教科書 青山学院大学 理工学部 物理・数理学科 西尾研究室 denen09

2009

^✌

8

^✍

15

^✎

By I.Nishio

✏✞✑ ✒✔✓✖✕

✗✙✘✛✚✢✜✤✣✦✥★✧✦✩

Z

✪✬✫✮✭

S

A · ˆndS = ~

Z

S

^✯✱✰

^{∇ · ~}

AdV

✚✳✲✵✴✢✶✷✚✢✜✤✧✦✩

I

✪✸✫✮✹

c

A · ˆtds = ~

Z

c

^{✺✼✻✾✽❀✿❂❁}

✭

(∇ × ~ A) · ˆndS

∇ · ( ~ E × ~ ^{B) = ~} B · (∇ × ~ E) − ~ E(∇ × ~ ^B)

❃❅❄✦❆

✜✤❇✢❈❊❉●❋■❍✦❏❊✶❑✲❂▲

S = ~ ¹

µ 0

( ~ _{E × ~} B)

∇ · (∇ × A) = 0

∇ × (∇ × ~ A) = ∇ · (∇ · ~ A) − ∇ ^{2 A ~}

❏❊✶❑✲❂▲✢▼❅◆P❖✷◗❙❘✛▲❚✜✤✧✦❯

∇ × ~ ^{A = ~ B}

1

❲ ❳ ❨

❩ ✜

❃✦❄✦❬❪❭❅❫❵❴✙❛✛❜❵❝❊❞✠❡

A

^❢

❭■❣✷❤❪❭❥✐

✩✦❦

❭❵❧❥✐♥♠

✩

✐♣♦

✩

❭❅q❅r

✜ts❪✉❚✜✢✈●✇

❃❵❄✦❬❵❭■①t②❵③❊④⑥⑤⑧⑦■⑨✷⑩

❶✳❷

✜

⑩✦❸★❹❻❺❻❼■①❾❽❅❿■➀❾❽❚⑦✷➁

❉❙➂✛❍➄➃➆➅

①❑➇➉➈❊⑦✦➊★⑦■➋➍➌

✜ ➀ ✇

❤❅➎P➏➑➐

✜❚➒

⑦ ❶❙➓

✇

❤❅➎P➏➑➐

✜❚➒

⑦ ❡➔❃

❃➍r ➓ ❡➣→❅↔➍r ➓❊↕

✜t➙⑥✜

❭❙q

✜

❶✳❷ ❹★❞➜➛

❏❅▲

①❻➝

❍P➞

➋❾➟P➌❙➠

❈✙➡t➢✢➤❪➥

➠➑➟➦➠➨➧❙➝

✜ ➛

❏❅▲P❉➑➩

⑨t➊❅➫➄➭

❩

⑤➑❞✛➯✙⑦

❩

⑤⑧➀✢❞✤➲❚⑦

✜ ➀

✇❙➳

①✢➵

❈■➸⑥➺✷❉●➻

➠➼➊★⑦❙⑩❅④★➽★➋❊⑤✵➾❚➀

❍t➥

❩ ❈ ❞✤➚❊➪❪➀✢❞❅➵ ➃●➡t➢❙➤❻✇❪➶❊✜ ⑩✦❸➘➹➣➹➣➹

➅ ➫ ✇ ❩ ✜

❡➔❃❅❄❅❬✦❭❚➴❾➝➷➛

❏✦▲

❫✦❴❊❛✤❜❅❝✢r➍❞❻❺❻❼

✜❚✘❪➬✦➮t➱■✴★✃

❹❅➵❙➋➼❐★q✦➈⑥❹✛❒❅❮❰➠➼➊★⑦

➡✷❍t➥

2

❲ Ï Ð Ñ Ò Ó

2.1

ÔÖÕ✂×ÙØ✞Ú✔ÕÛÔÖÕ

2.1.1

^{ÜÞÝ➄ß✳à❰ß}

1

➎✦♦

✜❙á

♦

y = f (x)

✜✤â❅ã

➫

➒tä❵◆✢✴✢å❪✴✤æ➍ç

f (x + ∆x) = f (x) + f ^′ (x)∆x + ¹

2! ^f

′′ _(x)∆x 2 _{+ ··}

è

❹❻é❙⑦✦➊✢❞tê✙ë ✜✤ì✦í ① ➅ ⑦➦⑤ïî❚⑦ ➡✷❍✷➥✷ð➍❈✙ñ❙ò ⑨❻⑩❚ó ✲ ❞❅ô

♦

✩⑥✜✛õ

↔ ❶ ❹ ➩ ➇ ❈

⑩ö➇✵÷Þ➇

❈ ➋ ➓ ➃ ❶

➥❾õ■ø✤✇❵æ➍ç

➀❊ù❪➋❊⑤❵➠➼➊

f (a + x) = c 0 + c 1 x + c 2 x ² + c 3 x ³ + c 4 x ⁴ _{+ ··}

⑤✵➈✙⑦✦➊

✇

x = 0

⑤⑧➫❰➭❻⑤

f (a + 0) = c 0 + c 1 × 0 + c 2 × 0 + · · ·

= c 0

⑤ ➅

⑨t➊

c 0 = f (a)

①✷ú

➡➦➃●➡✷❍t➥❪û

❹✤ü✦ý

❉

x

➀

â❅ã

➠➼➊

f ^′ (a + x) = c 1 + 2c 2 x + 3c 3 x ² + 4c 4 x ³ _{+ · · ··}

❩❅❩

➀❾ü✦ý

x = 0

⑤⑧➫❰➭❻⑤

f ^′ (a + 0) = c 1 + 2c 2 _{× 0 + 3c} 3 × 0 + · · ·

= c 1

⑤ ➅

⑨t➊

c 1 = f ^′ (a)

①✷ú

➡➦➃●➡✷❍✷➥❻û

❹ ➡

⑩✤ü✦ý

❉

x

➀

â❅ã

➠➼➊

f ^′′ (a + x) = 2c 2 + 3 × 2c 3 x + 4 × 3c 4 x ² _{+ · · ·}

➡ ⑩

x = 0

⑤⑧➫★⑦✦➊

f ^′′ (a + 0) = 2c 2 + 3 · 2 · 0 + 4 · 3 · c ⁴ · 0 + · · ·

= 2 · c ²

➒❊➃

c 2 = ^{f ′′} ₂ ^(a)

❉➼þP➡✷❍t➥

ÿ⑥❹✤ü✦ý

❉

x

➀

â❅ã

➠➼➊

f ^′′′ (a + x) = 3 · 2 · 1 · c ³ + 4 · 3 · 2 · c ⁴ x + · · ·

➒❊➃

x = 0

⑤⑧➫★⑦✦➊

c 3 = ^{f ′′′} _3! ^(a)

❉ ➏ ❍ ❩

⑤⑧①❅➀❊ù ➡✷❍t➥

✂✁☎✄

➃✝✆

➠➼➀

c n = ^f

n (a)

n!

⑤ ➅ ➋ ❩

⑤⑧①

ã ➟

➃●➡✷❍t➥

➡ ⑩ ❩

✜✤æ➍ç➍✜✟✞

⑤✡✠☞☛

✞ ➀

f (a + x) = f (a) + f ^′ (a)x + ¹

2 ^f

′′ _(a)x 2 ₊ ¹

3! ^f

′′′ _(a)x 3

+ ··

➂ ➇➼❹

❩

✜✟✞ ➀

X = a + x

⑤⑧➫★⑦✦➊

f (X) = f (a) + f ^′ (a)(X − a) + ¹ ₂ ^{f ′′} (a)(X − a) ^{2 +} _3! ^{1 f ′′′} (a)(X − a) ³ + ··

➎✦♦

❉✟✌

❹✎✍❑➠➼➊

f (x) = f (a) + f ^′ (a)(x − a) + ¹ ₂ ^{f ′′} (a)(x − a) ^{2 +} _3! ^{1 f ′′′} (a)(x − a) ³ + ··

✏ ❹ ❩

✜✟✞➦❉

f (x)

✜

x = a

✜✎✑✳➃ ✜✤æ➍ç ⑤✓✒➍⑦ ➡✷❍t➥

❛✤❜

f (x) = 1 + 3x + 3x ² + x ³

❉

x = −1

✜✎✑✳➃

➀

3

ûP➡

➀

æ➍ç

➠

➅❚➂

⑦ ➥

ó ❖➦✲☞✔✕✞

f (x) = f (a) + f ^′ (a)(x − a) + ¹

2 ^f

′′ _{(a)(x − a)} 2 ₊ ¹

3! ^f

′′′ _{(a)(x − a)} 3 _{+ ··}

❹❪➫★⑦✦➊

a = −1

⑤⑧➫★⑦✦➊ ✇✗✖✗✘ ❉✚✙✗✛ ➠ ➅❚➂ ⑦ ➥

➡ ⑩ ✇

4

^û

✢✜

✜✤æ➍ç

❞✤✣✦✥

➅

➋❾➟★✧✦✩

➅❚➂

⑦ ➥

❛✤❜

2-1-1-1

✂✁ ✜❙á ♦ ❉✫✪★✧ ➂➑❈ ⑩✎✬ ✜✎✑✳➃ ➀

4

ûP➡ ➀ æ➍ç ➠ ➅❚➂ ⑦ ➥

(1)ln x(ln x

❞

log _e x

✜✎✭✯✮☎✰ ➀ ❍

)

❉

x = 1

✜✎✑✳➃ ➀ ➥

(2)sin x

❉

x = 0

✜✎✑✳➃

➀ ➥

(3)ln(1 − x)

❉

x = 0

✜✎✑✳➃ ➀ ➥

(4)tan x

^❉

x = 0

^{✜✎✑✳➃}

➀ ➥

2.1.2

^{✱➄Ý➄ß✳à❰ß}

z = f (x, y)

è ✜✟✲ é ✂✳ ✜ ➎✦♦ ✜❙á ♦ ❉✡✴ ➎✦♦ á ♦Þ⑤✓✒➍⑦ ➡✷❍✷➥☎✵✂✶➍➅✸✷ ⑤❵➠➼➊

z = exp(−x ² − y ^{2 )}

❉✡✹ ➊✦✺ ➡ ➠✯✻✗✥ ➥

-2

-1

0

1

2 ^-2

-1

0

1

2

-2

-1

0

1

2 ^-2

-1

0

1

2

✼ ✴☞✽

⑤❵➠➼➊✿✾❁❀★❂■➀

✹

➊★⑦➍➋❊⑤

3D

❹ ➅ ➃●➡✷❍t➥

❩

➤❙➅✸❃

☛➼➀

❍✷➥✤❄✚❅

➀ ❍

①❾➎✦♦❚①

3

✂✳

✜❙á

♦

❉❇❆

➀

❋■❍

❩

⑤➑❞❪➀❊ù

➡t➢✢➤✢✜

➀

✇❉❈★❊

➠●❋✙⑦

✜

➀☎❋❰➠❚➭

❃ ☛

➋❾➟⑥➌❙➠ ❈✙➡t➢❙➤❪➥

2.1.3

^{❍❏■▲❑}

❩

✜■➒ ✥ ➅✟✴ ➎❵♦ á ♦ ✜✛â✦ã ❞ ✇

2

^▼

✢✳ ✜ ➎❵♦ ❉ ✠❖◆★❹✟P➍➟P➠●➊⑥➠ ➡ ✥❵⑤❘◗✿❙❚①❵é❅➟ ➅ ➭ ➅ ➃ ➡t❍ ➟ ➇ ✇❚✖ ❷

✜

➎✦♦

✜ ✺ ❉

P■➟ö➠➼➊★✧✦✩➍➋ ❡

â❅ã❯✔❲❱❅â❅ã

r❙①❇❳✂❨

❈✙➡✷❍t➥❵✧✦❯

❞

,

✷

⑤❵➠➼➊

f (x, y)

❉

✧✦✩✦➊

∂f

∂x ^{= lim ∆x→0}

f (x + ∆x, y) − f(x, y)

∆x

❩ ❈ ❞

y

^❉

❡✧

♦❙r➜⑤❩✧★✩❵➊❇❬

➻ ❹

x

➀

â✦ã

➠✛⑩

✜

⑤❪❭✳➭✓✠❫☛⑧➀

❍ ➟ ➇

✇❁❴

❭✦↔❊❹❻➀❊➌➼➀✙ù❻➋

(

❤❪❭✦↔

➅

➇●④

❈ ❹

➀❊➌

)

^❩

⑤➼❹ ➅➜➃ ➡t❍t➥✿❱✦â✦ã❚✜✚❵✦í✙➅✎❛✗❜ ⑤❪➠●➊ ❝

❍❞■❡❑❣❢✐❤❦❥♠❧▲♥❣♦q♣❏rts★✉❣✈♠✇

①❙➵ ➃⑧➡t❍t➥✿✵✢✶❙➅

①✙ë❅❞✟②★③⑥❹❅➵ ➃●➡✷❍ ➟➜➇

❩❅❩

➀✢❞✸①✙ë❅❞❊➠ ➡t➢❙➤❪➥

✷ ❜

1 exp(−x ² − y ^{2 )}

❉

x

➀ ❱❅â❅ã ➠ ➅❚➂ ⑦ ➥

❩

✜✎✷

❹ á

➠➼➊✢❞

exp(−x ² − y ² ) = exp(−x ² ) × exp(−y ^{2 )}

⑤✵➈P➽

➡✷❍

➟➜➇

x

➀

❱❅â❅ã➍❍

➋❊⑤❾ù❾❹❅❞

exp(−y ^{2 )}

❞

✧

♦Þ⑤④✧✦✩ ➇

❈✙➡✷❍

➟➜➇

∂

∂x ^exp(−x

2 − y ^{2 )} = exp(−y ^{2 )} _∂x ^∂ exp(−x ^{2 )} = exp(−y ² ) × (−2x) exp(−x ^{2 )}

= −2x exp(−x ² − y ^{2 )}

❉➼þP➡✷❍✷➥

❛✤❜

2-1-3-1

✂✁

✜❙á

♦

❉✫✪★✧ ➂➑❈

⑩✤➎✦♦❚➀

❱❅â❅ã

➠

➅❚➂

⑦ ➥

(1)f (x, y, z) = px ² + y ² + z ²

❉

x

➀ ➥

∂f

∂x ⁼

(2)f (x, x ^′ , y, y ^′ , z, z ^′ ) = _{p(x − x} ^′ ) ² _{+ (y − y} ^′ ) ² _{+ (z − z} ^′ ) ²

^❉

x

➀ ➥

∂f

∂x ⁼

(3)f (x, x ^′ , y, y ^′ , z, z ^′ ) = _{p(x − x} ^′ ) ² _{+ (y − y} ^′ ) ² _{+ (z − z} ^′ ) ²

❉

x ^′

➀ ➥

∂f

∂x ^{′ =}

(4)f (x, x ^′ , y, y ^′ , z, z ^′ ) = px ^′2 _{+ (y − y} ^′ ) ² _{+ (z − z} ^′ ) ²

^❉

x

➀ ➥

∂f

∂x ⁼

(5)f (x, x ^′ , y, y ^′ , z, z ^′ ) = _{p(x − x} ^′ ) ² _{+ (y − y} ^′ ) ² _{+ (z − z} ^′ ) ²

❉

z ^′

➀ ➥

∂f

∂z ^{′ =}

(6)sin ^x+y ₂

❉

y

➀ ➥

∂

∂y ^sin

x+y

2
⁼

(7)cos(x ² + y ² )

^❉

x

➀ ➥

∂

∂x ^cos(x

2 _{+ y} 2 _)
=

(8)ln(x ² + y ² )

❉

y

➀ ➥

∂

∂x ^ln(x

2 _{+ y} 2 _)
=

⑤✗⑥★⑦✯⑧☎⑨★⑩✦⑦

❩❅❩

➀❖✒❦✥ ✧★❶❅ã❊✜✎❱❅â❅ã ⑤➑❞

I =

Z ∞

−∞

f (x, x ^′ )dx ^′

❹❻é❙⑦✦➊

∂I

∂x

❉ ✧✦✩ ➒ ✥✦⑤●⑦❷✥

❩

⑤⑧➀ ❍✷➥

➌❵⑨❚⑤❘❸★❹

➅

❆✦❺❊❹❅❞

I =

Z ∞

−∞

Z ∞

−∞

Z ∞

−∞

f (x, x ^′ , y, y ^′ , z, z ^′ )dx ^′ dy ^′ dz ^′

➅ ✣➑❹❻é❙⑦✦➊

∂I

∂x

❻

_∂I

∂y

❼

_∂I

∂z

❩ ➅ ➃●➡✷❍✷➥

❩ ❈ ✶❪✴✢❾■❖t✜ ➅ ✧★❶❅ã

E(x, y, z) = ~ ¹

4πε 0

Z ∞

−∞

Z ∞

−∞

Z ∞

−∞

ρ(x ^′ , y ^′ , z ^′ _{){(x − x} ^′ _{x + (y − y} ˆ ^′ )ˆ _{y + (z − z} ^′ )ˆ _z)}

{(x − x ^{′ ) 2} + (y − y ^{′ ) 2 + (z + z ′ ) 2} } ^{3 2 dx}

′ _dy ′ _dz ′

➀✿➀❫✩➜➇

❈ ➋ ❩ ⑤ ➅

✣✛❹

á

➠➑➊❻❤★➎

❵❙í

❹

➅❑➃●➡❾❍t➥

❩

✜■✶❵✴✂❾❚❖✷✜

❂✦❿⑥❞✸➁

(x, y, z)

❹✦➫■➽❙➋✛❃❅❆❊①❻❃★➂✢➃✦➄

ρ(x ^′ , y ^′ , z ^′ )

❉✡➅■➤ ④ ✧★❶❅ã❊✜✟✰ ➀✷➈P➽❅➋

❩ ⑤ ❉●❋ ➠●➊❅➫ ➃ ✇ ➠➼➟⑥➌ ↕ ✜✎➆❁➇ ①❇➁

(x, y, z)

✜❙á ♦⑥❹ ➅ ➋

(

❄➦➃❪➈ ➀

❍❁➉

)

➠✛⑩❾①■⑨✛➊

✇ ❩ ❈ ❞

❱✦â✦ã✙✜✟➊❁➋

❹ ➅ ➋ ❩ ⑤

❉⑧❋

➠●➊✦⑦

➡t❍t➥❙❏✙✶ ✲ ▲

④❊⑤✎➌

❖➎➍✾✶✡➏❚➐❅✜

➀✷❃❪❆

✜

x

^{➑ ã}

④★➽ ✜✟✞

(

✶❪✴✢❾■❖t✜ ❂★❿ ✜✟➒★➓✦➔

)

❹ ❍ ➋❊⑤

E x (x, y, z) = ¹

4πε 0

Z ∞

−∞

Z ∞

−∞

Z ∞

−∞

ρ(x ^′ , y ^′ , z ^′ _{)(x − x} ^′ )

{(x − x ^{′ ) 2} + (y − y ^{′ ) 2 + (z + z ′ ) 2} } ^{3 2 dx}

′ _dy ′ _dz ′

❍✢➅ ❨ ñ

I(x, y, z) =

Z ∞

−∞

Z ∞

−∞

Z ∞

−∞

f (x, x ^′ , y, y ^′ , z, z ^′ )dx ^′ dy ^′ dz ^′

✜✟✰➦❉

➠➼➊★⑦➍➋

❩

⑤⑧①

ã ➟

➃●➡✷❍✷➥

(

^➉

?)

❩

✜❙á

♦

✜✎❱❅â❅ã

❞

✇✗❶❅ã

①⑥➌➍⑤❻➌➍⑤Ù❡➣→❙r

(

^✎

②★❿■➀✢❞

➅ ⑦

➥❪➶❊✜

⑩✦❸

)

✜✎↔q↕

I(x, y, z) =

Z ∞

−∞

Z ∞

−∞

Z ∞

−∞

f (x, x ^′ , y, y ^′ , z, z ^′ )dx ^′ dy ^′ dz ^′ = lim

n→∞

n

X

i

f (x, x i , y, y i , z, z i )∆x i ∆y i ∆z i

➀✢➵❙➋

❩ ⑤ ❉ ✧✦✩➍➋❊⑤ ✇

∂I(x, y, z)

∂x ⁼

∂

∂x

(

n→∞ lim

n

X

i

f (x, x i , y, y i , z, z i )∆x i ∆y i ∆z i

)

➂

➇➼❹

✇

❡➣→

✜✤â❅ã

r❚❞ ❡

â❅ã❊✜

→❙r★➀

❍

➟➜➇

= lim

n→∞

n

X

i

∂

∂x ^{f (x, x i , y, y i , z, z i )∆x i ∆y i ∆z i}

=

Z ∞

−∞

Z ∞

−∞

Z ∞

−∞

∂

∂x ^{f (x, x}

′ _{, y, y} ′ _{, z, z} ′ _)dx ′ _dy ′ _dz ′

❝

❍❏■▲❑➛➙✯➜❡❑➝❢✤❧➞♥➠➟❦➡❞➢❞➤❣➥❣➦q➢❣➧➨✇

⑤●⑦❷✥

➆❁➇

❹ ➅ ➃●➡✷❍✷➥

➂ ➇➼❹ ➮q✽④➩✙❘☎➫❅◆ ✒❦✥✦⑤ ✇ ❃❅❄❅❬■➀✢❞ ✴ ➭ ✜ ❆✦❺★➭➜➌✟✧✦✩ ø ❹

❝

❍❏■➞❑❣➯♠➜❏❑➞➲❡➳▲❥▲➵➞♦♠➡❏➢➺➸➻s☞➼❦➥

_!!!

✇

➀

❍✷➥

✷ ❜

1

E x (x, y, z) = ¹

4πε 0

Z ∞

−∞

Z ∞

−∞

Z ∞

−∞

ρ(x ^′ , y ^′ , z ^′ _{)(x − x} ^′ )

{(x − x ^{′ ) 2} + (y − y ^{′ ) 2 + (z + z ′ ) 2} } ^{3 2 dx}

′ _dy ′ _dz ′

❉

x

➀

❱❅â❅ã

➠

➅❚➂

⑦ ➥

➽★➾

â❅ã ⑤ ❶❅ã❊✜✎➚★➪ ❞✎➶ ❈✿➹ ✩➍➋

❩

⑤⑧①❅➀❊ù❪➋❾➟➜➇

∂E x (x, y, z)

∂x ⁼

1

4πε 0

Z ∞

−∞

Z ∞

−∞

Z ∞

−∞

∂

∂x

 _ρ(x ′ _{, y} ′ _{, z} ′ _{)(x − x} ′ ₎

{(x − x ^{′ ) 2} + (y − y ^{′ ) 2 + (z + z ′ ) 2} } ^{3 2}



dx ^′ dy ^′ dz ^′

❩❅❩

➀

ρ(x ^′ , y ^′ , z ^′ )

❞

x

❹✎➘★➴❑➠

➅

⑦✦➟➜➇

✇❪â❅ã❊✜✎➷

❹❙➏

❍ ❩

⑤⑧①❅➀❊ù✤➊

= ¹

4πε 0

Z ∞

−∞

Z ∞

−∞

Z ∞

−∞

ρ(x ^′ , y ^′ , z ^′ ) ^∂

∂x

 (x − x ^{′ )}

{(x − x ^{′ ) 2} + (y − y ^{′ ) 2 + (z + z ′ ) 2} } ^{3 2}



dx ^′ dy ^′ dz ^′

❽❊❞

∂

∂x

 (x − x ^{′ )}

{(x − x ^{′ ) 2} + (y − y ^{′ ) 2 + (z + z ′ ) 2} } ^{3 2}



❉✚✙✗✛

➠➼➊

= ¹

{(x − x ^{′ ) 2} + (y − y ^{′ ) 2 + (z + z ′ ) 2} } ^{3 2 −}

3(x − x ^{′ ) 2}

{(x − x ^{′ ) 2} + (y − y ^{′ ) 2 + (z + z ′ ) 2} } ^{5 2}

➒

⑨t➊

∂E x (x, y, z)

∂x ⁼

1

4πε 0

Z ∞

−∞

Z ∞

−∞

Z ∞

−∞

ρ(x ^′ , y ^′ , z ^′ )

 1

{(x − x ^{′ ) 2} + (y − y ^{′ ) 2 + (z + z ′ ) 2} } ^{3 2}

− ^{3(x − x}

′ ₎ 2

{(x − x ^{′ ) 2} + (y − y ^{′ ) 2 + (z + z ′ ) 2} } ^{5 2}



dx ^′ dy ^′ dz ^′

⑤ ➅ ➋ ➥

✷ ❜

2

❡ ✗✙✘✛✚❇❶❅ã r

Z ∞

−∞ ^exp(−ax

2 _{)dx =} ^{r π}

a

✜

ü✦ý

❉

a

➀

❱❅â❅ã➍❍

➋ ❩

⑤➑❹

➒

⑨t➊

Z ∞

−∞

x ² _exp(−ax ² )dx

❉

ú❊❸

➒●➥

➽★➾

:

❡

✗✙✘✛✚❇❶❅ã

r

✜✟✞⑥✜ ü✦ý

❉

a

➀

❱❅â❅ã➍❍

➋❊⑤

∂

∂a

Z ∞

−∞ ^exp(−ax

2 _{)dx =} ^∂

∂a

r π

a

❩❅❩

➀✿➬❵ý ✜ ❡❶❅ã ⑤ ❱❅â❅ã r ✜✎➚★➮ö❉ ➶ ❈✿➹ ✩✦➊

Z ∞

−∞

∂

∂a ^exp(−ax

2 )dx = −

√ π

2a ^{2 3}

Z ∞

−∞ ^−x

2 exp(−ax ² )dx = −

√ π

2a ^{2 3}

➒ ⑨t➊

Z ∞

−∞

x ² _exp(−ax ² )dx =

√ _π

2a ^{2 3}

❉➼þP➡✷❍✷➥

❛✤❜

2-1-3-2 I 4 = ^R _−∞ ^∞ x ⁴ _exp(−ax ² )dx

❉ ú❊❸ ➅❚➂ ⑦ ➥

❛✤❜

2-1-3-3 I 2n = ^R _−∞ ^∞ x ²ⁿ _exp(−ax ² )dx

^❉

ú❊❸

➅❚➂

⑦ ➥

2.1.4

✱➄Ý➄ß✳à❰ß➞❥♠➱❣✃

✴ ➎✦♦ á ♦ ➌❒❐q❮t➎✦♦ ✜❙á ♦Þ⑤✡✠ ✉ ❹ æ➍ç❅❍ ➋ ❩

⑤⑧①❅➀❊ù ➡✷❍✷➥

f (x 0 +∆x, y 0 +∆y) = f (x 0 , y 0 )+ ^{ ∂f}

∂x







 ₀

∆x + ^∂f

∂y







 ₀

∆y



+ ¹

2!

 ∂ ² f

∂x ²







 ₀

∆x ² + 2 ^∂

2

∂x∂y







 ₀

∆x∆y + ^∂

2 _f

∂y ²







 ₀

∆y ²



+···

+ ¹

n!



n C 0

∂ ⁿ f

∂x ⁿ







 0

∆x ⁿ + n C 1

∂ ⁿ f

∂x ⁿ⁻¹ ∂y







 0

∆x ⁿ⁻¹ ∆y + · · +nC k

∂ ⁿ f

∂x ^n−k ∂y ^k







 0

∆x ^n−k ∆y ^k _{+ · · +} n C n

∂ ⁿ f

∂y ⁿ







 0

∆y ⁿ



+··

⑤ ➅ ➃●➡✷❍✷➥

❩❅❩

➀

| 0

❞

(x 0 , y 0 )

❹❪➫➍➽★➋✚✬

❉✫❰

➪❊➠

➡✷❍t➥❻❍✢➅

❨ ñ

∂f

∂x





 ₀

❞

∂f

∂x

✜

(x 0

^Ï

y 0 )

❹❪➫➍➽★➋✚✬❊❹

➅ ➃●➡

❍✷➥

n

^{û❊✜☎Ð❖Ñ❁✘}

❞

1

n!

" _n

X

k=0

n C k

∂ ⁿ f

∂x ^n−k ∂y ^k







 0

∆x ^n−k ∆y ^k

#

➀

❍✷➥

2

➎✦♦❚➀

❩ ➤❙➅

➀ ❍

➟➜➇

✇

3

➎✦♦

✇

4

➎✦♦

✇

k

➎✦♦

✜☎Ð❖Ñ❁✘

❞✛ÿ⑥❹✦❸

➤

✣➍➭

➂ ➭ ➅ ➋ ✜ ➀

❩❅❩

➀✢❞

✭✯✮

➠ ➡

❍ ①

✇✦Ð❵û❊✜✚✘➦➡

➀✢❞

ì✦í❊➅★✜

➀

3

➎✦♦⑥❹❻é❙⑦✦➊

❩❅❩

❹✛➈✙⑦✦➊❅➫❊ù

➡✷❍✷➥

f (x 0 + ∆x, y 0 + ∆y, z 0 + ∆z) = f (x 0 , y 0 , z 0 ) + ^{ ∂f}

∂x







 ₀

∆x + ^∂f

∂y







 ₀

∆y + ^∂f

∂z ^∆z







 ₀



+ ¹

2!

 ∂ ² f

∂x ²







 ₀

∆x ² + ^∂

2 _f

∂y ²







 ₀

∆y ² + ^∂

2 _f

∂z ²







 ₀

∆z ² + 2

 ∂ ² f

∂x∂y







 ₀

∆x∆y + ^∂

2 _f

∂y∂z







 ₀

∆y∆z + ^∂

2 _f

∂z∂x







 ₀

∆z∆z



+(

^Ò✸Ó

⑧✟Ô

)

✷

❜Þ⑤❵➠➼➊

õ☞Õ

✣ ✜

exp(−(x ^{2 + y 2 ))}

^❉

^{(0, 0)}

^{❉❁Ö✚×}

❹

2

^ûP➡

➀

æ➍ç

➠➼➊✦✺

➡

➠✯✻✗✥

➥

❩❅❩

➀

ì✦í❊➅

➌ ✜ ❞

f (x, y) = exp(−(x ^{2 + y 2 ))}

⑤✡❮✦⑦❙⑩P⑤❾ù

f (x, y)

∂f

∂x = −2x exp(−(x ^{2 + y 2 ))}

∂f

∂y = −2y exp(−(x ^{2 + y 2 ))}

∂ ² f

∂x ² = −2 exp(−(x ^{2 + y 2 )) + 4x 2} exp(−(x ^{2 + y 2 ))}

∂ ² f

∂y ² = −2 exp(−(x ^{2 + y 2 )) + 4y 2} exp(−(x ^{2 + y 2 ))}

∂ ² f

∂x∂y = 4xy exp(−(x ^{2 + y 2 ))}

✜

(0, 0)

❹❪➫➍➽★➋✚✬■➀

❍t➥

f (0, 0) = 1

∂f

∂x = −2x exp(−(x ^{2 + y 2 ))}

➒ ⑨t➊

∂f

∂x





 0 ^{= 0}

∂f

∂y = −2y exp(−(x ^{2 + y 2 ))}

➒

⑨t➊

∂f

∂y





 0 ^{= 0}

∂ ² f

∂x ² = −2 exp(−(x ^{2 + y 2 )) + 4x 2} exp(−(x ^{2 + y 2 ))}

➒❊➃

∂ ² f

∂x ²





 0 ^{= −2}

∂ ² f

∂y ² = −2 exp(−(x ^{2 + y 2 )) + 4y 2} exp(−(x ^{2 + y 2 ))}

➒❊➃

∂ ² f

∂y ²





 0 ^{= −2}

∂ ² f

∂x∂y = 4xy exp(−(x ^{2 + y 2 ))}

➒❊➃

∂ ² f

∂x∂y





 0 ^{= 0}

❉➼þ ➋ ❩

⑤⑧①❅➀❊ù ➡✷❍t➥ ✂✳ ❉✦➡ ⑤➼❸❵➊ ✇ ✂✁ ✜ ❡æ➍ç➍✜✟✞ r❚❹✟Ø ➃

❩✎Ù

⑤

f (0 + x, 0 + y) = f (0, 0) + ^{ ∂f}

∂x







 ₀

x + ^∂f

∂y







 ₀

y



+ ¹

2!

 ∂ ² f

∂x ²







 ₀

x ² + 2 ^∂

2

∂x∂y







 ₀

xy + ^∂

2 _f

∂y ²







 ₀

y ²



+ (

^Ò✗Ó

⑧✿Ô

)

exp(−(x ^{2 + y 2 )) = 1} + [0 × x + 0 × y] + ¹ ₂ [−2 × x ² + 2 × 0 × xy − 2 × y ^{2 ) + (}

^Ò✗Ó

⑧✿Ô

)

= 1 − (x ^{2 + y 2 ) + (}

^Ò✗Ó

⑧✿Ô

)

❉➼þ

➋ ❩

⑤⑧①❅➀❊ù

➡✷❍✷➥

❩

✜✟✞

➀☎Ú✦Û❑➠✤⑩✂Ü

å★Ý

❞

-0.4

-0.2

0

0.2

0.4 ^-0.4

-0.2

0

0.2

0.4

-0.4

-0.2

0

0.2

0.4 ^-0.4

-0.2

0

0.2

0.4

❩

➤❙➅✸❃

☛➼➀

(0, 0)

✜✎✑✳➃

➀❇Þ❚⑦

Ð❖ß➦❉✡✹✢➢

➊★⑦➍➋

❩

⑤⑧①✗❨❙➟

➃●➡✷❍✷➥

(

✁ ✜

➁★à■➀☎á❚➟

❈

⑩❚➌

✜ ① ✳ ➀☎â❚ã■⑩✟Ú

Û ✞ ➀ ❍✷➥

)

❄✚❅

➀ ❍ ①

(0, 0)

➟➜➇❪ä

❈

➋❊⑤

-1.5 -2

-0.5 -1

0.5 0

1.5 1

-2 2

-1 -1.5

0 -0.5

1 0.5

2 1.5

✜✤✉

❹✟å❚①❾❤ ù⑥➭

➅

⑨t➊★⑦✙ù

➡✷❍✷➥

✏ ❹ ❩ ✜

❆✦❺❊❹❅❞

|x|

➓

|y|

①❾❤ ù⑥➭

➅

➋❊⑤

0

❹✸◗✗æ

❍ ➋ á ♦

❉➑✇❖ç

❹

✣✦✥■❍

➋ ✴❖✘★✞ ➀✎Ú★Û ➠●➊❅⑦❙➋ ✜ ➀ ❍ ➟➦➇ ❄✡❅ ⑤⑧⑦✗✩ ➡t❍t➥ ⑩✦④✙➠ æ✢ç✢✜✦Ö✡×❚✜ ➁

(0, 0)

✜ Ú❁è➍➀ ➒ ➭ Ð❇ß ➠●➊❅⑦✢➋

❩ ⑤

① ❵❅í ➀ ❍✷➥

❛✤❜

2-1-4-1

✂✁

✜❙á ♦ ❉✫✪★✧ ➂➑❈ ⑩✎➁ ✜✎✑✳➃ ➀

1

ûP➡ ➀ æ➍ç ➠ ➅❚➂ ⑦ ➥

❛✤❜

2-1-4-2

✂✁

✜❙á

♦

❉✫✪★✧ ➂➑❈

⑩✎➁

✜✎✑✳➃

➀

2

^ûP➡

➀

æ➍ç

➠

➅❚➂

⑦ ➥

❛✤❜

2-1-4-3

✂✁

✜❙á

♦

❉✫✪★✧ ➂➑❈

⑩✎➁

✜✎✑✳➃

➀

3

ûP➡

➀

æ➍ç

➠

➅❚➂

⑦ ➥

2.1.5

^é❞ê à❑ß➠❥♠■▲❑ìë➠í✐î➎❍❏■▲❑

_I

❩❅❩

➀☎ï❡✥

✜ ❞

f (r)

^ð★ð❇ñ

r = px ² + y ² + z ² )

➓

g(ρ)

^ð★ð❇ñ

ρ = ^p x ² + y ²

⑤●⑦■⑨❾⑩ ✰⑥✜❙á ♦❚➀ ❍t➥ ❃❅❄❅❬✦❭⑥❹❅❞

❩

✜✟✰⑥✜❙á ♦❚①✢⑩➜➭ ➂✛➤❇ò ❆❰➠ ➡t❍ ➟ ➇

❩❅❩

➀❖ó✿ô⑥❹❊➠●➊❅➫❅⑦✦➊❑➭ ④ ➂ ⑦ ➥

✏ ❹

✇❖õ★ö❫÷✦➅✎✰

➀✢❞

➅ ➭

φ(r)

^✜❚➒

✥

➅✎✰⑥✜❙á

♦

❉✡❱❅â❅ã

➠➼➊ ❡

â❅ã

❶★ø

✞❙✇✿❱❅â❅ã

❶★ø

✞ r

❉✫ù

➭

ì✦í

①✢➵

➃

➡✷❍ ➟➜➇ ✇ ❩

✜❚➒ ✥ ➅ ❡ûú ➋q÷✦➅✎✞

(?)

r ✜✤â❅ã ➓ ❱❅â❅ã ❹ ➅❻❈ ➊❰➭ ④ ➂ ⑦ ➥

➀✢❞

f (g(x, y))

^❉

x

➀

❱❅â❅ã

➠➼➊✦✺

➡

➠❚✻❖✥

➥

❩❅❩

➀❇ü

❰■❍

➋ ❩

⑤➑❞

f ())

❞❡ý✸➎✦♦

✜❙á

♦❚➀

✇

g(x, y)

❞ ✹

⑩P⑤⑧➫

➃❂✇

2

➎✦♦ ✜❙á ♦❚➀✢➵❙➋

❩

⑤⑧➀ ❍t➥

✷ ❜

1

➡✷øt✇✦✧✦❯

❹❊➠✤⑩❪①✙⑨t➊✷➈✙⑦✦➊✦✺➍➋❊⑤

∂f

∂x ^{= lim ∆x→0}

f (g(x + ∆x, y)) − f(g(x, y)

∆x

⑤ ➅ ➃●➡✷❍✷➥

❩❅❩

➀ ❡æ➍ç r ❉ ❳❏✥✦⑤

g(x + ∆x, y) = g(x, y) + ^∂g

∂x ^∆x

(

Ð❵ûP➡ ➀ ✜✤æ➍ç

)

⑤ ➅ ➃●➡✷❍t➥✙➡ ⑩ ❩

✜✎➆❁➇➦❉ ✂✁ ✜❚➒ ✥➼❹✛➈➜ù☎þ⑥➠➼➊

= g(x, y) + ∆g

^ð★ð❇ñ

∆g = ^∂g

∂x ^∆x

✜✎❱❅â❅ã❊✜✟✞ ❍

∂f

∂x ^{= lim ∆x→0}

f (g(x, y) + ∆g) − f(g(x, y)

∆x

⑤ ➅ ➃●➡✷❍✷➥

❩❅❩

➀

f (g(x, y) + ∆g)

❹❻é❙⑦✦➊⑥➌

1

➎✦♦

á ♦

f ()

^{✜✤æ➍ç✙❉}

❳❏✥✦⑤

= lim

∆x→0

f (g(x, y)) + f ^′ (g(x, y))∆g − f(g(x, y)

∆x

= lim

∆x→0

f ^′ (g(x, y))∆g

∆x

❩❅❩

➀

∆g

^✜✁✦ö

= _∂x ^∂g ∆x

^❉

Ø ➃ ❩✎Ù

⑤

= lim

∆x→0

f ^′ (g(x, y) ^∂g _∂x ∆x

∆x

= f ^′ (g(x, y)) ^∂g

∂x

❩✦❩

➀ ✇

g(x, y)

è

✜✚✰

①

õ❁öq÷

❹❵➵✦⑩❖✩➦➇

❈

➊❅⑦✢➋●❆★❺❚❹✦❞

✇ ❩ ✜✟❱✦â❵ãP❉✄✂

➴❙❹✚❀P⑨✛➊ ➭➆④

➂ ⑦

➥⑥➡

⑩

g()

①

3

➎✦♦

✜❙á

♦❚➀⑥➌✸✠

✉ ❹

✙✗✛

➀❊ù

➡✷❍✷➥

✷ ❜

2 f (ρ)

^❉

x

➀

❱❅â❅ã

➠

➅❚➂

⑦ ➥

❩❅❩

➀

ρ = px ² + y ² (

^{☎✁✆✞✝✠✟✠✡ ✜}

ρ)

❩❅❩

➀✢❞

➈❚✜✎✷

❜

1

✜✎➆❁➇➦❉

❳❏✥

➥ ↕

✜✎➆❁➇

➟➜➇

∂f

∂x ^{= f}

′ _(ρ) ^∂ρ

∂x

⑤ ➅ ➋ ✜ ❞❅ëö➇ ➟❅➀✢➵❙➋ ➥ ➠✤⑩❪①✙⑨t➊❇❽❊❞ ✇

∂ρ

∂x

❉✚✙✗✛

➀❊ù

❈■➸❚➒

⑦ ➥

∂ρ

∂x ⁼

∂

∂x

p x ² + y ² = ^∂

∂x ^(x

2 _{+ y} 2 ₎ ¹ 2 ₌ ¹

2 ^(x

2 _{+ y} 2 ₎ ¹ 2 −1 _(2x)

= ^x

px ² + y ^{2 =}

x

ρ

⑤ ➅ ➋❾➟➜➇ ✇✗➆❁➇ ❞

∂f

∂x ^{= f}

′ _(ρ) ^x

ρ ⁼

df

dρ

 x

ρ



⑤ ➅ ➋ ➥

❛✤❜

2-1-5-1r = px ² + y ² + z ²

⑤❵➠✤⑩P⑤❾ù

(1) ¹ _r

❉

x

➀

❱❅â❅ã

➠

➅❚➂

⑦ ➥

(2) _r ^{1 2}

❉

x

➀ ❱❅â❅ã ➠ ➅❚➂ ⑦ ➥

❛✤❜

2-1-5-2

^✷

❜

1

❹❻é❙⑦✦➊

∂f

∂y

❉

ú❊❸

➅❚➂

⑦ ➥

❛✤❜

2-1-5-3

✷ ❜

2

❹❻é❙⑦✦➊

∂f

∂y

❉ ú❊❸ ➅❚➂ ⑦ ➥

❛✤❜

2-1-5-4

✷ ❜

2

❹❻é❙⑦✦➊

∂ ² f

∂x ^{2 +}

∂ ² f

∂y ²

❉

ú❊❸

➅❚➂

⑦ ➥

ó ❖➦✲

:f ^′ (ρ) = _dρ ^df

➌

ρ

④★➽ ✜

1

➎✦♦ ✜❙á ♦❚➀✢➵❙➋❾➟➜➇ ✇ ❺ ➑ á ♦Þ⑤④✧✦✩✦➊ ❱❅â❅ã ①❅➀❊ù❪➋ ➥ ➌ ñ✠☛❾➤

f ^′ (ρ) = _dρ ^df

❉❙➂

➇➼❹

ρ

➀✿❬

➻ ❹ â❅ã

➠✤⑩❚➌

✜ ❞

f ^′′ (ρ) = ^d _dρ ^{2 f 2}

⑤✵➈✙⑦✦➊

➒ ⑦ ➥

❛✤❜

2-1-5-5 2

➎✦♦

✜❙á

♦

f ()

①

f (g(x, y), h(x, y))

⑤✵➈P➽★➋❊⑤❾ù

✇

∂f

∂x

❉

ú❊❸

➅❚➂

⑦ ➥

(

ó

❖➦✲

:

^✴

➎✦♦

á ♦

✜✤æ➍ç

❉

❳❏✥

)

❛✤❜

2-1-5-6 φ(r)

⑩❅④❊➠

r = px ² + y ² + z ²

❹❻é❙⑦✦➊

(1) ^∂φ _∂x

^❉

ú❊❸

➅❚➂

⑦ ➥

(2) ^∂ _∂x ^{2 φ 2}

❉ ú❊❸ ➅❚➂ ⑦ ➥

(3) _∂x ^{∂ 2 φ} 2 + ^∂ _∂y ^{2 φ} 2 + ^∂ _∂z ^{2 φ} 2

❉

ú❊❸

➅❚➂

⑦ ➥

(4) (3)

✜✎➆❁➇

①

1

r

d ²

dr ^{2 (rφ(r))}

⑤ è ➠➑⑦➦⑤

❉●❋

➠

➅❚➂

⑦ ➥

2.1.6

^é❞ê à❑ß➠❥♠❍❏■▲❑

_II

❺

➑✌☞

♦

(←

^{✍✏✎✒✑}

^!)

á ♦

✜✎❱❅â❅ã

I

➀✢❞

f (r)

è

✜☎Ð

➎✦♦

á ♦ ❉ ï✙⑦

➡

➠✤⑩❪①

❩❅❩

➀✢❞ ↕

❈❊❉✔✓✏✕

➠➼➊

f (r(x, y, z), θ(x, y, z), φ(x, y, z))

è ❉ ï✙⑦

➡✷❍✷➥

❩

✜✞✖

♦

❉➼✧✦❯

❹❊➠✤⑩❪①✙⑨t➊

x

➀

❱❅â❅ã➍❍

➋❊⑤

(

^{✧✦❯➦❉}

➈ ➭❻⑤

)

∂f

∂x ^{= lim ∆x→0}

f (r(x + ∆x, y, z), θ(x + ∆x, y, z), φ(x + ∆x, y, z)) − f(r(x, y, z), θ(x, y, z), φ(x, y, z))

∆x

⑤ ➅ ➃●➡✷❍✷➥

❩❅❩

➀ ✴ ➎✦♦ á ♦ ✜✤æ➍ç

(

Ð❵ûP➡ ➀

)

❉ ❳➦⑨t➊

r(x + ∆x, y, x) = r(x, y, z) + ^∂r

∂x ^∆x

θ(x + ∆x, y, x) = θ(x, y, z) + ^∂θ

∂x ^∆x

φ(x + ∆x, y, x) = φ(x, y, z) + ^∂φ

∂x ^∆x

❩

❈❊❉

✳

✜✎❱❅â❅ã❊✜✤✧✦❯⑥✜✟✞

❹✎ÿ★➶❑➠➼➊

= lim

∆x→0

f (r(x, y, z) + ^∂r _∂x ∆x, θ(x, y, z) + ^∂θ _∂x ∆x, φ(x, y, z) + ^∂φ _∂x ∆x) − f(r(x, y, z), θ(x, y, z), φ(x, y, z))

∆x

❩❅❩

➀ ➂ ➇➼❹

f (r + ∆r, θ + ∆θ, φ + ∆φ)

➌ æ➍ç ➠➼➊

f (r + ∆r, θ + ∆θ, φ + ∆φ) = f (r, θ, φ) + ^∂f

∂r ^{∆r +}

∂f

∂θ ^{∆θ +}

∂f

∂φ ^∆φ

➒ ⑨t➊

f (r + ∆r, θ + ∆θ, φ + ∆φ) − f(r, θ, φ) = ^∂f _∂r ^{∆r + ∂f} _∂θ ^{∆θ + ∂f} _∂φ ^∆φ

❩❅❩

➀

∆r = ^∂r

∂x ^∆x

è ❉ ÿ★➶❑➠➼➊ ✙✗✛■❍ ➋❊⑤

∂f

∂x ^{= lim ∆x→0}

∂f

∂r

∂r

∂x ^{∆x +}

∂f

∂θ

∂θ

∂x ^{∆x +}

∂f

∂φ

∂φ

∂x ^∆x

∆x

= ^∂f

∂r

∂r

∂x ⁺

∂f

∂θ

∂θ

∂x ⁺

∂f

∂φ

∂φ

∂x

❉➼þ

➋ ❩

⑤⑧①❅➀❊ù

➡✷❍✷➥

2-1-6-1

f (x, y, z)

❹❻é❙⑦✦➊

x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ

✜❙á✘✗

①✢➵❙➋❊⑤❾ù

(1) ^∂f _∂r

❉ ú❊❸ ➒●➥

(2) ^∂f _∂θ

^❉

ú❊❸

➒●➥

(3) ^∂f _∂φ

❉ ú❊❸ ➒●➥

❛✤❜

2-1-6-2

f (x, y, z)

❹❻é❙⑦✦➊

x = ρ cos φ, y = ρ sin φ, z = z

✜❙á✘✗ ①✢➵❙➋❊⑤❾ù

(1) ^∂f _∂ρ

❉

ú❊❸

➒●➥

(2) ^∂f _∂φ

❉ ú❊❸ ➒●➥

(3) ^∂f _∂z

^❉

ú❊❸

➒●➥

❩ ❈ ❹ á✷❍

➋❪❛✤❜⑥❞

2-3-4

❹❅➵

➃●➡✷❍➍✜

➀ ↕ ❩ ➀ ➓ ⑨t➊❰➭ ④ ➂ ⑦ ➥

2.2

❏✙✶ ✲ ▲ ❞➑❤➦ù ➂ ⑤✔✤➍ù ❉✄✥ ⑨t⑩✘✦■➀ ❍t➥★✧ ➓ ❡❃❪❆❙r ➓ ❡❄❵❆★r è ① ↕ ✜✟✷ ➀ ❍t➥❖Ð✿Ñ ❹ ❏✙✶➜✲ ▲ ❞✘✩✫✪t➀✭✬

➂➼❈❚➡t❍t➥❾❏✙✶ ✲ ▲

❞✎✾✗❀✞✮✗P✙❹

➒

⑨➑➊✦➫✭✯❚⑦❪①

❵❚➉

❺✢❨

➂✦➂✒✰

➋✙⑤tù

✇

↕✲✱

✲ é ✱✴✳✌✵✷✶✹✸ ❞✗✠❫☛★⑤

✹■➅■➂✒✰❚➡

✺ ➥ ➠✤⑩❪①✙⑨t➊ ✇

✁✻✱

❆ ✱

❭❚➊

✱✼✳✽✵✾✶✿✸

❞❁✠☞☛

✳✽✵✾✶✿✸

❹ ➅ ➃●➡ ✺ ➥

(

^❩

✱ ❆ ❞✸✾❁❀★❂■➀ ✼❁❀ ✽ ⑤❵➠➼➊ ✹ ➋❊⑤

3D

❹ ➅ ➃●➡ ✺

)

✳✽✵✾✶✿✸❂✱ →❊❞✤➁

➂✘✰

➋ ✳✽✵✾✶✿✸❂✱

õ ❹ ✇ ➁ ✺ ✳✽✵✾✶✿✸❂✱

✌❄❃ é■➽❻➊

✇

➏➑➐ ✳ ①✙⑨❻⑩✛➧★❅ ✱✼✳✽✵✾✶✿✸❂✱

✌

⑤❇❆★➁

❃✡➆■➤ ④✁✩❈✪

(

✳✽✵✾✶✿✸

)

❹ ➅ ➃●➡ ✺ ➥ ♦★❉

✱✼✳✽✵✾✶✿✸

❃ ➁❑➠❋❊✂❨ ➢ ➋❊⑤ ✇

✁✻✱

❆ ✱ ➒ ✥ ➅ ❡❃ ☛✤❹✷r ➅ ➃●➡ ✺ ➥

OP

D ^E D

B C

O

P

A

B

OP

E

C

O _A

P

❩

✱✁● ❺❅➁❑➠❋❊✂❨

➢ ➋

➚★➮❈❃

➎❉✩✦➊⑥➌

➆❁➇

①❾➎✂❨❑➇

➅ ⑦ ❩

⑤➼❞ ✁✻✱

❆

➟➜➇❾➌✷ëP➇ ➟❅➀ö➠❚✻✗✥

➥

(

^❩

✱ ✲ é ✱ ❆

❞■❍

Û

3D

➀ ✺ ➥❑❏ å★▲✏▼ ✹✏◆✣❖◗P✏❘✭❙❯❚ ✥✁❱ ❖ ❀✏❲✻❳ ❙❩❨ ù✼❬❑❭✼❪ ❘■❫ ❲

50

^❴

✂✳✻✱✁❵★❛ ❲❝❜❡❞❣❢✲❤ ❃✒✐❦❥✔❧✽❨ ✥

❱ ❖ ❀ ù❑❱ ✺ ❚

)

A ^A

C _C

B B

OP OP

E

D

E

D

O

P

O

P

✳✽✵✾✶✿✸❂✱ →✏♠✼♥ ✺

➚★➮✽♦❂♣

❜✔❳

❙✽q❣❨

❬

~ _{A + ~ B}

r

~ _{B + ~ A}

^{❃❯s❄t❋✉ ➁✏▼☎á} ❖❯❨❡✈✞✇❩①✄②✠③

♠✠▼✽④

◆❣⑤

✱■⑥✠⑦ à✏♠

⑧✏⑨ ♦ ◆★❙❣❧❣q❣❨ ❚

B _A

A ^B

B A + A + B

q✒✰✽❃✒❼❦❥❾❽ ✺ q❣❨✘♦❂♣ ❶ ◆✼❿✞➀ ✱✁➁❄✱✼✳✽✵✾✶✿✸❂✱■⑩ ❷◗♥✾❹❋➂❁➃✠➄ ❧✘➅✠➆✽♦■➇ ❜✔❳ ❙✽q❣❨ ♠✴➈ ❘✘✰ ❱ ✺ ❚

2.2.1

➉➋➊➍➌❑➎➐➏✣➑✣➒✣➓→➔↔➣✽➑✣➒✾↕➛➙

n

❉ ✱■✳❂✵➜✶➝✸

~

a 1 , ~ a 2 ,

^➞

, ~ a n

♦ ⑨ ❙❑◆✻⑤

✱✘➟❯➠✞➡ ➂ ♦ ⑨ ❙❑◆

c 1 a ~ 1 + c 2 a ~ 2 +

^➞

c n a ~ n = 0

♠✭➢❑➤ ✺ ❧ ✱ ♠

c 1 = c 2 =

➞

_{= c n = 0}

✱✁● ➂

P★➥

✱ ❨

④✔➦

n

❉ ✱✼✳✽✵✾✶✿✸

~

a 1 , ~ a 2 ,

^➞

, ~ a n

♠❑➧✲➨✠➩✞➤❂▼❁➫

❧✽❨✔❙✠❙

❱ ✺ ❚

➧✲➨✠➩✞➤❂▼✻❳

❙

✳✽✵

✶✿✸❂✱■➭ ❃ ➧✲➨✠➯✠➲✌❳ ✳✽✵✾✶✿✸ ❨ ✳✽✵✾✶✹✸ ❨✄❙✠❙ ❱ ✺ ❚

2

➨✻➳ ✱✁➵★➸ ▼❁❬✴➧✲➨✠➩✞➤✽❳ ✳✽✵✾✶✿✸ ❬✁➺✠➻

2

❉★➦

3

➨✻➳

▼❁❬✴➧✲➨✠➩✞➤✽❳ ✳✽✵✾✶✿✸ ❬✁➺★➻

3

❉✏▼ ✺

(

➨✻➳ ✱✁➁

❨✒s❄t

)

^❚

a a

b b

c c

q ✱✲➼ ❬

3

❉ ✱ ➧✲➨✠➩✞➤✽❳ ✳✽✵✾✶✿✸❂✱■➽ ▼

(3

➨✻➳❁▼✻❳ ❙❩❨

3

❉ ✱✼✳✽✵✾✶✿✸ ❬✴➧✲➨✠➯✠➲ ♦ ❳❣❶ ◆ ❹✴❱ ❙ ❱ ✺ ❲➜❜ ➽

♦❂♣

❶ ◆

3D

▼ ✺ ❚

)

^➾✞➚

➥

▼✴➈✣❹■➪ ✱ ♠ ✳✽✵✷✶✿✸

~b,~c

^{♠✞➶ ❧✁✈✲➹}

(

✱ ➧❑➘

)

▼❯➦ ✳✽✵✾✶✿✸

~a

❬ q ✱

✈✲➹

✱■➴

❃✔➷

❹

◆★❙ ❱ ✺ ❚✫q ✱ ➧✲➨✠➩✞➤✽❳

3

➬❄➮✼➱✽✃✾❐✿❒

♦✏❘ ❜ ♦ ❷✏❻❡➧

➬✠❮✌❰

◆★Ï❣❧✽❨

d d

P

S Q

Q

R

S

P

O O R

a a

b b

c

c

➮ ♣ ❻ ♦ ❳ ❥

❱✴Ð

❚✫q

➮ ➼ ▼❁❬

➱✽✃✾❐✿❒

_d ~

➮■Ñ✠Ò

_P

❲❝❜

➱✽✃✾❐✿❒

_~a

❨❡✈✞✇✽♦✞Ó

➟❈Ô✒Õ✽Ö Ð

❨❇×❣Ø

~b,~c

^{➮ ➶ ❧✒➹❄♦}

Ù ⑨ ❲ ❥ ❱✴Ð★❲❝❜ ⑤ ➮ ❏ Ò Ô✼➼

➮ ♣ ❻ ♦

Q

❨ ❹Ú➦

Q

❲❝❜

➱✽✃✾❐✿❒

_~b

♦✭✈✞✇✽♦

➱✽✃✾❐✿❒

_~c

❭ ♣✭Û✻⑤

➮✁Ü✠Ý

➟ ♦✞Þ✻➥

◆✻Ó

➟❈Ô✒Õ✽Ö ❹ ❏ Ò

_R

Ô ➦

Q

❲❝❜

➱✽✃✾❐✿❒

_~c

♦✭✈✞✇✽♦✞Ó

➟❈Ô✒Õ✽Ö ❹ ◆

➱✽✃✾❐✿❒

_~b

❨ ➮ ❏ Ò

_S

Ô✔ß✽à

❧❣q❣❨

♠✠▼✽④

❱✴Ð ❚❩q

➮

➡✞á ❲❝❜

OP = ~ ~ OR + ~ OR + ~ QP

❨ãâ✫➥ ➦

OR k ~c ~

RQ = ~ ~ _{OS k ~b}

QP k ~a ~

▼✞Ð★❲❝❜ ⑤✴ä❂å✴ä

➮■æ✞ç

❱✼▼✼è à ➪✴é ➽✠ê✞➁❩Ô

d c , d b , d a

❨ Ð ❧✽❨

d = d ~ a ~a + d b ~b + d c ~c

Ð❁❳❯➃✌ë

d − d ~ a ~a − d b ~b − d c ~c = 0

❨

❳❣❶

◆

~a,~b, ~c

❬✴➧✲➨✠➩✞➤❂▼✞Ð★♠

d ~

❱✼▼✼è à ➪

~a,~b, ~c, ~ d

❬✴➧✲➨✠➩✞➤❂▼❁❬✞❳

❙✽q❣❨

♠✼ì✏❲

❥

❱■Ð

❚✔í❄♦✲❙

❰ Ö ➦

3

➨

➳ ➮

❿✞➀

➮✼➱✽✃✾❐✿❒

❬✁î❈ï❑❳✼➧✲➨✠➩✞➤✽❳

3

ð✽➮✼➱✽✃✾❐✿❒❂➮

➟✞➠★➡ ➂ ❨ ❹ ◆✴â✫➥★❧❣q❣❨✘♦ ❳ ❥ ❱✴Ð ❚

ñ

➣❈ò

3

➒➐ó➐➏✾➉➋➊➍➌❑➎→ô

₃

õö➏✣➑✣➒✣➓÷➔ùøù➉➋➊➍➌✲➎ú➏✷û➐ü✣ý➐þ→ÿ✾➑✁✄✂✆☎✞✝✠✟

_!!

q ➮

➧✲➨✠➩✞➤✽❳

➱✽✃✾❐✿❒❂➮

➭ ❨ ❹ ◆ î❈ï✼▼✻❳ ❙

(

❙✠❙

❮☛✡

❳

)

➭❈Ô✌☞ Ù ❨ã➹☛✍ ▼✞Ð★❲❝❜❇➦✲➧✏✎ ♦ ❭✒✑ ❙✻♦✞Ó■❏ Ð ❧

✓✏✔

➱✽✃✾❐✿❒

₍

Ý ❘ ♠

1

^{➮✼➱✽✃✾❐✿❒}

)

^➮

➭❈Ô✌☞

Û

❱✴Ð

❚❈q

➮

➭❈Ô✖✕

➬❣➱✽✃✾❐✿❒✘✗

❨✔❙✠❙

❱✴Ð

❚✚✙✜✛

♠✣✢➛❻✥✤

❖ ➮ ✕

➬❣➱✽✃✾❐✿❒✘✗

❬✧✦✩★

✗ ❨ ❤ ❖✌✪

ê ❹ ◆★❙ ❱✴Ð ❚

2.3

✫✭✬✯✮✱✰✳✲✵✴✷✶✹✸ ✺✼✻

2.3.1

^{✽✜✾❩➎ ➌❀✿❂❁✜❃}

(XYZ

^{✿❂❁✜❃}

,

❄✼❅❆✿❂❁✜❃✼❇✳✾❉❈✼❊✜❋❂●■❍❏✿❂❁✜❃ ➣❆❑❆▲✁▼

₎

➏❖◆✼Pú➉ö➊

➌❑➎

Cartesian Coordinate ¹

❷✲❶❘◗❯❷❚❙

✓❱❯❲✕

➬❣➱✽✃✾❐✿❒✘✗❨❳

q

➮✏❩❭❬✼❒ ❐

✦✩★

✗✽➮

✕

➬❣➱✽✃✾❐✿❒✘✗❫❪

x

^{❴❄➮❛❵❄➮✧❜ Þ}

Ô

Þ❞❝

➪ ✓✏✔

➱✽✃✾❐✿❒

_{x ˆ}

y

^{❴❄➮❛❵❄➮✧❜}

Þ Ô Þ❞❝

➪ ✓✏✔

➱✽✃✾❐✿❒

_{y ˆ}

z

^{❴❄➮❛❵❄➮✧❜}

Þ Ô Þ❞❝ ➪ ✓✏✔

➱✽✃✾❐✿❒

_{z ˆ}

❡❣❢✥❤

➢❖✐

ä❦❥

Ð✒❧✚♠✩✦✩★

❴❘♥ Ó✧♦

❹✖♣

❝❱q❣q

◗

❡❣❢

x · ˆx = 1 ˆ

y · ˆy = 1 ˆ

z · ˆz = 1 ˆ

x · ˆy = ˆy · ˆx = 0 ˆ

y · ˆz = ˆz · ˆy = 0 ˆ

z · ˆx = ˆx · ˆz = 0 ˆ

♥ Ó ë

♦❛r ❡ ä❦❥ Ð✒❧

1

s✉t❛✈①✇✖②①③⑤④⑦⑥⑨⑧✥⑩⑨❶❸❷❺❹⑤❻✣❼❾❽✖✈✉✇➀❿✉➁✌➂❺➃➅➄➇➆➉➈✥➊

(United Federation of Planets)

➋❛➌➎➍⑤➏✧➐➒➑❚➓①➔

➆

➱✽✃✾❐✿❒❂➮

♦ ❳ ♥ ❥

➱✽✃✾❐✿❒❂➮❲➝✘➞

q ➥

➝❞➞

Ð q

➮✏♥✣➠✩➡❫❪ Ð✒❧

A

(A ,A ,A ) x y z

(0,0,A ) _z

x y

(A ,A ,0)

A

(A ,A ,A ) x y z

(0,0,A ) _z

(A ,0,0) _x (A ,A ,0) x y ^{(A ,0,0)} x

(0,A ,0) y ^y

z

(0,A ,0) y

x x

y

z

A = A ~ x x + A ˆ y y + A ˆ z z ˆ

➢✧➤

2-3-1-1

^{➱✽✃✾❐✿❒}

Ô ➢✠ì

❪✒➥

➈✣❹■➪➦◗✼④✔➦

⑤

➮❛➥

➈ ♥

➧✏➧✜➛✴❹ ❡ ❯

❝❨➨

◗ Ô ❹ à ➄❞❧

➩

~ _{A = A x ˆ x + A y ˆ y + A z ˆ z = a x ˆ x + a y ˆ y + a z ˆ z}

◗ â✫➥ ➪➦◗◗Ð q ◗

A x = a x , A y = a y , A z = a z

Ô ➈❣➄ Ö❘➫ ❝ ❧

➨ ➮

_{A x}

Ô

A ~

^➮

x

➢✠ì❂◗➀➭

❝❱❥ Ð❲❧

➢✧➤

2-3-1-2 ~ B = B x x + B ˆ y y + B ˆ z z ˆ

◗◗Ð

q

◗✼④

(1) ~ A + ~ B

Ô ➢✠ì Ô ✢❩❶❲♣

â✫➥

❧

(2) ~ _{A · ~} B

Ô ➢✠ì Ô ✢❩❶❲♣ â✫➥ ❧

(3) ~ _{A · ˆy}

Ô ➢✠ì Ô ✢❩❶❲♣

â✫➥

❧

(4) | ~ B| = ^{p ~} B · ~ ^B

Ô ➢✠ì Ô ✢❩❶❲♣ â✫➥ ❧

➱✽✃✾❐✿❒❂➮

➴ ➣

➢✠ì Ô ✢❩❶❯➪

➱✽✃✾❐✿❒❂➮

➴ ➣

➮❲➝✘➞➯❳

x × ˆx = 0 ˆ

y × ˆy = 0 ˆ

z × ˆz = 0 ˆ

x × ˆy = ˆz, ˆy × ˆx = −ˆz ˆ

y × ˆz = ˆx, ˆz × ˆy = −ˆx ˆ

z × ˆx = ˆy, ˆx × ˆz = −ˆy ˆ

Ô ✢ ❝ ➦ ➡ ➂✠ì❞↔☛↕✩➙ Ô ✢❩❶❲♣✘➲❏➳

➮ ➫ ❻✖➟

➝✘➞

✐ ä❦❥ Ð❲❧

A × ~ ~ ^{B = (A} x x + A ˆ y y + A ˆ z _{z) × (B} ˆ x x + B ˆ y y + B ˆ z z) ˆ

= A x _{x × (B} ˆ x x + B ˆ y y + B ˆ z z) + A ˆ y _{y × (B} ˆ x x + B ˆ y y + B ˆ z ˆ z) + A z _{z × (B} ˆ x x + B ˆ y y + B ˆ z z) ˆ

= A x _{x × (B} ˆ y y + B ˆ z ˆ z) + A y _{y × (B} ˆ x x + B ˆ z ˆ z) + A z _{z × (B} ˆ x x + B ˆ y y) ˆ

= A x _{x × B} ˆ y y + A ˆ x _{x × B} ˆ z z + A ˆ y _{y × B} ˆ x x + A ˆ y _{y × B} ˆ z ˆ z + A z _{z × B} ˆ x x + A ˆ z _{z × B} ˆ y y ˆ

= A x B y _{x × ˆy + A} ˆ x B z _{x × ˆz + A} ˆ y B x _{y × ˆx + A} ˆ y B z _{y × ˆz + A} ˆ z B x _{z × ˆx + A} ˆ z B y _{z × ˆy} ˆ

= A x B y _{z − A} ˆ x B z _{y − A} ˆ y B x z + A ˆ y B z x + A ˆ z B x _{y − A} ˆ z B y x ˆ

= (A y B z − A z B y )ˆ x + (A z B x − A x B z )ˆ y + (A x B y − A y B x )ˆ z

➢✧➤

2-3-1-2

^{➱✽✃✾❐✿❒}

A ^~

^{➵❏➸✏➺}

B ^~

^♥

_{A × ~} ^~ B

◗ Ó✧♦ ❹✖♣ ❝❱q❱➨ ◗ Ô ➈✣❹ ❯ ✐ ❝ ❧

➢✧➤

2-3-1-3

➲❏➳

➮✼➱✽✃✾❐✿❒

₃

ð✽➮

➭ ➟ ⑨ ❝

♣❯➦✠➧✲➨✠➩✞➤

❪ ❯ ❝

➭❈Ô✌➻ ❹ ❯ ✐ ❝ ❧

(1)

2.3.2

^{➼❖➽❉✿✠❁➾❃}

(

➼➦➚❂✿✠❁➪❃✠❇➜➼✆➚❂➶✠✿✠❁➪❃↔➣❖❑❉➹✞➘

₎

Cylindrical coordinate

➳ ➮ ➼ ➮ ➫ ❻✌➟■➦ ✔❘➴❁Ô

z

^❴

❡❖❢

➮❛➷✏➬

_{ρ = px}

2 _{+ y} 2

^{◗ ⑤}

➮✁Ò

◗

z

^❴

❡❖❢❺➮✘➱ ✐ ä❘q✌✃☛❐❯➹

(

➼❏❒

➮✁➾✲➚✌➮

➘✠ì

)

◗

x

^❴❄➮

❯ Ð ⑦

φ

^❮

➫

Û✻⑤

➮■Ò✽➮

_z

✦✩★

❪✒➥ Ð✚✦✩★

✗

Ô❞❰❛Ï ✦✩★

✗

₍

❰❀Ð ✦✩★

✗

₎

◗➀➭

❝❱❥

Ð❲❧✒♠✩Ñ ➁ ➮

➮❞➱✩Ò

❳

0 ≤ ρ < ∞)

(ρ, φ , z) R

(ρ, φ , z) ^Q

(ρ, φ , z) P

φ R

φ Q

φ P

φ P ^{φ Q} _{φ P} ^{φ Q}

φ R φ _R

x _{x x}

y

y

z z

Q

R

P

Q

R

P

(ρ, ^{, z)}

(ρ, ^{, z)}

(ρ, ^{, z)}

❹❑➪

♥

❶❲♣

❩❦❬✞❒ ❐

✦❫★

✗➯❪

_{(x, y, z)}

❪❞Ó❩❰

❢ ä✆q

Ò❖❳

❰✚Ï ✦❫★

✗➦❪❨❳

_{(ρ, φ, z)}

➨❱➨

❪

_{ρ = px}

2 _{+ y} 2 _{, φ =}

arctan ^{ x} _y ^

❥ ➪ ❳

_{φ = arcsin}

 x

ρ



^Ô

❳

_{φ = arccos}

 y

ρ



❪ ➦

z

^{❳✘Õ❨ÖØ×}

➟ ❯ ➛ ❥ Ð❲❧ í ➟ ❰❀Ï ✦❞★

✗❱❪

_{(ρ, φ, z)}

❪ Ó✌❰

❢ ä❦q

Ò❨❳✏❩❭❬✼❒ ❐

✦✩★

✗❫❪❏❳

(ρ cos φ, ρ sin φ, z)

◗ ❯ ➛ ❥ Ð❲❧

➢✧➤

2-3-2-1

^{Ù✣Ú❱❮}

➫ Û ➳ ➮

➼❱❒

➮✧Û

✐ ❯✧Ü✆Ý

❪

➈❣✐

ä ➪ Ò

_(2.0,

π

2 ^{, 3.0)}

Ô✌Þ❘ß ➟✽❹✖♣

(ρ, φ, z)

^♥

➲❏➳

➮✧×❫❪

➫ q

Ò

Ô✼➼❱❒ ➟ Ú ❹ ❯ ✐ ❝ ❧

(3

➨✻➳❘à✩➟✧á

❰ q ➫ ❻✖➟

!!)

2-3-2-2

^{❩❭❬✼❒ ❐ ✗✽➮ ♥ ➮ ❰ ❢ ä ❝❱q ⑤ ➮■Ò✽➮ ➮ ❝}

➨ ➮

✦✩★

✗✽➮

✕

➬❣➱✽✃✾❐✿❒✘✗❨❳

➳ ➮ ➼ ➮ ➫ ❻✖➟ ❯ ➛ ❥ Ð✒❧

z

z

z

z

ρ

ρ ρ

ρ ^ρ

φ

ρ

φ

z φ φ

φ φ

z

x _x

y y

z z

ã✩ä

ρ

➟ ⑥✩å ❹■➪ ✕

➬❣➱✽✃✾❐✿❒

_{ρ ˆ}

❳ ➦ ⑤

➮■Ò✽➮

_ρ

♥✒æ✠❮

Ð q ❜ Þ Ô Þ❞❝ ➪ ✓✏✔

➱✽✃✾❐✿❒

❧

⑦

φ

➟ ⑥✩å ❹■➪ ✕

➬❣➱✽✃✾❐✿❒

_φ ˆ

❳ ➦ ⑤

➮■Ò✽➮

_φ

♥✒æ✠❮

Ð q ❜ Þ Ô

Þ❞❝

➪ ✓✏✔

➱✽✃✾❐✿❒

❧

z

➟

⑥✩å

❹■➪

✕

➬❣➱✽✃✾❐✿❒

_{z ˆ}

❳ ➦ ⑤

➮■Ò✽➮

_z

♥✒æ✠❮

Ð q ❜ Þ Ô Þ❞❝ ➪

✓✏✔

➱✽✃✾❐✿❒

❧

❪

Ð✒❧❝❹■➪

♥

❶❲♣❯➦

⑤

➮■Ò✽➮

✔❦➴

(ρ, φ, z)

^ç✚è❭é

(x, y, z)

➟

➇✩ê

❹ ❥ Ð❲❧

(

ï❀ë

❪ Ð ♥

_{ˆ z}

❳✘ì

❝ ♣

!!!)

➢✧➤

2-3-2-3 ˆ ρ

Ô

ˆ

x, ˆ y.ˆ z

◗ ⑤

➮■Ò✽➮

✦✩★

(ρ, φ, z)

Ô ✢❩❶❲♣

â❣í

❯ ✐ ❝ ❧