0 8 cX S y ð ; c" e s ð ' [ A 0V Ä õ m Í W Ä Ò ÞX ì Äß Ã Å Ò ÷ »4 Æ U Ø Ó Þ

(1)

8 0 8 cX S y ð ; c" e s ð ' [ A 0V Ä õ m Í W Ä Ò ÞX ì Äß Ã Å Ò ÷ »4 Æ U Ø Ó Þ

*

× <r )^ ï B

^∗

3 l q" é ¶ @ / < Æ § F g Ó ü t o < Æõ , @ / 302-729 (2005¸ 2 Z 4 16{ 9 ~ Ã Î6 £ §)

ý

a³ ð> \ " f 7 ' K $ 3 É r < ÆÂ Ò_ Ó ü t o < Æ l í < Æ_ þ v õ & ñ \ " f © Ù þ d & h õ ] j_ . Ó ü t o < Æ r

Û ¼% 7 É r @ /Â Òì r { 9 & ñ ô Ç @ /g A$ í õ : £ ¤ Ã ºô Ç : £ ¤ s & h ì r í\ ¦ ° ú H ½ ¨ ¸s Ù ¼ Ð Õ ª r Û ¼% 7 ` ¦ © & h ] X

>

l Õ ü t H ý a³ ð> \ ¦ & ñ H כ s Ó ü t o ë H ] j\ ¦ Ø Ô> É Ò H \ P û Z ) a . 0 A& h Ó ü t o r Û ¼% 7 É r f

§ý a³ ð> \ " f ~ 1 > l Õ ü t ÷ & ¦ Û ¦ o t ë ß @ /Â Òì r > _ r Û ¼% 7 É r f §ý a³ ð> \ " f Ð H : £ ¤ Ã ºô Ç /

B

G ý a³ ð> \ " f & h ] X y l Õ ü t ÷ & ¦ ë H ] j ¸ ç ß é ß y Û ¦ 2 ; . Ä ºo H B Û ¼B jw ¨ 8 â \ " f @ /g A-q @ /g A Õ

ªÒ ¨_ 8 ú x 14- ý a³ ð> \ " f_ 7 ' > í ß õ s [ þ t ý a³ ðî r6 x` ¦ z ´] j §Ã º < Æ_ þ v õ & ñ \ & h 6 x H ~ ½ ÓZ O \ @ /

# \ Vr ü < < Êa ] jr % i .

PACS numbers: 01.40, 03.50.De

Keywords:BÛ¼Bjw, 7'K$3, ýa³ð¨8, 14-ýa³ð>

I. " e Â ] Ø

{ 9

ì ø Í& h Ü ¼ Ð Û ¼º ú < Ê É r 7 ' © É r f §ý a³ ð> \ " f ¼ # o

> l Õ ü t ÷ & ¦ s K l ¸ ~ 1 t ë ß , M : Ð H É r / B G ý

a³ ð> \ " f l Õ ü t H כ s s ` ¼ # o ¦ ¸y 9 ç ß é ß

½ +

É M : ´ ú § [1,2]. < ÆÂ Ò_ Ó ü t o < Æ_ þ v õ & ñ \ " f H 7 ' ,J $

"

f, 7 ' © , p ì r í ß \ ¦ s 6 xô Ç 7 ' K $ 3 1 p x Ó ü t o r Û ¼

% 7

_ l Õ ü t õ K $ 3 s & h Ü ¼ Ð ý a³ ð> \ " fÂ Ò' Ø ¦µ 1 Ï ¦ e

6 £ § Ü ¼ Ð ý a³ ð> \ @ /ô Ç l : r < Æ_ þ v s Á º% Á Ð ¸ ' ÷ &

#

Q 9 × æ כ ¹ > À Ò# Q 4 R ô Ç [3–6]. t F K õ ° ú s

l íõ < Æ < Æ_ þ v # | s Ô ¦ o ô Ç & ³z ´ 5 Å q \ " f z ´6 x$ í §

¹

¢

¤ s Ä º ÷ &# Q ô Ç H כ ¹' õ Aõ · ú § 4 1 p x Ü ¼ Ð # , § Ã

º < Æ_ þ v l í < Æ_ þ v õ & ñ É r ~ 1 > Ò q t| Ä Ì ¦ ´ òÖ ¦& h

$ í õ \ ¦ À 1 Ïo % 3 ¦ H $ í / å L < Ês < ÆÂ Ò_ l íõ < Æ

§¹ ¢ ¤_ d y ô Ç ë H ] j& h s . Õ ª Q §Ã º < Æ_ þ v \ @ /ô Ç _ 6

¤ õ _ t ë ß e , Ô ¦ o ô Ç # | 5 Å q \ " f ¸ % i 1 l x& h < Æ _

þ

v¨ 8 â ` ¦ ¸$ í # l í < Æ_ þ v` ¦ t ¦ ½ Ó_ $ í ` ¦ ¦ ª

H < Æ_ þ v ´ òõ \ ¦ Ñ ü t Ã º e H ~ ½ ÓZ O ` ¦ s 7 Hë H \ " f ] jr

¦ ô Ç . @ /Â Òì r_ Ó ü t o < Æ ¸[ þ t ¸ f §ý a³ ð>

É r ý a³ ð> [ þ t \ @ / # H e ¸ n q t · ú § . f §ý a³ ð>

É r ý a³ ð> \ ¦ 6 x½ + É M : & ñ S X ô Ç ý a³ ð Ã º[ þ t, ý

a³ ð> _ © Ã º[ þ t, ý a³ ð_ # 3 0 A, Jacobian, ¨ 8 í ß , ý a

³

ð ¨ 8 / B Nd [ þ t` ¦ S X z ´ > · ú H è ß ' a \ Â Òv 9 u

>

) a . Ä ºo Ó ü t o < Æ ¸[ þ t É r S X z ´ô Ç t d ` ¦ ° ú ¦ e t · ú §

∗E-mail: [email protected]

8 ¸ ë H ] jK \ ¸¹ ¡ §` ¦ ~ Ã Î` ¦ Ã º e H ¸½ ¨[ þ t` ¦ a ¦

6 x ¦ e . ¸Z þ t± ú Ä ºo H s p ¦ ¸ Ð µ 1 Ï² ú ) a ( É Ó '

r Û ¼% 7 s ° ú Æ Ò# Q ¨ 8 â \ " f Ò q t Ö ¸ ¦ e . s Qô Ç

¨ 8

â ` ¦ & h ] X y Ö ¸6 x < H ) a < Æ_ þ v l r\ ¦ ÐØ æ ½ + É Ã º e

Ü ¼ 9 Ä »6 xô Ç í ß ` ¦ ° ú Ø ¦ Ã º ¸ e . Õ ª Qô Ç í ß ` ¦ ° ú

>

) a Ó ü t o < Æ ¸[ þ t É r Â Ò{ Û ¼ Qî r \ O ¸ _ ü @ Ð ~ 1 > K

½ + É Ã º e l M :ë H s . l ñ> í ß õ Ã ºu > í ß s Ã Ì Z 4ô Ç B

Û ¼B jw r Û ¼% 7 É r # Qt 4 ¤¸ ú ô Ç ý a³ ðr Û ¼% 7 \ " f 7 ' > í ß õ ý a³ ð> î r6 x \ @ /ô Ç Ä »6 xô Ç ¸½ ¨[ þ t` ¦ ] j/ B N

#

§Ã º ü < < Æ_ þ v \ > ¼ # o ¦ % i 1 l x& h < Æ_ þ v¨ 8 â ` ¦ ]

j/ B Nô Ç [7,8]. l ñ í ß õ < Êa ý a³ ðî r6 x \ ¸¹ ¡ §` ¦ Å Ò

H ¸½ ¨[ þ t s ´ ú §s · ú 9$ e t ë ß , Ó ü t o < Æõ © 2 ; o& h

á Ô ÐÕ ªÏ þ # Q Ð B Û ¼B jw Ã Ì Z 4ô Ç r Û ¼% 7 s H

כ

É r ¸ ú · ú 9 z ´s [12–15]. : r 7 Hë H \ " f H B Û ¼ B

jw r Û ¼% 7 \ " f 14-ý a³ ð> \ " f_ l : r& h 7 ' K $ 3 õ

¼ # o ô Ç ý a³ ð> î r6 x ¸½ ¨[ þ t` ¦ Ö ¸6 x # Ó ü t o < Æ r Û ¼

% 7

` ¦ l Õ ü t K H ý a³ ð> î r6 x \ @ / # y ` ¦ ° ú

¸2 ¤ Õ ª_ Ö ¸6 x~ ½ ÓZ O ` ¦ l Õ ü t ¦ 7 H_ ô Ç .

II. 8 0 8 cX 8 ý 14-Ò ÷ »4 S y ð ; c" e s ð

' [A 0V Ä õ m Í Ò ÷ »4 Æ U Ø Ó Þ8 ý ÷ m Ç< g

Helmholtz ~ ½ Ó& ñ d s © p ì r~ ½ Ó& ñ d Ü ¼ Ð ì r o | ¨ c M : @ /

g AÕ ªÒ ¨ \ 5 Å q H 11 > h_ ý a³ ð> & ñ _ ) a . ¢ ¸ q @ /

g A Õ ªÒ ¨_ Confocal Ellipsoidal, Confocal Paraboloidal,

-134-

(2)

Conical ý a³ ð> 1 p x 3 > h_ ý a³ ð> \ ¦ ½ + Ë , Ó ü t o < Æ r Û ¼

% 7

` ¦ l Õ ü t l 0 Aô Ç 14> h_ ý a³ ð> [ O & ñ ) a [5,6,8].

Õ

ª X < s 14> h_ ý a³ ð> H f §ý a³ ð> ü < # Q " ¨ 8 ' a

>

\ ¦ ° ú H \ ¦ ½ Ó © ¦ 9 > ) a . f §ý a³ ð> \ " f $ í ì

r ý a³ ð\ ¦ x, y, z Ð ¦ É r ý a³ ð> _ { 9 ì ø Í o ý a³ ð[ þ t` ¦ ξ

1

, ξ

2

, ξ

3

Ü ¼ Ð s [ þ t s _ ' a > H ¨ 8 í ß [ þ t Ð ½ ©

&

ñ ) a . 7 £ ¤ h

n

= q (

_∂ξ^∂x

n

)

²

+ (

_∂ξ^∂y

n

)

²

+ (

_∂ξ^∂z

n

)

²

Ð h

ⁿ

s & ñ K

t ¦, norm É r ds

²

= dx

²

+ dy

²

+ dz

²

= P h

n2

(dξ

_n

)

²

s

) a . s ] j Helmholtz ~ ½ Ó& ñ d ∇

²

Ψ + k

²₁

Ψ = 0 É r

X

m

1 h

1

h

2

h

3

∂

m

[ h

₁

h

₂

h

₃

h

m2

∂Ψ

∂ξ

m

] + k

²₁

Ψ = 0 (1)

s

) a . # l " f k

1²

= 0 â Ä º á Ô Û ¼ ~ ½ Ó& ñ d , k

₁²

= const â Ä º H 1 l x~ ½ Ó& ñ d , k

₁²

=

1

− V (ξ)

â

Ä º Schr¨odinger ~ ½ Ó& ñ d s . 7 á § 8 [ jô Ç [ O " î É r @ / Â

Òì r_ §F \ " f ¸ ú [ O " î ÷ &# Q e [5, 6]. & ³ ©

`

¦ l Õ ü t ¦ K $ 3 H כ É r ô Ç n Ð ´ ú s ü < ° ú s

Helmholtz ~ ½ Ó& ñ d ` ¦ & h ] X ô Ç ý a³ ð> \ " f É Ò H \ O s

¦ ´ ú ½ + É Ã º e . Õ ª QÙ ¼ Ð Ó ü t o < Æ ¸[ þ t É r Á º% Á Ð ¸ ý

a³ ð> \ @ /ô Ç l : r t d õ î r6 x0 p x§ 4 ` ¦ ° ú Æ Ò# Q ô Ç .

Õ

ª Q & ³z ´& h Ü ¼ Ð Ä ºo Ó ü t o < Æ ¸[ þ t s s 14ý a³ ð> r Û

¼% 7 \ @ /ô Ç t d ` ¦ ¢ - a# 4 > ° ú Æ Òl ê ø Í Ô ¦ 0 p x ¦ ¢ ¸ Õ

ªX O 9 כ ¹ ¸ \ O . © ´ ú §s 6 x ÷ & H r Û ¼% 7 7 £ ¤, f y ý

a³ ð, " é ¶: x ý a³ ð, ½ ¨ ý a³ ð\ @ / # l : r& h t d ` ¦ ° ú Æ

Ò ¦ è ß K ô Ç É r ý a³ ð> \ @ /K " f H B Û ¼B jw ü < ° ú

É

r ¸½ ¨1 p x` ¦ ¸ ú Ö ¸6 x ) a [7, 8]. B Û ¼B jw r Û ¼

% 7

\ " f H 14- ý a³ ð> \ ¦ î r6 x H X < 9 כ ¹ô Ç l ½ ¨[ þ t` ¦ ¸

¿

º í < Ê ¦ e l M :ë H \ s \ ¦ & h ] X y 6 x ) a .

B

Û ¼B jw r Û ¼% 7 \ " f H Ä ºo " é ¶ H ý a³ ð> \ ¦ l

: r ý a³ ð> Ð Ë ¨ 13> h_ ý a³ ð> ü < f §ý a³ ð> ü <

¨ 8

s F K~ ½ Ó s À Ò# Q t 9 y ý a³ ð> r Û ¼% 7 \ " f p ì r í

ß \ ¦ s 6 xô Ç 7 ' ¢ ¸ H J $ " f K $ 3 s 0 p x > ) a .

Ä

ºo H Å Ò Ð Calculus‘VectorAnalysis‘ J v t \ ¦ Ô ¦ Qü <

"

f SetCoordinates , CoordinateToCartesian , Grad, Div, Curl, Lapalacian 1 p x` ¦ 6 x # ¼ # o ¦ ¸ 5 Å q > y

7 á x ý a³ ð> \ " f > í ß ` ¦ Ã º' ¦ S X ½ + É Ã º e [12, 13].

1. 8 0 8 cX ; c" e s ð ' [U ê sX N ËÅ k Ä Ö «ã _ Ë

VectorAnalysis J v t \ ¦ s 6 x # 7 ' p ì r í ß \ ¦ í

<

Êô Ç 7 ' ~ ½ Ó& ñ d ` ¦ 7 £ x" î ½ + É Ã º e . # l " f H @ /Â Òì r _ §F \ " f / B N: x& h Ü ¼ Ð 2 [/ å L H l : r& h 7 ' d ` ¦ Y

J B Û ¼B jw r Û ¼% 7 \ " f 7 £ x" î % i [1, 3]. B Û

¼B jw \ Ò q t èô Ç Ó ü t o < Æ ¸{ 9 t ¸ 6 £ § \ \ Vr ô Ç B

Û ¼B jw ï` ç õ Ã º' õ \ ¦ Ð [ O " î ` ¦ t î ß K

¸ ~ 1 > s K ½ + É Ã º e ` ¦ & ñ ¸ . B Û ¼B jw " î § î

#

Q Cross, Dot, Grad, Div, Curl, Lapalacian s Ó ü t o < Æ

\

" f 6 x H 7 ' í ß l ñ[ þ t Cross, Dot, Gradient, Divergence, Curl, Lapalacian ü < ° ú q 5 p w l M :ë H

\

Ó ü t o < Æ ¸[ þ t É r ~ 1 > B Û ¼B jw ï` ç ` ¦ s K ½ + É Ã º e

. Õ ª Ql M :ë H \ B Û ¼B jw H Ó ü t o 2 ; o& h á Ô ÐÕ ªÏ þ s

¦ ¸ Â Ò É r [7,12]. 7 ' ~ ½ Ó& ñ d ` ¦ 7 £ x" î l 0 AK " f

$

Calculus‘VectorAnalysis‘ " î § î # Q Ð ‘VectorAnalysis‘

J

v t \ ¦ 6 x ' ` s Û ¼ Front End\ Ô ¦ Q[ þ t

Ê ê\ SetCoordinates [Cartesian[x,y,z]] _ " î § î # Q

Ð l : r ý a³ ð> \ ¦ Cartesian[x,y,z] Ü ¼ Ð [ O & ñ K Z ~ É r 6

£

§ B Û ¼B jw " î § î # Q\ ¦ 6 x # 7 ' ~ ½ Ó& ñ d ` ¦ 7 £ x" î K

ç ß . 7 £ x" î É r K $ 3 & h Ü ¼ Ð < Ê É r 7 H o & h Ü ¼ Ð S X

"

f 7 ' d ` ¦ § K y Ü ¼ Ð+ 7 ' _ $ í | 9 ` ¦ 7 á §

8 " î S X > s K ½ + É Ã º e > ô Ç . 7 ' d _ ý a õ Ä º _ s \ ¦ > í ß # % ò 7 ' {0,0,0}\ ¦ S X ô Ç . 7 ' d

`

¦ Ã º' ô Ç õ 0 s ¸ Õ ª 7 ' d _ õ Û ¼º ú e

` ¦ · ú Ã º e . 7 ' d _ > í ß õ True ¸ 7 H o

& h Ü ¼ Ð Õ ª 7 ' d s Ã Ð כ ` ¦ · p . 6 £ § É r 13 > h _ 7 ' d ` ¦ 7 £ x" î ô Ç B Û ¼B jw á Ô ÐÕ ªÏ þ ` ¦ Ð כ s

. { 9 § 4 õ Ø ¦§ 4 ` ¦ Ð 7 £ x" î õ & ñ ` ¦ ~ 1 > s K ½ + É Ã º e

`

¦ כ s .

In[1]:=

Calculus‘VectorAnalysis‘;

In[2]:=

SetCoordinates[Cartesian[x,y,z]]

Out[2]=

Cartesian[x,y,z]

In[4]:=

Unprotect[C]

Out[4]=

{C}

In[5]:=

a= {ax[x,y,z],ay[x,y,z],az[x,y,z]};

b= {bx[x,y,z],by[x,y,z],bz[x,y,z]};

c= {cx[x,y,z],cy[x,y,z],cz[x,y,z]};

f= {fx[x,y,z],fy[x,y,z],fz[x,y,z]};

g= {gx[x,y,z],gy[x,y,z],gz[x,y,z]};

r= {x,y,z}; u={1,1,1};

phi=func[x,y,z]; psi=func[x,y,z];

1. ~ ∇ · ~ ∇ϕ = (~ ∇)

²

ϕ

In[15]:=

Div[Grad[phi]][Equal]Laplacian[phi]

Out[15]=

True

2. ~ ∇ · (~ ∇ × ~ A) = 0

In[16]:=

Div[Curl[a]]

Out[16]=

0 3. ~ ∇ × (~ ∇ϕ) = 0

In[17]:=

Curl[Grad[phi]]

Out[17]=

{ 0,0,0 }

4. ~ ∇ × (~ ∇ × ~ A) = ~ ∇(~ ∇ · ~ A) − ~ ∇

²

A ~

In[18]:=

Curl[Curl[a]]-Grad[Div[a]]+Laplacian/@a

Out[18]=

{ 0,0,0 }

(3)

5. ~ ∇(ϕψ) = (~ ∇ϕ)ψ + (~ ∇ψ)ϕ

In[19]:=

Grad[ phi psi]

== Grad[phi] psi + Grad[psi] phi

Out[19]=

True

6. ∇( ~ ~ F · ~ G) = ( ~ F · ~ ∇) ~ G+ ~ F ×(~ ∇× ~ G)+( ~ G · ~ ∇) ~ F + ~ G ×(~ ∇× ~ F )

In[20]:=

Grad[Dot[f,g]]

-Cross[f,Curl[g]]-Cross[g,Curl[f]]

-(Dot[f,D[#,x],D[#,y],D[#,z]]&)/@g -(Dot[g,D[#,x],D[#,y],D[#,z]]&)/@f //Simplify

Out[20]=

{ 0,0,0 }

7. ~ ∇ · (ϕ) ~ F ) = ( ~ ∇ϕ) · ~ F + ϕ ~ ∇ · ~ F

In[21]:=

(Div[phi f]

-phi Div[f]-Dot[f,Grad[phi]]) //Simplify

Out[21]=

0 8. ~ ∇ · ( ~ F × ~ G) = ( ~ ∇ × ~ F ) · ~ G − (~ ∇ × ~ G) · ~ F

In[22]:=

(Div[Cross[f,g]]

- Dot[Curl[f],g]+ Dot[Curl[g],f]) //Simplify

Out[22]=

0

In[23]:=

Div[Cross[f,g]]

== (Dot[Curl[f],g]-Dot[Curl[g],f]) //Simplify

Out[23]=

True

9. ~ ∇ × (ϕ ~ F ) = ( ~ ∇ϕ) × ~ F + ϕ ~ ∇ × ~ F

In[24]:=

Curl[ phi f]-Cross[Grad[phi],f]

-phi Curl[f]//Simplify

Out[24]=

{ 0,0,0 }

In[25]:=

Curl[ phi f]

== Cross[Grad[phi],f]+ phi Curl[f]

//Simplify

Out[25]=

True

10. ∇×( ~ ~ F × ~ G) = ( ~ ∇· ~ G) ~ F ) −(~ ∇· ~ F ) ~ G + ( ~ G · ~ ∇) ~ F −( ~ F · ~ ∇) ~ G

In[26]:=

Curl[Cross[f,g]]-(Div[g]f)+(Div[f]g) -(Dot[g, {D[#,x],D[#,y],D[#,z]}]&)/@f +(Dot[f, {D[#,x],D[#,y],D[#,z]}]&)/@g //Simplify

Out[26]=

{ 0,0,0 }

11. ~ A × ~ B × ~ C = ~ B( ~ A · ~ C) − ~ C( ~ A · ~ B)

In[27]:=

Cross[a,Cross[b,c]]

-b Dot[a,c]+ c Dot[a,b]

//Simplify

Out[27]=

{ 0,0,0 }

12. ( ~ G · ~ ∇)~r = ~ G

In[28]:=

(Dot[g, {D[#,x],D[#,y],D[#,z]}]&)/@r-g

Out[28]=

{ 0,0,0 }

In[29]:=

(Dot[g, {D[#,x],D[#,y],D[#,z]}]&)/@r==g

Out[29]=

True

13. (~u · ~ ∇)~r = ~u, where ~u is a unit vector.

In[30]:=

Dot[u, {D[#,x],D[#,y],D[#,z]}]&)/@r-u

Out[30]=

{ 0,0,0 }

In[31]:=

(Dot[u, {D[#,x],D[#,y],D[#,z]}]&)/@r==u

Out[31]=

True

In[36]:=

Clear[Global‘*;]

2. 8 0 8 cX ; c" e Ò ÷ » ì Åò & ÿ s

] j VectorAnalysis J v t \ [8] e H " î § î # Q ^SetCoordi nates , CoordinatesToCartesian , CoordinateFromCartesian

\

¦ s 6 x K " f 13> h_ ý a³ ð> ü < f y ý a³ ðç ß _ ý a³ ð ¨ 8 s

~ 1

> s À Ò# Q t H כ ` ¦ Ðs x .

SetCoordinates [coordsys[vars] ] H Ã º varss ¦ ý a

³

ð> coodsys ÷ & H l : r ý a³ ð> ÷ &> ô Ç . ¢ ¸ CoordinatesToCartesian [pt, coordsys] H coordsys_ pt\ ¦ f

y ý a³ ð Ð Ë ¨# Q Å Ò 9 CoordinatesFromCartesian [pt, coordsys] H f y ý a³ ð> _ pt\ ¦ coodsys ý a³ ð Ð Ë ¨# Q ï r

. s H 13 > h_ coodsysü < f y ý a³ ð\ Õ ª@ / Ð & h 6 x ) a .

ý a³ ð ¨ 8 ½ ©g Ë :` ¦ [ jÄ º ¦ > í ß H Â Ò{ Û ¼ Qî r \ O ` ¦ B Û

¼B jw @ / K " f Ã º' K ï r . Fig. 1,2\ ¦ Ð " f 6

£

§ ý a³ ð ¨ 8 B Û ¼B jw ï` ç ` ¦ ¼ # o > ý a

³

ð ¨ 8 õ \ ¦ s K ½ + É Ã º e ` ¦ כ s . ë ß # Qì r s B Û

¼B jw r Û ¼% 7 ` ¦ ° ú ¦ e Õ ª@ / Ð s i ç # Ã º '

r v ° ú É r õ \ ¦ % 3 > | ¨ c כ s . ¢ ¸ É r ý a³ ð>

\

@ /K " f ¸ Ã º' K Ð ý a³ ð> \ @ /ô Ç ´ ú § É r t d ` ¦ % 3

>

| ¨ c כ s .

In[37]:=

x2rRule = {r,ϑ, φ}->

CoordinateFromCartesian[ {x,y,z},Spherical]

//Threads

Out[37]=

{ r-> px

²

+ y

²

+ z

²

Fig. 1. Spherical polar coordinates depicting on the base

of rectangular coordinates.

(4)

Fig. 2. Circular cylindrical coordinates depicting on the base of rectangular coordinates.

,ϑ -> ArcCos[ √

^z

x²+y²+z²

] ,φ->ArcTan[x,y] }

In[38]:=

r2xRule = {x,y,z}->

CoordinateToCartesian[ {r,ϑ, φ},Spherical]

//Threads

Out[37]=

{ x-> r Sin[ϑ] Cos[φ]

,y-> r Sin[ϑ] Sin[φ]

,z-> r Cos[ ϑ] }

In[38]:=

x2cRule= {r,ϑ,z}->

CoordinateFromCartesian[ {x,y,z},Cylindrical]

//Threads

Out[38]=

{ r-> px

²

+ y

²

,ϑ -> ArcTan[x,y]]

,z-> z }

In[39]:=

c2xRule= {x,y,z}->

CoordinateToCartesian[ {r,ϑ,z},Cylindrical]

//Threads

Out[39]=

{ x-> r Cos[ϑ]

, y ->r Sin[ϑ]

, z-> z }

In[40]:=

x2pRule= {u,v,φ}->

CoordinateFromCartesian[ {x,y,z},Paraboloidal]

//Threads

Out[40]=

{ u->

√

x²+y² q

−z+

√

x²+y²+z²

,v ->

q

−z + px

²

+ y

²

+ z

²

,φ -> ArcTan[x,y] }

In[41]:=

p2xRule= {x,y,z }->

CoordinateToCartesian[ {u,v,φ},Paraboloidal]

Out[41]=

{ x-> u v Cos[φ]

,y -> u v Sin[φ]

,z -> 1/2 (u

²

− v

²

)

#

l " f x2rRule É r f y ý a³ ð{x,y,z}\ ¦ ½ ¨ ý a³ ð{r,ϑ, φ} Ð

¨ 8 K Å Ò H " î § î s ¦ r2xRule H Õ ª_ ì ø Í@ /s . ð ø Í

t

Ð x2cRule H á Ô ÐÕ ªÏ þ \ " f Ðs H ü < ° ú s f y ý a

³

ð\ ¦ " é ¶: x ý a³ ð Ð Ë ¨# QÅ Ò 9 p2xRule H Paraboloidal ý a

³

ð\ ¦ f y ý a³ ð Ð Ë ¨# Qï r . ¢ ¸ô Ç s \ ¦ & h 6 x # In[47]

\

" f Ð ü < ° ú s ½ ¨ ý a³ ð_ {1,

^π₂

,

^π₄

}\ ¦ f y ý a³ ð {

√1 2

,

√¹

2

, 0 } Ð Ð ¨ 8 K ï r . É r ý a³ ð> \ " f ¸ ° ú

É

r ~ ½ Ód Ü ¼ Ð > í ß K ^ ¦ Ã º e . s > í ß ` ¦ f ] X < H Ü ¼

Ð 9 © { © ô Ç Â Ò{ ` ¦ ° ú H . É r 10 > h_ ý a³ ð> \

"

f H _ Ô ¦ 0 p x K Ð .

In[47]:=

CoordinateToCartesian[ {1,

^π2

,

^π₄

},Spherical]

Out[47]=

{

^√¹₂

,

√¹ 2

, 0 }

In[48]:=

CoordinateFromCartesian[ {

^√¹₂

,

√¹

2

,0 },Spherical]

Out[48]=

{ 1,

^π₂

,

^π₄

}

In[49]:=

CoordinateToCartesian[ {1,

^π2

,5 },Cylindrical]

Out[49 ]=

{ 0,1,5 }

In[50]:=

CoordinateFromCartesian[ {0,1,5},Cylindrical]

Out[50]=

{ 1,

^π₂

,5 }

3. 8 0 8 cX ; c" e s ð ' [A 0V Ä

s

] j ý a³ ð Ã º {r,θ, φ } ½ ¨ ý a³ ð> \ ¦ l : r ý a³ ð>

Ð ô Ç r Û ¼% 7 \ " f 6 £ § õ ° ú s 7 ' < ÊÃ º[ þ t` ¦ > í ß K Ð

.

In[51]:=

SetCoordinates[Spherical[r,θ, φ]]

Out[51]=

{ Spherical[r,θ, φ] }

In[52]:=

Grad[

¹_r

]

Out[52]=

{ -

_r¹2

, 0, 0 }

In[53]:=

Div[ {r,0,0}]

Out[53]=

3

In[60]:=

Laplacian[r

²

]

Out[60]=

6

In[58]:=

Div[r

ⁿ

{r,0,0}]

Out[58]=

(3 + n)r

ⁿ

In[59]:=

Curl[r

ⁿ

{r,0,0}]

Out[59]=

{ 0,0,0 }

s

] j l : r ý a³ ð> \ ¦ ý a³ ð Ã º\ ¦ {r,θ,z} Ð H " é ¶: x ý a³ ð

>

Ð Ë ¨# Q 7 ' < ÊÃ º\ ¦ > í ß K : r .

(5)

In[61]:=

SetCoordinates[Cylindrical[r,θ,z]]

Out[61]=

Cylindrical[r,θ,z]

In[62]:=

Laplacian[r

²

]

Out[62]=

4 r ½ ¨ ý a³ ð> Ð Ë ¨# Q > í ß ô Ç .

In[63]:=

SetCoordinates[Spherical[r,θ,φ]]

Out[63]=

Spherical[r,θ,φ]

In[64]:=

Laplacian[r

²

Cos[θ]Sin[φ]//Simplify

Out[64]=

-Cos[3θ] Csc[θ]

²

Sin[φ]

In[65]:=

SetCoordinates[Cartesian[x,y,z]]

Out[65]=

Cartesian[x,y,z]

In[66]:=

Laplacian[x

²

+ y

²

+ z

²

]

Out[66]=

6

In[1]:=

potential = (

^a³_r^E0₂

-E0 r) Cos[θ]

In[79]:=

eField=-Grad[potential

,Spherical[r, θ, φ]] //ExpandAll

Out[66]=

{E0 Cos[θ] +

^2a³^E0_r3^Cos[θ]

, −E0 Sin[θ] +

a³E0 Sin[θ]

r³

, 0 }

In[65] \ " f l : r ý a³ ð> \ ¦ f y ý a³ ð Ð õ H © I \ " f In[79] \ " fü < ° ú s -Grad[potential] > í ß É r ½ ¨ ý a³ ð>

\

" f > í ß ¸2 ¤ " î § î # Q ? /Â Ò\ " f ½ ©& ñ # > í ß ) a כ

\

Ä »_ ½ + É 9 כ ¹ e . s H ] j~ t 9 כ ¹ l : r ý a

³

ð> Ð " î r # > í ß ½ + É Ã º e # Q" f B Û ¼B jw r Û ¼% 7 É r B

Ä º y © § 4 ô Ç ý a³ ð> î r6 x r Û ¼% 7 s ) a . 6 £ §_ Lapla- cian > í ß õ \ ¦ Ð .

In[67]:=

Laplacian[x

²

+ y

²

+ z

²

,Cartesian[x,y,z]]

Out[67]=

6

In[68]:=

Laplacian[r

²

, Spherical[r, θ, φ]]

Out[68]=

6

In[69]:=

Laplacian[r

²

,Cylindrical[r, θ, z]]

Out[69]=

4

In[70]:=

Laplacian[u

²

+ v

²

+ φ

²

,Parabolidal[u, v, φ]]

Out[70]= ^8uv+

2(u2 +v2 ) uv uv(u²+v²)

s

ü <° ú s [ O & ñ ) a r Û ¼% 7 s # Q* ô Ç ý a³ ð> { 9 t ¸ 9 כ

¹½ + É M : " î § î # Q ? /Â Ò\ " f > í ß ½ + É ý a³ ð> \ ¦ t & ñ K º ¡ § Ü ¼

Ð+ t & ñ ) a ý a³ ð> _ > í ß ° ú כ` ¦ ½ ¨½ + É Ã º e . # l " f ° ú

É r 7 ' < ÊÃ º { 9 t ¸ ý a³ ð> \ > í ß ° ú כs Ø Ô

H כ ` ¦ Ä »_ ½ + É 9 כ ¹ e . Õ ª QÙ ¼ Ð ý a³ ð> " î r ÷ &

#

Q e t · ú § É r > í ß ° ú כ É r Á º_ p ô Ç > í ß e ` ¦ " î d K ô Ç

. B Û ¼B jw \ " f H l : r ý a³ ð> \ " f f y ý a³ ð> Ð

¨ 8

½ + É M :_ p ì r > Ã º_ ' § > = 7 £ ¤ JacobianMatrix[] \ ¦ ~ 1 >

½

¨½ + É Ã º e . s ' § > =_ ' § > =d ` ¦ Jacobian determinant s

¦ Â ÒØ Ô ¦ ç ß é ß y Jacobians ô Ç . s Jacobian

`

¦ s 6 x # ì ø Ít 2 £ § s a ½ ¨_ ^ & h ` ¦ ~ 1 > ½ ¨½ + É Ã º e

(Out[91]).

In[89]:=

JacobianMatrix[Spherical[r,θ, φ]]

In[90]:=

JacobianDeterminant[Spherical[r,θ, φ]]

Out[90]=

r

²

Sin[θ]

In[91]:=

Integrate[

JacobianDeterminant[Spherical[r,θ, φ]]

, {r,0,a},{θ,0,π},{φ, −π, π}]

Out[91]= ^4πa³ 3

s

p t & h ô Ç ü < ° ú s l ñ> í ß s H Ã ºu > í ß s H ý a³ ð

>

Ø Ô > í ß õ Ø Ô H כ ` ¦ S X % i . Õ ª

QÙ ¼ Ð B Û ¼B jw H ½ Ó © # Q " ý a³ ð> \ " f > í ß ½ + É כ

\ ¦ ¶ n s > ) a . 6 £ § õ ° ú s ý a³ ð> \ > í ß õ

É r כ ` ¦ · ú Ã º e .

In[93]:=

DotProduct[ {1.5,1.2,1.0},{5.3,-2,3}

,Cartesian]

DotProduct[ {1.5,1.2,1.0},{5.3,-2,3}

,Spherical]

DotProduct[ {1.5,1.2,1.0},{5.3,-2,3}

,Cylindrical]

DotProduct[ {1.5,1.2,1.0},{5.3,-2,3}

,ParabolicCylindrical]

DotProduct[ {1.5,1.2,1.0},{5.3,-2,3}

,ProlateSpheroidal]

Out[93]=

8.55

Out[94]=

1.60503

Out[95]=

-4.93644

Out[96]=

-11.2018

Out[97]=

39.6875

Verifying Sadiku’s #3-6,7 [3]

In[100]:=

Div[ {x

²

yz, 0, xz }

,Cartesian[x,y,z]]//Simplify Div[ {ρSin[φ], ρ

²

z, zCos[φ] }

,Cylindrical[ρ, φ, z]]//Simplify Div[ {

_r¹2

Cos[θ], rSin[θ]Cos[φ], Cos[θ] }

,Spherical[r, θ, φ]]//Simplify

Out[100]=

x + 2xyz

Out[101]=

Cos[φ] + 2Sin[φ]

Out[102]=

2Cos[θ]Cos[φ]

In[105]:=

Curl[ {x

²

yz, 0, xz }

,Cartesian[x,y,z]]//Simplify Curl[ {ρSin[φ], ρ

²

z, zCos[φ] }

,Cylindrical[ρ, φ, z]]//Simplify

Curl[ {

_r¹2

Cos[θ], rSin[θ]Cos[φ], Cos[θ] }

(6)

,Spherical[r, θ, φ]]//Simplify Curl[ {

r¹²

Cos[θ], rSin[θ]Cos[φ], Cos[θ] }

,Cylindrical[r, θ, φ]]//Simplify

Out[105]=

{0, x

²

y − z, −x

²

z }

Out[106]=

{−

^ρ³^+zSin[φ]_ρ

, 0, 3zρ − Cos[φ]}

Out[107]=

{

Cos[θ]Cot[θ]−Sin[θ]+rSin[φ]

r

, −

^Cos[θ]r

, (

_r¹3

+ 2Cos[φ])Sin[θ] }

Out[108]=

{

^Sin[θ](^−1+rr ²^Sin[φ])

, 0, (

_r¹₃

+ 2Cos[φ])Sin[θ] }

s

H §õ " f_ ô Ç \ V] j\ ¦ B Û ¼B jw \ " f ç ß é ß y > í ß

# S X ô Ç כ s . §Ã º < Æ_ þ v H 4 ¤¸ ú ô Ç > í ß ë H ] j\ ¦ ç

ß é ß y S X ¦ ¢ ¸ B j 1 p x` ¦ Ë ¨# Q õ ] j\ ¦ ? / H X

<\ ¸ ¼ # o . s \ ¦ ¸ ú Ö ¸6 x §Ã º < Æ_ þ v \ Ä »6 xô Ç

íà Ôe ¦ o ¸\ ¦ ë ß [ þ t # Q H X <\ ¸ ¼ # o [9–15].

4. Ò ÷ » ì Å ¤Ñ ÷ Ò ÷ » ì Å ¤ Q ' [ ß e Èt

ý

a³ ð> \ " f \ O ½ + É M : ý a³ ð> _ Ã º[ þ t s # Qb G> & ñ _ ÷ & 9 Ã º[ þ t \ @ /ô Ç p ' [ þ t s e H t \ O H t

¢

¸ H # 3 0 A # Qb G> ÷ & H t S X z ´y · ú ô Ç .

CoordinateRanges [coordsys] H ý a³ ð> coordsys Ã º[ þ t_

#

3 0 A\ ¦ Ø ¦§ 4 K Å Ò ¦ ParameterRange [coordsys] H ý a³ ð> coordsys_ ý a³ ð p ü < p _ # 3 0 A\ ¦ Ø ¦§ 4 K ï

r .

In[110]:=

{Coordinates[Cylindrical]

,CoordinateRanges[Cylindrical] } {Coordinates[Spherical]

,CoordinateRanges[Spherical] } {Coordinates[Bispherical[u,v,φ]]

,CoordinateRanges[Bispherical[u,v,φ]] }

Out[110]=

{{ r,θ,z } , {0 ≤ r < ∞ , -π ≤ θ ≤ π ,

−∞ < z < ∞}}

Out[111]=

{{ r,θ,φ } , {0 ≤ r < ∞ , 0 ≤ θ ≤ π ,

−π < φ < π}}

Out[112]=

{{ u,v,φ } , { 0 ≤ u ≤ π ,

−∞ < v < ∞ , −π < φ < π}}

In[113]:=

{Coordinates[Conical[λ, µ, ν]

,CoordinateRanges[Conical[λ, µ, ν]] } {Coordinates[ConfocalEllipsoidal[λ, µ, ν]

,CoordinateRanges[ConfocalEllipsoidal[λ, µ, ν]] } {Coordinates[ConfocalParaboloidal[λ, µ, ν]

,CoordinateRanges[ConfocalParaboloidal[λ, µ, ν]] }

Out[113]=

{{λ, µ, ν} , {−∞ < λ < ∞ , 1 < µ

²

< 4, ν

²

< 1 }}

Out[114]=

{{λ, µ, ν} , {−∞ < λ < 1 , 1 < µ < 4, 4 < ν < 9 }}

Out[115]=

{{λ, µ, ν} , {−∞ < λ < 1 , 1 < µ < 4, 4 < ν < ∞}}

In[116]:=

Parameters[Conical]

,ParameterRanges[Conical]

Parameters[Spherical]

,ParameterRanges[Spherical]

Parameters[Cartesian]

,ParameterRanges[Cartesian]

Out[116]=

{{ 1,2 } , 0<#1<#2< ∞}

Out[117]=

{{ } , Null }

Out[118]=

{{ } , Null }

¿

º " î § î # Q CoordinateRanges [coordsys] ü < ParameterRange [coordsys] H Ó ü t o < Æ ¸[ þ t s ý a³ ð> _ & ñ S X ô Ç & ñ _ \ ¦ ] t

è ß y K ½ + É M : s \ ¦ s 6 x B Ä º ¼ # o ô Ç ¸½ ¨s

. 0 A\ " f q @ /g AÕ ªÒ ¨_ ý a³ ð> Ã º[ þ t s y y Ø Ô

H כ ¸ Ð · ú Ã º e . É r 14 > h_ ý a³ ð> \ @ / K

" f ¸ ð ø Ít Ð S X K Ð ´ ú § É r / B N Â Ò | ¨ c כ s

. s ] j B Û ¼B jw r Û ¼% 7 _ ø @ ¸\ ¦ · ú Ðl 0 A K

14> h_ ý a³ ð> \ @ / # CoodinatesToCartesian ü <

CoodinatesFromCartesian \ ¦ s 6 x # § % i .

s

p · ú ¡] X \ " f 7 H_ % i t ë ß Ó ü t o < Æ ¸[ þ t \ > e ¸ n qô Ç f y

ý a³ ð, " é ¶: x ý a³ ð, ½ ¨ ý a³ ð\ " f H 9 Ð > í ß # ¸

~ 1

> S X ½ + É Ã º e t ë ß É r ý a³ ð> \ @ /K " f H 9

Ð > í ß K " f S X K Ðl ê ø Í ~ 1 t · ú § . § l 0 A

# @ /g AÕ ªÒ ¨ \ 5 Å q H 11 > h_ ý a³ ð> \ @ /K " f H & h {1.5, 1.2, 1.1}\ ¦ × þ % i Ü ¼ 9 q @ /g A ý a³ ð> \ @ /K " f

H · ú ¡\ " f Ð ü < ° ú s ý a³ ð Ã º# 3 0 A y l Ø ÔÙ ¼

Ð 3 & h {0.5, 1.2, 5.1},{0.5, 1.2, 5.1},{0.5, 1.2, 0.2}` ¦

& h Ü ¼ Ð & ñ % i .

In[200]:=

CoordinatesToCartesian[ {1.5,1.2,1.1}

,Bipolar]

CoordinatesToCartesian[ {1.5,1.2,1.1}

,Bispherical]

CoordinatesToCartesian[ {0.5,1.2,5.1 } ,ConfocalEllipsoidal]

CoordinatesToCartesian[ {0.5,1.2,5.1 } ,ConfocalParaboloidal]

CoordinatesToCartesian[ {0.5,1.2,0.2}

,Conical]

CoordinatesToCartesian[ {1.5,1.2,1.1}

,Cylindrical]

CoordinatesToCartesian[ {1.5,1.2,1.1}

,EllipticCylindrical]

CoordinatesToCartesian[ {1.5,1.2,1.1}

,OblateSpheroidal]

CoordinatesToCartesian[ {1.5,1.2,1.1}

,ParabolicCylindrical]

CoordinatesToCartesian[ {1.5,1.2,1.1}

,Paraboloidal]

CoordinatesToCartesian[ {1.5,1.2,1.1}

(7)

,ProlateSpheroidal]

CoordinatesToCartesian[ {1.5,1.2,1.1}

,Spherical]

CoordinatesToCartesian[ {1.5,1.2,1.1}

,Toroidal]

CoordinatesToCartesian[ {1.5,1.2,1.1}

,Cartesian]

Out[200]=

{ 0.867547,0.5733,1.1 }

Out[201]=

{ 0.260047,0.510929,0.867547 }

Out[202]=

{ 2.54249,0.847742,0.130703 }

Out[203]=

{ 1.89561,0.369685,-0.90 }

Out[204]=

{ 0.18,0.56285,1.3787 }

Out[205]=

{ 0.06,0.187617, 0.459565 }

Out[206]=

{ 0.852414,1.98457,1.1 }

Out[207]=

{ 0.0580969,0.114146,1.50568 }

Out[208]=

{ 0.405,1.8,1.1 }

Out[209]=

{ 0.816473,1.60417,0.405 }

Out[210]=

{ 0.900194,1.76866,0.852414 }

Out[211]=

{ 0.634154,1.24596,0.543537 }

Out[212]=

{ 0.485331,0.953558,0.468349 }

Out[213]=

{ 1.5,1.2,1.1 }

In[220]:=

CoordinatesFromCartesian[

{0.867547,0.5733,1.1},Bipolar]

CoordinatesFromCartesian[

{0.260047,0.510929,0.867547},Bispherical]

CoordinatesFromCartesian[

{2.54249,0.847742,0.130703},ConfocalEllipsoidal]

CoordinatesFromCartesian[

{1.89561,0.369685,-0.90},ConfocalParaboloidal]

CoordinatesFromCartesian[

{0.18,0.56285,1.3787},Conical]

CoordinatesFromCartesian[

{0.543537,1.39806,1.1},Cylindrical]

CoordinatesFromCartesian[

{0.852414,1.98457,1.1},EllipticCylindrical]

CoordinatesFromCartesian[

{0.0580969,0.114146,1.50568},OblateSpheroidal]

CoordinatesFromCartesian[ {0.405,1.8,1.1}

,ParabolicCylindrical]

CoordinatesFromCartesian[

{0.816473,1.60417,0.405},Paraboloidal]

CoordinatesFromCartesian[

{0.900194,1.76866,0.852414},ProlateSpheroidal]

CoordinatesFromCartesian[

{0.634154,1.24596,0.543537},Spherical]

CoordinatesFromCartesian[

{0.485331,0.953558,0.468349},Toroidal]

CoordinatesFromCartesian[

{1.5,1.2,1.1},Cartesian]

Out[220]=

{ 1.5, 1.2, 1.1 }

Out[221]=

{ 1.5, 1.2, 1.1 }

Out[222]=

{ 0.499998, 1.2, 5.1 }

Out[223]=

{ 0.499998, 1.2, 5.1 }

Out[224]=

{ 0.5, 1.2, 0.2 }

Out[225]=

{ 1.5, 1.2, 1.1 }

Out[226]=

{ 1.5, 1.2, 1.1 }

Out[227]=

{ 1.5, 1.2, 1.1 }

Out[228]=

{ 1.5, 1.2, 1.1 }

Out[229]=

{ 1.5, 1.2, 1.1 }

Out[230]=

{ 1.5, 1.2, 1.1 }

Out[231]=

{ 1.5, 1.2, 1.1 }

Out[232]=

{ 1.5, 1.2, 1.1 }

Out[233]=

{ 1.5, 1.2, 1.1 }

Ð H ü < ° ú s > í ß õ H 14 > h_ ý a³ ð> \ " f ¢ - a# 4

>

¨ 8 õ % i ¨ 8 õ { 9 u < Ê` ¦ · ú Ã º e . Õ ª QÙ ¼ Ð B

Û ¼B jw r Û ¼% 7 \ " f H ¸ H 14- ý a³ ð> \ " f 7 ' í ß

`

¦ Ä »\ v > Ã º' ½ + É Ã º e . ý a³ ð> î r6 x \ @ /ô Ç Â Ò{ Ü ¼

ÐÂ Ò' Ä » ÐÖ ¦ Ã º e > ÷ &Ù ¼ Ð Ó ü t o < Æ ¸[ þ t s Ð : r

| 9

& h Ó ü t o ë H ] j\ ¦ K ¦ ½ Ó_ & h { 9 \ r ç ß ` ¦ ½ + É E

½ + É Ã º e > ) a .

III. + s Ç Â ] Ø

s

p 7 H_ ô Ç ü < ° ú s B Û ¼B jw r Û ¼% 7 \ " f H 14 > h _ ¸ H ý a³ ð> \ " f 7 ' K $ 3 õ ¼ # o ô Ç ý a³ ðî r6 x ¸½ ¨ [

þ

t` ¦ Ö ¸6 x½ + É Ã º e 6 £ §` ¦ S X % i . : £ ¤ y II-1 ] X \ " f,

[ þ t` ¦ K $ 3 & h < Ê É r 7 H o & h Ü ¼ Ð 7 £ x" î ¦ S X ½ + É Ã º e 6

£

§` ¦ ] jr % i . s H ¦ ¸_ ( É Ó' r Û ¼% 7 s ¸ ú ° ú Æ

Ò# Q ¸Z þ t_ < Æ_ þ v ¨ 8 â ` ¦ þ j@ /ô Ç Ö ¸6 x ¦ § î

r% ò r Û ¼% 7 ` ¦ 1 l x r v , < Æ_ þ v< É ª p \ ¦ Ä »µ 1 Ï ¦ < H

)

a < Æ_ þ v r ç ß ` ¦ Ð ¢ - a H X < \ ¸ Ä »e ô Ç D h Ðî r §Ã º < Æ _

þ

v J Q e Ü ¼ Ð Ö ¸6 x½ + É Ã º e 6 £ §` ¦ r K ï r . Ó ü t o

<

Æs $ l : r& h s ¦ Ù þ d & h §Ã º < Æ_ þ v \ × æ& h ` ¦ ¿ º

#

H ý a³ ð> < Æ_ þ v` ¦ ^ o =$ y K ½ + É כ s . Õ ª Q @ / Ã

º_ Ó ü t o < Æ ¸[ þ t É r ý a³ ð> \ @ /ô Ç s K ü < t d s p f

¨½ + É ÷ rë ß m ) ¹ 1 Þ ¦ è ß K ô Ç < Æ_ þ v õ ] j ¦ Ò q ty

#

t % ! Qo H â ¾ Ós e . Õ ª X < s p 7 H_ ô Ç @ /

Ð B Û ¼B jw r Û ¼% 7 \ " f H ~ 1 ¦ ¸ " î S X > ý a³ ð>

\

" f_ í ß õ K $ 3 s Ã º' ÷ & ¦ î r6 x½ + É Ã º e 6 £ §` ¦ Ð

%

i . ý a³ ð ¨ 8 s 9 כ ¹ CoordinatesFromCartesian , CoordinatesToCartesian ` ¦ s 6 x # ý a³ ð ¨ 8 ` ¦ Ã º'

¦ ¢ ¸ô Ç % i ¨ 8 ¸ Ð Ã º' r v ) a . ¢ ¸ ý a³ ð> _

Ã º\ ¦ ¸ ú ¸\ ¦ M :\ H Coordinates[coordsys ] \ ¦ Ã º' r

v Õ ª coordsys_ Ã º[ þ t s Á º% Á Ð ] jr K

Å Ò 9, Õ ª Ã º[ þ t õ Ã º[ þ t_ o p ' [ þ t_ # 3 0 A ¸

(8)

CoordinateRanges ü < ParameterRanges \ ¦ Ã º' r v Ð

· ú 9 Å Ò> H d Ü ¼ Ð ý a³ ð> î r6 x_ ë H | ¨ c Ã º e .

Ä

Ã

~

½ Ó ¾ Ó @ / or Û ¼% 7 Ü ¼ Ð 7 á § 8 % i 1 l x& h < Æ_ þ v` ¦ Ã º' < ÊÜ ¼

o ³ ð & ³, r Ð 3 x W 1l , Ã ºu > í ß 1 p x Ü ¼ Ð è ß K ¦ Ò q ty

÷

& H < ÆÂ Ò Ó ü t o < Æ_ < Æ_ þ v õ ] j\ ¦ < É ª p \ v ¦ ½ Ó_ & h Ü ¼ Ð Ã º '

K ° ú Ã º e 6 £ §` ¦ S X % i .

Y c

p w Ã U Ø ô

[1] John R. Reitz, Frederick J. Milford, and Robert W. Christy, Foundation of Electromagnetic Theory, (Addison-Wesley, New york, 1992), Chap 1.

[2] D. J. Griffiths, Introduction to electrodynamics, (Prentice Hall, Englewood Cliffs, 1989), Chap 1.

[3] M. N. O. Sadiku, Elements of electromagnetics, 3rd ed. (Oxford University Press, New york, 2001), p.

73. [4] J. D. Jackson, Classical Electrodynamics, (John Wiley & Soms, New york, 1975), Chap 11.

[5] G. B. Arfken, Mathematical methods for physicists, (Academic Press, San Diego, 1970), Chap 2.

[6] P. M. Morse, H. Feshbach, Methods of theoretical physics, (McGraw-Hill, Boston, 1953), Chap 5.

[7] S. Wolfram, The M athematica Book, 4th ed. ( McGraw-Hill, Champaign, 1953), Chap 5.

[8] Wolfram Research, The M athematica 4.0 Stan- dard Add-on Packages,(Wolfram Media, Cham- paign, 1999), Chap 2.

[9] H. J. Yun, SAE MULLI 40, 530 (2000).

[10] H. J. Yun, SAE MULLI 46, 249 (2003).

[11] H. J. Yun, Applied Surface Science 214, 312 (2003).

[12] R. L. Zimmerman and F. I. Olness, M athematica for Physics 2nd ed.,(Addison-Wesley, San Francisco, 2002), Chap 8.

[13] Partric T. Tamm, A Physicist’s Guide to Mathemat- ica, Academic Press, San Diego, 1997), Chap 2.

[14] Bernard Thaller, The M athematica Journal 7, 163 (1998).

[15] Phil Ramsden, The M athematica Journal 7, 308 (1999).

[16] http://home.mokwon.ac.kr/ ∼ heejy/

Interactive Vector Analysis and Coordinates Manipulation with M athematica

Hee-Joong Yun

^∗

Department of Optical and Electronic Physics, Mokwon University, Daejon 302-729 (Received 16 February 2005)

Vector analysis in a coordinate system is a main component of the physics curriculum in un- dergraduate school. Because physical systems exhibits structures with special symmetries and singularities that make particular coordinate system especially useful, it is essential to choose a proper coordinate system in which the physical system can be more easily analyzed. Obviously, curvilinear coordinate systems are appropriate for natural physical systems while the Cartesian rectangular coordinate system is more appropriate for continued objects. We have presented the capability of Mathematica for use in vector analysis and in manipulating the 14-coordinate systems.

PACS numbers: 01.40, 03.50.De

Keywords: M athematica, Vector analysis, Manipulation coordinates, 14-coordinstes

∗E-mail: [email protected]

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