · (∇r)

∇(~a · ~r) = ∇(axx + ayy + azz) = ax ex + ay ey + az ez = ~a

E��Æ ÔnX �� Mï� 17


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


r= ~A · (∇r)


∇ · (~a × ~r)

E��Æ ÔnX �� Mï� 17


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


r= ~A · (∇r)


∇ · (~a × ~r) = ∇r · (~a × ~r)|^ ~b · (~a × ~c) = −~a · (~b × ~c)

E��Æ ÔnX �� Mï� 17


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


r= ~A · (∇r)


∇ · (~a × ~r) = ∇r · (~a × ~r) = −~a · (∇× ~r)|^ ~b · (~a × ~c) = −~a · (~b × ~c)

E��Æ ÔnX �� Mï� 17


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


r= ~A · (∇r)


∇ · (~a × ~r) = ∇r · (~a × ~r) = −~a · (∇× ~r) = 0

E��Æ ÔnX �� Mï� 17


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


r= ~A · (∇r)


∇ · (~a × ~r) = ∇r · (~a × ~r) = −~a · (∇× ~r) = 0

∇× (~a × ~r)

E��Æ ÔnX �� Mï� 17


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


r= ~A · (∇r)


∇ · (~a × ~r) = ∇r · (~a × ~r) = −~a · (∇× ~r) = 0

∇× (~a × ~r) = ∇r × (~a × ~r)

E��Æ ÔnX �� Mï� 17


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


r= ~A · (∇r)


∇ · (~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


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


r= ~A · (∇r)


∇ · (~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


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


r= ~A · (∇r)


∇ · (~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


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


r= ~A · (∇r)


∇ · (~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


1�ÙµêÆÄ: § 1.2

∇ · [(~a · ~r)~r]

E��Æ ÔnX �� Mï� 18


1�ÙµêÆÄ: § 1.2

∇ · [(~a · ~r)~r]|^ ∇ · (f ~A) = (∇f) · ~A + f∇ · ~A

E��Æ ÔnX �� Mï� 18


1�ÙµêÆÄ: § 1.2

∇ · [(~a · ~r)~r] = [∇(~a · ~r)] · ~r + (~a · ~r)∇ · ~r

|^ ∇ · (f ~A) = (∇f) · ~A + f∇ · ~A

E��Æ ÔnX �� Mï� 18


1�ÙµêÆÄ: § 1.2

∇ · [(~a · ~r)~r] = [∇(~a · ~r)] · ~r + (~a · ~r)∇ · ~r

|^ ∇(~a · ~r) = ~a 9 ∇ · ~r = 3

E��Æ ÔnX �� Mï� 18


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


1�ÙµêÆÄ: § 1.2

∇ · [φ(r)(~a × ~r)]

E��Æ ÔnX �� Mï� 19


1�ÙµêÆÄ: § 1.2

∇ · [φ(r)(~a × ~r)]|^ ∇ · (f ~A) = (∇f) · ~A + f∇ · ~A

E��Æ ÔnX �� Mï� 19


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


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


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


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


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


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


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


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


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


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


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


|^ ∇f(u) =df

du∇u

E��Æ ÔnX �� Mï� 19


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


= − ~E0 × [cos(~k · ~r − ωt)]∇(~k · ~r)

|^ ∇f(u) =df

du∇u

E��Æ ÔnX �� Mï� 19


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


= − ~E0 × [cos(~k · ~r − ωt)]∇(~k · ~r)

|^ ∇(~k · ~r) = ~k

E��Æ ÔnX �� Mï� 19


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


= − ~E0 × [cos(~k · ~r − ωt)]∇(~k · ~r)

= ~k × ~E0 cos(~k · ~r − ωt)

|^ ∇(~k · ~r) = ~k

E��Æ ÔnX �� Mï� 19


1�ÙµêÆÄ: § 1.2

∇(

~a ·~r

r3

)

E��Æ ÔnX �� Mï� 20


1�ÙµêÆÄ: § 1.2

∇(

~a ·~r

r3

)|^� ~a �~¥þ�µ∇(~a · ~A) = (~a ·∇) ~A + ~a × (∇× ~A)

E��Æ ÔnX �� Mï� 20


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


1�ÙµêÆÄ: § 1.2

∇(

~a ·~r

r3

)= (~a ·∇)

(~r

r3

)+ ~a ×

(∇×

~r

r3

)

|^ ∇×~r

rn= 0

E��Æ ÔnX �� Mï� 20


1�ÙµêÆÄ: § 1.2

∇(

~a ·~r

r3

)= (~a ·∇)

(~r

r3

)+ ~a ×

(∇×

~r

r3

)= (~a ·∇)

(~r

r3

)|^ ∇×

~r

rn= 0

E��Æ ÔnX �� Mï� 20


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


1�ÙµêÆÄ: § 1.2

( ~A ×∇)× ~r

E��Æ ÔnX �� Mï� 21


1�ÙµêÆÄ: § 1.2

( ~A ×∇)× ~r =[ ~A × ∇ � k ©þ︷︸︸︷

εijk(Ai∂j) ek


×(xl el)

= εijk(Ai∂jxl)( ek × el)

E��Æ ÔnX �� Mï� 21


1�ÙµêÆÄ: § 1.2

( ~A ×∇)× ~r =[ ~A × ∇ � k ©þ︷︸︸︷

εijk(Ai∂j) ek


×(xl el)

= εijk(Ai∂jxl)( ek × el) |^

{∂jxl = δjl

ek × el = εklm em

E��Æ ÔnX �� Mï� 21


1�ÙµêÆÄ: § 1.2

( ~A ×∇)× ~r =[ ~A × ∇ � k ©þ︷︸︸︷

εijk(Ai∂j) ek


×(xl el)


{∂jxl = δjl

ek × el = εklm em

= εijk(Aiδjl)(εklm em)

E��Æ ÔnX �� Mï� 21


1�ÙµêÆÄ: § 1.2

( ~A ×∇)× ~r =[ ~A × ∇ � k ©þ︷︸︸︷

εijk(Ai∂j) ek


×(xl 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


1�ÙµêÆÄ: § 1.2

( ~A ×∇)× ~r =[ ~A × ∇ � k ©þ︷︸︸︷

εijk(Ai∂j) ek


×(xl el)


{∂jxl = δjl

ek × el = εklm em


{é l k¦Ú

�3 l = j �

= εijkAiεkjm em

E��Æ ÔnX �� Mï� 21


1�ÙµêÆÄ: § 1.2

( ~A ×∇)× ~r =[ ~A × ∇ � k ©þ︷︸︸︷

εijk(Ai∂j) ek


×(xl el)


{∂jxl = δjl

ek × el = εklm em


{é l k¦Ú

�3 l = j �

= εijkAiεkjm em ëY|^ εijk = −εikj

E��Æ ÔnX �� Mï� 21


1�ÙµêÆÄ: § 1.2

( ~A ×∇)× ~r =[ ~A × ∇ � k ©þ︷︸︸︷

εijk(Ai∂j) ek


×(xl el)


{∂jxl = δjl

ek × el = εklm em


{é l k¦Ú

�3 l = j �


= −[εijkεmjk]Ai em

E��Æ ÔnX �� Mï� 21


1�ÙµêÆÄ: § 1.2

( ~A ×∇)× ~r =[ ~A × ∇ � k ©þ︷︸︸︷

εijk(Ai∂j) ek


×(xl el)


{∂jxl = δjl

ek × el = εklm em


{é l k¦Ú

�3 l = j �


= −[εijkεmjk]Ai em [ ] S�Levi-Civita ÎÒ�ü¦Ú

E��Æ ÔnX �� Mï� 21


1�ÙµêÆÄ: § 1.2

( ~A ×∇)× ~r =[ ~A × ∇ � k ©þ︷︸︸︷

εijk(Ai∂j) ek


×(xl el)


{∂jxl = δjl

ek × el = εklm em


{é l k¦Ú

�3 l = j �



= −2δimAi em ÷v εijkεmjk = 2δim

E��Æ ÔnX �� Mï� 21


1�ÙµêÆÄ: § 1.2

( ~A ×∇)× ~r =[ ~A × ∇ � k ©þ︷︸︸︷

εijk(Ai∂j) ek


×(xl el)


{∂jxl = δjl

ek × el = εklm em


{é l k¦Ú

�3 l = j �




= −2Ai ei = −2 ~A

E��Æ ÔnX �� Mï� 21


1�ÙµêÆÄ: § 1.2

( ~A ×∇)× ~r =[ ~A × ∇ � k ©þ︷︸︸︷

εijk(Ai∂j) ek


×(xl el)


{∂jxl = δjl

ek × el = εklm em


{é l k¦Ú

�3 l = j �




= −2Ai ei = −2 ~A

( ~A ×∇) × ~r = −2 ~A

E��Æ ÔnX �� Mï� 21


1�ÙµêÆÄ: § 1.2

( ~A ×∇)× ~r =[ ~A × ∇ � k ©þ︷︸︸︷

εijk(Ai∂j) ek


×(xl el)


{∂jxl = δjl

ek × el = εklm em


{é l k¦Ú

�3 l = j �




= −2Ai ei = −2 ~A

( ~A ×∇) × ~r = −2 ~A ~A �±�?¿¥þ

E��Æ ÔnX �� Mï� 21


1�ÙµêÆÄ: § 1.2

( ~A ×∇) · ~r

E��Æ ÔnX �� Mï� 22


1�ÙµêÆÄ: § 1.2

( ~A ×∇) · ~r =[ ~A × ∇ � k ©þ︷︸︸︷

εijk(Ai∂j) ek


·(xl el)

= εijk(Ai∂jxl)( ek · el)

E��Æ ÔnX �� Mï� 22


1�ÙµêÆÄ: § 1.2

( ~A ×∇) · ~r =[ ~A × ∇ � k ©þ︷︸︸︷

εijk(Ai∂j) ek


·(xl el)

= εijk(Ai∂jxl)( ek · el) |^

{∂jxl = δjl

ek · el = δkl

E��Æ ÔnX �� Mï� 22


1�ÙµêÆÄ: § 1.2

( ~A ×∇) · ~r =[ ~A × ∇ � k ©þ︷︸︸︷

εijk(Ai∂j) ek


·(xl el)


{∂jxl = δjl

ek · el = δkl

= εijk(Aiδjl)δkl

E��Æ ÔnX �� Mï� 22


1�ÙµêÆÄ: § 1.2

( ~A ×∇) · ~r =[ ~A × ∇ � k ©þ︷︸︸︷

εijk(Ai∂j) ek


·(xl el)


{∂jxl = δjl

ek · el = δkl

= εijk(Aiδjl)δkl = εillAi = 0

( ~A ×∇) · ~r = 0

E��Æ ÔnX �� Mï� 22


1�ÙµêÆÄ: § 1.2

( ~A ×∇) · ~r =[ ~A × ∇ � k ©þ︷︸︸︷

εijk(Ai∂j) ek


·(xl el)


{∂jxl = δjl

ek · el = δkl


( ~A ×∇) · ~r = 0 ~A �±�?¿¥þ

E��Æ ÔnX �� Mï� 22


1�ÙµêÆÄ: § 1.2

( ~A ×∇) · ~r =[ ~A × ∇ � k ©þ︷︸︸︷

εijk(Ai∂j) ek


·(xl el)


{∂jxl = δjl

ek · el = δkl


( ~A ×∇) · ~r = 0 ~A �±�?¿¥þ

,y

E��Æ ÔnX �� Mï� 22


1�ÙµêÆÄ: § 1.2

( ~A ×∇) · ~r =[ ~A × ∇ � k ©þ︷︸︸︷

εijk(Ai∂j) ek


·(xl el)


{∂jxl = δjl

ek · el = δkl


( ~A ×∇) · ~r = 0 ~A �±�?¿¥þ

,y ( ~A ×∇) · ~r = ~A · (∇× ~r) = 0

E��Æ ÔnX �� Mï� 22


1�ÙµêÆÄ: § 1.2

( ~A ×∇) · ~r =[ ~A × ∇ � k ©þ︷︸︸︷

εijk(Ai∂j) ek


·(xl el)


{∂jxl = δjl

ek · el = δkl


( ~A ×∇) · ~r = 0 ~A �±�?¿¥þ

,y ( ~A ×∇) · ~r = ~A · (∇× ~r) = 0

( ~A ×∇) × ~r = ( ~A ×∇r) × ~r

E��Æ ÔnX �� Mï� 22


1�ÙµêÆÄ: § 1.2

( ~A ×∇) · ~r =[ ~A × ∇ � k ©þ︷︸︸︷

εijk(Ai∂j) ek


·(xl el)


{∂jxl = δjl

ek · el = δkl


( ~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


1�ÙµêÆÄ: § 1.2

( ~A ×∇) · ~r =[ ~A × ∇ � k ©þ︷︸︸︷

εijk(Ai∂j) ek


·(xl el)


{∂jxl = δjl

ek · el = δkl


( ~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


1�ÙµêÆÄ: § 1.2

( ~A ×∇) · ~r =[ ~A × ∇ � k ©þ︷︸︸︷

εijk(Ai∂j) ek


·(xl el)


{∂jxl = δjl

ek · el = δkl


( ~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


1�ÙµêÆÄ: § 1.2

( ~A ×∇) · ~r =[ ~A × ∇ � k ©þ︷︸︸︷

εijk(Ai∂j) ek


·(xl el)


{∂jxl = δjl

ek · el = δkl


( ~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)


= −2 ~A

E��Æ ÔnX �� Mï� 22


1�ÙµêÆÄ: § 1.2

( ~A ×∇) · ~r =[ ~A × ∇ � k ©þ︷︸︸︷

εijk(Ai∂j) ek


·(xl el)


{∂jxl = δjl

ek · el = δkl


( ~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)


= −2 ~A ( ~A ×∇) × ~r = −2 ~A

E��Æ ÔnX �� Mï� 22

classical electrodynamics - fudan...

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