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❉r❛❢t■♥st✐t✉t♦ P♦❧✐té❝♥✐❝♦ ◆❛❝✐♦♥❛❧❊s❝✉❡❧❛ ❙✉♣❡r✐♦r❞❡❋ís✐❝❛②▼❛t❡♠át✐❝❛s✏❚r✐♣❛s ❞❡ ❣❛t♦ ② ❛♥á❧✐s✐s ✈❡❝t♦r✐❛❧✑❆rt✉r♦❩úñ✐❣❛✲❙❡❣✉♥❞♦▼é①✐❝♦✱❈✐✉❞❛❞❞❡▼é①✐❝♦✱❛❜r✐❧❞❡❧✷✵✷✶❉r❛❢t❮♥❞✐❝❡❣❡♥❡r❛❧✶✳ ■♥tr♦ ❞✉❝❝✐ó♥ ✶✵ ✶✳✶✳ ❏✉❡❣♦❚r✐♣❛s❞❡❣❛t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✷✳ ❘❡♣r❡s❡♥t❛...

Description

 A  +B  A

o

 B

o

p(1, 2, 3) A = (1, 2, 3)

p(r, θ, z) aˆr aˆθ aˆz r θ z p(r, θ, ϕ) aˆr aˆθ r θ

aˆϕ ϕ

 = AB cos θ  •B A

A A

B  A

AB sin θ

θ

 B θ  B

A

B

 ×B  |= | A

 = (A, B, C) D  . ro = (xo , yo , zo ) r = ro + Dt x − xo = tA , y − yo = tB , z − zo = tC , t p Ax + By + Cz + D = 0 , A, B, C D po (xo , yo , zo ) D = −(Axo + Byo + Czo ) p(xo , yo , zo ) Ax + By + Cz + D = 0 , | Axo + Byo + Czo + D | √ d= A2 + B 2 + C 2 b a ab = a • ˆb ˆ ˆ ˆb(= b/b) b ab = (a •b)b a b

aˆr aˆx aˆy π/2

aˆz

θ π/2 − θ

aˆz aˆθ aˆx aˆy

aˆz

π/2 + θ θ

π/2

aˆz aˆr

aˆr

xy n ˆ sin θ n ˆ sin θ aˆx aˆz θ aˆθ θ A α β

γ

→

aˆy

ϕ

n ˆ π/2 − θ

π/2− ϕ

n ˆ n ˆ cos θ n ˆ cos θ aˆx aˆy aˆθ aˆz π/2 + θ aˆx aˆy aˆz

ϕ

A →

π/2 − ϕ

+q

ℓmn

ℓmn

−q

uvw vw nvw r

→

r

→

r′

q

→

x=a Rxy

Rxy x=b

y = f1 (x)

Rxy

b

Rxy

R

2

A = b /2 2

A = πR /4

Rrθ

Rxy

y = f2 (x)

r J = r 2 sin θ Ruv Ruv p Ruv 1 ≤ z ≤ 2 1 − x2 − y 2 x2 + y 2 ≤

Rxy Rxy Rxy

3 4

z r⊥ = r sin θ δ(x − x0 ) δ(r − R) δ(z − 0) δ(z − 0) δ(r − R)

R

R R

(a) (b) x=t b a → b

a

a

b

θ : 0 → 2π y = 1 − t2 t : 1 → −1 y=t t:0→1 3

(2, 0, π ) 0

y = t4 + t3 t : 0 → 1

x = cos(πt) y = sin(πt) z = 6t t t : 1/3 → 1 x = (1) cos θ y = √12 sin θ θ C1 : x(t) = t y = 0 t : −1 → 1 C2 : x(t) = t C1 : x(t) = t y = 0 t : 1 → 2 C2 : x(t) = 2 x = 1−cos t y = sin t a b

z = t3

(0, 0, 0) t

π x = t+1 y = t+1 z = t+1 (1, 1, 1) (2, 2, 2)

t:0→1

C x = 3t y = 4t z = t3 0 ≤ t ≤ 1 C P1 P2 P0 a

A →

C1

C3

a b a F →

b

b

· dr

F

→

→

C A →

C dS

C

→

dS

→

C

dr

→

n ˆ

dS x

y

C dS = dxdy aˆz →

dS = dxdy aˆz →

dxdy y=0 x=1

y = 1 − x2

x = −1

x = r cos θ y = 2r sin θ

θ : 0 → 2π

r:0→1 C

P1

P2

P0

dS xy xy

z = z(x, y)

S

V

dS →

r =1 −dydz aˆx

dS = dS aˆr →

dS aˆr

xz

xy

dS1 = −dxdy aˆz

dS3 = −dxdz →

xy

yz

→

r=1

dSˆ ar dS1 = dxdyˆ az →

dS = →

dS1 = −dxdy aˆz →

z = z = r−1

dS2 =

→ aˆy

θ : 0 → 2π

r = 1

z = ±1 C

√

1 − r2 dS = →

dS2 = dxdyˆ az →

P1

P2

P0 1/2

1

1

2

2

F

R

V

F F V

V

F

V V

u

v

u v u

V

V w

u+v =v+u. V

(u + v) + w = u + (v + w) . −u

V

u + (−u) = 0 .

V

0+u = u+0= u.

c

c(u + v) = cu + cv . (a + b)v = av + bv . (ab)v = a(bv) . u

V

1·u = u

Rn

1

n u = (u1 , u2 , · · · , un ) , n

Rn

V c1 , · · · , cn

u1 , · · · , un

u1 , · · · , un

V

c 1 u1 + c 2 u2 + · · · + c n un . V u1 , · · · , un 0

V F

F F

u1 , · · · , un

c 1 u1 + c 2 u2 + · · · + c n un = 0 . u1 , · · · , un V

u1 , · · · , un {u1 , · · · , un }

u1 , · · · , un V

u1 , · · · , un

n u

V {u1 , · · · , un }

V

V

u = c 1 u1 + c 2 u2 + · · · + c n un , c1 , · · · , cn

A A

|A|

A →

→

 A o

+ B  A

o

 B

o o o

A

B

→

→

A+B = B+A, →

→

→

→

A + (B + C ) = (A + B ) + C . →

u↔A

→

(x, y, z)

x y

→

→

→

→

→

v↔B w↔C →

→

z

x y (−∞, ∞)

A = (Ax , Ay , Az ) , →

Ax Ay

AZ

x y

z

z

p(1, 2, 3) x b(1, 2, 0)

y p(1, 2, 3)

z

a(1, 0, 0) o

p(1, 2, 3)

p(1, 2, 3)  = (1, 2, 3) A

z

p(1, 2, 3) c(0, 0, 3)

d(0, 2, 3)

y x

p(1, 2, 3)

A = (1, 2, 3) →

p(1, 2, 3) x

Ax = 1 y

Ay = 2

z A = (1, 2, 3) →

p(1, 2, 3)

Az = 3

p(1, 2, 3)

p(1, 2, 3) x

y z

aˆx , aˆy , aˆz , x y

z A →

A = Ax aˆx + Ay aˆy + Az aˆz , →

Ax Ay

= 1ˆ ax + 2ˆ ay + 3ˆ az ,

Az

{ˆ ax , aˆy , aˆz } ,

ˆ . {ˆi , ˆj , k}

q(−1, 4, −3) 3 −1 q(−1, 4, −3)

B →

ax + 4ˆ ay − 3ˆ az . B = −1ˆ →

C A+C →

→

C = B− A . →

A = (Ax , Ay , Az )

 = (Bx , By , Bz ) B

→

→

C

→

C = (Bx − Ax , By − Ay , Bz − Az ) , →

→

B →

p

x2 + y 2 ,  y θ = arctan , x z = z

r =

xy

0 ≤ r < ∞ 0 ≤ θ < 2π −∞ < z < ∞ r o p(r, θ, z) xy θ x r

p(r, θ, 0) z

p(r, θ, z) x

r

θ z

z

z=0 xy z =

(r, θ) x = r cos θ , aˆr aˆθ

aˆz

y = r sin θ ,

z=z.

p(r, θ, z) aˆr aˆθ

aˆz

r θ

z r θ

z aˆz aˆr θ r

θ

aˆθ

θ

θ + π/2 aˆz z

θ aˆx

aˆr aˆθ

aˆy x

y

aˆz A = Ar aˆr + Aθ aˆθ + Az aˆz , →

Ar Aθ

Az

r θ

ϕ

p(r, θ, ϕ) 0 ≤ r < ∞ 0 ≤ θ < π 0 ≤ ϕ < 2π r o z

p(r, θ, ϕ)

θ r

xy

x

z r

ϕ xy

p(r, θ, ϕ) aˆr aˆθ

aˆϕ

r θ

ϕ

x = r sin θ cos ϕ ,

y = r sin θ sin ϕ ,

aˆr aˆθ aˆϕ r θ ϕ

aˆr aˆθ

aˆϕ A = Ar aˆr + Aθ aˆθ + Aϕ aˆϕ , →

Ar Aθ

Aϕ

z = r cos θ .

z

r ϕ

θ

3 : 25

c c

A

→

B →

A •B = B•A →

→

→ →

= |A||B| cos θ = AB cos θ , → →

θ

A →

B →

A  • B = AB cos θ A A →

A

B

 B θ θ = π/2 = 90o

B →

A →

B

→

A=B →

→

2 2 A • A = A = |A| , →

→

→

aˆx • aˆy = aˆy • aˆz = aˆz • aˆx = 0 , aˆx • aˆx = aˆy • aˆy = aˆz • aˆz = 1 ,

A = Ax aˆx + Ay aˆy + Az aˆz →

B = Bx aˆx + By aˆy + Bz aˆz →

A • B = (Ax aˆx + Ay aˆy + Az aˆz ) • (Bx aˆx + By aˆy + Bz aˆz ) , →

→

= Ax Bx + Ay By + Az Bz ,

A →

|A| = →

q

A2x + Ay2 + A2z ,

A →

A×B →

→

B

→

A →

B →

C

→

 A

A

 B  |= AB sin θ | A × B

B

| A × B |= AB sin θ →

→

B × A = A × B →

A →

→

→

→

B → A×B →

→

θ

A × B = −B × A , →

→

→

→

A × A = −A × A = 0 →

→

aˆx × aˆy = aˆz ,

→

→

aˆy × aˆx = −ˆ az ,

aˆy × aˆz = aˆx , aˆz × aˆy = −ˆ ax , aˆz × aˆx = aˆy , aˆx × aˆz = −ˆ ay , aˆx × aˆx = aˆy × aˆy = aˆz × aˆz = 0 . A = Ax aˆx + Ay aˆy + Az aˆz →

B = Bx aˆx + By aˆy + Bz aˆz →

A × B = (Ax aˆx + Ay aˆy + Az aˆz ) × (Bx aˆx + By aˆy + Bz aˆz ) , →

→

= (Ay Bz − Az By ) aˆx + (Az Bx − Ax Bz ) aˆy + (Ax By − Ay Bx ) aˆz ,

D = (A, B, C) po (xo , yo , zo )

→

p(x, y, z) tD

D →

D →

D

→

t

t p

po

t = 0 p = po

→

 = (A, B, C) D  . r = ro + Dt ro = (xo , yo , zo ) x−xo = tA , y−yo = tB , z −zo = tC , t p r − r o = tD

→

→

→

r = r o + tD ,

→

→

→

r = (x, y, z)

ro = (xo , yo , zo ) →

D = (A, B, C) →

(x − xo , y − yo , z − zo ) = t(A, B, C) . x − xo z − zo y − yo =t, = = C B A

→

C = 0 (x − xo , y − yo , z − zo ) = t(A, B, 0) y − yo x − xo = = t , z = zo , A B t

xy   B y − yo = (x − xo ) . A

Ax + By + Cz + D = 0 , D D = −(Axo + Byo + Czo )

A, B, C po (xo , yo , zo )

p(x, y, z) E = (A, B, C) →

r

ro

→

E →

( r − ro) • E = 0 , →

→

→

→

r −ro

→

→

p (2,13) Ax + By + Cz + D = 0 , D = −(Axo + Byo + Czo ) A B C y

D

A B D

z

C

x

po = (xo , yo , zo ) D = −(Axo + Byo + Czo ) p(xo , yo , zo )

p(xo , yo , zo )

Ax + By + Cz + D = 0

Ax + By + Cz + D = 0 ,

| Axo + Byo + Czo + D | √ A2 + B 2 + C 2 q(x′ , y′ , z ′ ) p(xo , yo , zo ) p

q d=

p(xo , yo , zo )

p

(x′ − xo )2 + (y ′ − yo )2 + (z ′ − zo )2 . Ax + By + Cz + D = 0 (A, B, C) (A, B, C)

x − xo y − yo z − zo =t. = = C A B

d =

q(x′ , y′ , z ′ ) x′ − xo = At ,

y ′ − yo = Bt , z ′ − zo = Ct , Ax′ + By ′ + Cz ′ + D = 0 . q

d=

p

p

(x′ − xo )2 + (y ′ − yo )2 + (z ′ − zo )2 =| t |

t=−

d=

Axo + Byo + Czo + D , A2 + B 2 + C 2

| Axo + Byo + Czo + D | √ . A2 + B 2 + C 2

√

A2 + B 2 + C 2 .

a

AB

b

→

→

a

AB

b

→

→

b

→

a

→

a

b

→

→

b

ab = a • →

a

→

→

b

,

b

→

b b a = ab → = ( a • b) →2 , → → b →b b ˆb = b /b →

b

a b

ab = (a ˆb)bˆ •

A →

a

b ab = a • ˆb ˆb(= b/b)

b

→

A = Ax aˆx + Ay aˆy + Az aˆz →

x

→

aˆx

A

→

Ax

A aˆx

A • aˆx = (Ax aˆx + Ay aˆy + Az aˆz ) • aˆx = Ax , →

Ay = A • aˆy

Az = A • aˆz

→

→

A →

A = Ar aˆr + Aθ aˆθ + Az aˆz , →

Ar = A • aˆr = (Ax aˆx + Ay aˆy + Az aˆz ) • aˆr , →

Aθ = A • aˆθ = (Ax aˆx + Ay aˆy + Az aˆz ) • aˆθ , →

Az = A • aˆz = (Ax aˆx + Ay aˆy + Az aˆz ) • aˆz . →

Ar

aˆx • aˆr aˆy • aˆr

aˆz • aˆr

aˆr aˆx aˆy aˆz

aˆz

θ π/2 − θ

π/2

aˆr cos θ sin θ 0

•

aˆx aˆy aˆz

aˆθ − sin θ cos θ 0

aˆz 0 0 1

A aˆr aˆx • aˆr = (1)(1) cos θ = cos θ , θ aˆy • aˆr = (1)(1) cos(π/2 − θ) = sin θ , π/2 − θ cos(π/2) cos θ + sin(π/2) sin θ = sin θ

cos(π/2 − θ) =

aˆz • aˆr = (1)(1) cos(π/2) = 0 , aˆz

aˆr aˆθ aˆx aˆy

π/2 + θ θ

π/2

aˆx • aˆθ = cos(π/2 + θ) = − sin θ ,

aˆy • aˆθ = cos θ ,

aˆz • aˆθ = 0 . A →

Ar = Ax (r, θ, z ) cos θ + Ay (r, θ, z ) sin θ , Aθ = −Ax (r, θ, z ) sin θ + Ay (r, θ, z ) cos θ ,

Az = Az (r, θ, z ) . Ax Ay r θ

Az

z Ax = Ar (x, y, z) cos θ − Aθ (x, y, z) sin θ , Ay = Ar (x, y, z ) sin θ + Aθ (x, y, z ) cos θ , Az = Az (x, y, z ) .

aˆz

aˆx aˆy

aˆz

aˆθ π/2 + θ θ

π/2

aˆz

A →

A = Ar aˆr + Aθ aˆθ + Aϕ aˆϕ , →

Ar = A • aˆr = (Ax aˆx + Ay aˆy + Az aˆz ) • aˆr , →

Aθ = A • aˆθ = (Ax aˆx + Ay aˆy + Az aˆz ) • aˆθ , →

Aϕ = A • aˆϕ = (Ax aˆx + Ay aˆy + Az aˆz ) • aˆϕ . →

aˆr n ˆ π/2 − θ

xy

(ˆ ar • n ˆ )ˆ n=n ˆ cos(π/2 − θ) = n ˆ sin θ , xy aˆx • aˆr = aˆx • n ˆ sin θ = sin θ cos ϕ , • • aˆy aˆr = aˆy n ˆ sin θ = sin θ sin ϕ , aˆz • aˆr = cos θ ,

n ˆ sin θ aˆr

aˆz

aˆx

aˆy

ϕ

π/2 − ϕ

θ

aˆr n ˆ sin θ

xy n ˆ sin θ aˆx aˆy ϕ

n ˆ π/2 − θ aˆr

π/2 − ϕ

aˆz aˆθ

n ˆ θ (ˆ aθ • n ˆ )ˆ n=n ˆ cos θ , aˆθ

xy

aˆx • aˆθ = aˆx • n ˆ cos θ = cos θ cos ϕ , aˆy • aˆθ = aˆy • n ˆ cos θ = cos θ sin ϕ , aˆz • aˆθ = − sin θ , aˆθ

aˆz

n ˆ cos θ π/2 + θ

aˆx

aˆy

ϕ

π/2− ϕ

aˆϕ

θ

aˆr sin θ cos ϕ sin θ sin ϕ cos θ

•

aˆx aˆy aˆz

aˆθ cos θ cos ϕ cos θ sin ϕ − sin θ

aˆϕ − sin ϕ cos ϕ 0

A aˆθ aˆx • aˆϕ = − sin ϕ ,

aˆy • aˆϕ = cos ϕ , aˆz • aˆϕ = 0 .

aˆθ n ˆ cos θ

θ aˆθ

aˆz

π/2 + θ

n ˆ aˆx

aˆy

n ˆ cos θ ϕ π/2 − ϕ

A

→

Ar = Ax (r, θ, ϕ) sin θ cos ϕ + Ay (r, θ, ϕ) sin θ sin ϕ + Az (r, θ, ϕ) cos θ , Aθ = Ax (r, θ, ϕ) cos θ cos ϕ + Ay (r, θ, ϕ) cos θ sin ϕ − Az (r, θ, ϕ) sin θ , Aϕ = −Ax (r, θ, ϕ) sin ϕ + Ay (r, θ, ϕ) cos ϕ .

Ax Ay θ

Az

r

ϕ

Ax = Ar (x, y, z) sin θ cos ϕ + Aθ (x, y, z) cos θ cos ϕ − Aϕ (x, y, z) sin ϕ , Ay = Ar (x, y, z ) sin θ sin ϕ + Aθ (x, y, z ) cos θ sin ϕ − Aϕ (x, y, z) cos ϕ , Az = Ar (x, y, z ) cos θ − Aθ (x, y, z ) sin θ .

A = A1 aˆ1 + A2 aˆ2 + A3 aˆ3 , →

1, 2, 3

x, y, z

1, 2, 3 → x, y, z 1, 2, 3 → r, θ, z

aˆ1 • aˆ2 = aˆ2 • aˆ3 = aˆ3 • aˆ1 = 0 , aˆ1 • aˆ1 = aˆ2 • aˆ2 = aˆ3 • aˆ3 = 1 ,

aˆ1 × aˆ2 = aˆ3 ,

aˆ2 × aˆ1 = −ˆ a3 ,

aˆ2 × aˆ3 = aˆ1 , aˆ3 × aˆ2 = −ˆ a1 , aˆ3 × aˆ1 = aˆ2 , aˆ1 × aˆ3 = −ˆ a2 , aˆ1 × aˆ1 = aˆ2 × aˆ2 = aˆ3 × aˆ3 = 0 .

1, 2, 3 → r, θ, ϕ

A = A1 aˆ1 + A2 aˆ2 + A3 aˆ3

B = B1 aˆ1 + B2 aˆ2 + B3 aˆ3

→

→

A • B = (A1 aˆ1 + A2 aˆ2 + A3 aˆ3 ) • (B1 aˆ1 + B2 aˆ2 + B3 aˆ3 ) , →

→

= A1 B1 + A2 B2 + A3 B3 ,

A × B = (A1 aˆ1 + A2 aˆ2 + A3 aˆ3 ) × (B1 aˆ1 + B2 aˆ2 + B3 aˆ3 ) , →

→

= (A2 B3 − A3 B2 ) aˆ1 + (A3 B1 − A1 B3 ) aˆ2 + (A1 B2 − A2 B1 ) aˆ3 (2,29) |A| = →

q

A21 + A22 + A32 .

   Ar cos θ sin θ Aθ =  − sin θ cos θ Az 0 0    Ax cos θ − sin θ Ay  =  sin θ cos θ Az 0 0

A

A

  Ax =  Ay , Az   Ar = Aθ  , Az

(2,30)

At

At

  0 Ax 0  Ay 1 Az   0 Ar   0 Aθ 1 Az

  = Ax Ay Az ,   = Ar Aθ Az ,

A →



 cos θ sin θ 0 R = − sin θ cos θ 0 , 0 0 1

    1 0 0 cos θ − sin θ 0 Rt =  sin θ cos θ 0 , I =  0 1 0 , 0 0 1 0 0 1 t

A

=RA = Rt A

A

A A

= R Rt A = Rt RA

I

, .

= IA

= IA

=A

,

=A

,

R Rt = Rt R = R−1 R = I A = Ax aˆx + Ay aˆy + Az aˆz →

A • B = Ax Bx + Ay By + Az Bz , → →     Bx = Ax Ay Az  By  , Bz = At

B

= (RA

t

) RB

= At Rt RB

,     Br = Ar Aθ Az Bθ  , Bz = Ar Br + Aθ Bθ + Az Bz ,

R

B = Bx aˆx + By aˆy + Bz aˆz →

,

{ˆ ax , aˆy , aˆz }

{ˆ ar , aˆθ , aˆz }

R

(1, 0, 0)

x

z θ = π/6

R

    √3  cos θ sin θ 0 1 2  − sin θ cos θ 0 0 = − 1  , 2 0 0 1 0 0 

(1, 0, 0) Rt

θ = 30o

θ

z

A α β

γ

A = Ax aˆx + Ay aˆy + Az aˆz →

z

→

x y

aˆx aˆy

aˆz

z

α β

γ

Ax , + A2y + Az2 Ay , cos β = p 2 Ax + A2y + Az2 Az , cos γ = p 2 Ax + A2y + Az2

cos α = p

A2x

A

Az

→

cos2 α + cos2 β + cos2 γ = 1 .

Ax Ay

g(x)

dg/dx

∇· E = 4πρ , →

1 ∂ ∇× E = − B, → c ∂t →

∇·

∂/∂t

x g(x)

d/dx

∇×

∇· B = 0 , →

1 ∂ 4π ∇× B = E+ J. → c→ c ∂t →

t E →

B →

E = −∇ϕ− →

ϕ

1 ∂ A , B = ∇× A , → c ∂t → → A →

ϕ

E B A → →

→

∇ϕ

∇

ϕ ϕ 1 ∂2 ∇2 − 2 2 c ∂t

A →

!

ϕ = −4πρ ,

1 ∂2 ∇2 − 2 2 c ∂t

∇2 = ∇ · ∇

!

A= →

4π J . c → A2 = A • A → →

ρ ϕ

J

c

→

∇· A + →

4−

A →

1 ∂ ϕ=0, c ∂t

4−

z1

z2

zn

m

mgz

z g

z1 < z2 < · · · < zn

z1 mgz1 z2...