Generalization of Differential Operators by Using Differential Forms = Diferansiyel Formlar ile Diferansiyel Operatörlerin Genelleştirilmesi

Journal home page: http://fbd.beun.edu.tr

DOI: 10.7212%2Fzkufbd.v8i1.739 Received / Geliş tarihi : 25.11.2016 Accepted / Kabul tarihi : 27.11.2017

*Corresponding author: [email protected]

Let S be a scalar field such that , ,...,

S=S q q^ 1 2 qnh (1)

and F be a vector field which can be defined as ... F F ei i F e F e F en n i n 1 1 2 2 1 = = + + =

/

(2)

where q_i’s are the components of the coordinate system and e_i’s are the basis one-forms (vectors) for the considered coordinate system. We then define the basis one-forms such as

ei _dqdr dq i i

= (3)

where r is any vector [1]. The Hodge Dual (star) operator * should be defined for the basis one-forms, such that [2-4]

*ei=fijkej/ek (4)

* e^ i/ej/ekh=1 (5)

Generally, we can derive the following derivative operators in different coordinates, on fields as follows

S dS d = (6) F d F $ ) ) d = (7) F dF # ) d = ₍₈₎

1. Introduction

In physics, differential operators are widely used in order to obtain the dependency of a physical quantity with respect to some coordinates. From classical mechanics to electrodynamics and general relativity to quantum mechanics, the differential operators are all in use in different forms. For instance, the gradient of the potential energy scalar yields as the conservative force in classical mechanics, or the divergence of the electric vector field yields as the scalar charge density in electrodynamics, or so on. With the change of the symmetry of the space in which we investigate the change of the physical quantities, we alter the symmetry of the coordinate, which changes to obtain the differential of the physical quantity, such as, from Cartesian coordinate to spherical coordinates for a spherically symmetric quantity. Therefore, we need to make the coordinate transformation for the differential operator in use.

For this purpose, we derive a simple approach to obtain the differential operators in different coordinates by using the differential forms. Some general definitions and properties are given as follows.

Generalization of Differential Operators by Using Differential Forms

Diferansiyel Formlar ile Diferansiyel Operatörlerin Genelleştirilmesi

Emre Dil

*

Department of Energy Systems Engineering, Beykent University, İstanbul, Turkey

Öz

Bu çalışmada, diferansiyel formları kullanarak farklı koordinat sistemlerinde; Kartezyen, silindirik ve küresel koordinat sistemlerinde, gradyan, diverjans, rotasyonel ve Laplasyen gibi fizikte en çok kullanılan diferansiyel operatörleri türeteceğiz. Son olarak bu diferansiyel operatörleri genelleştirilmiş koordinatlar için türeceğiz.

Anahtar Kelimeler: Diferansiyel formlar, Diferansiyel operatörler, Vektör kalkülüs Abstract

In this study, we derive the mostly used differential operators in physics, such as gradient, divergence, curl and Laplacian in different coordinate systems; Cartesian, cylindrical and spherical coordinate systems by using the differential forms. Also, we finally derive these differential operators for the generalized coordinates.

(6) means the gradient of the scalar S, (7) is the divergence, (8) is for the curl of a vector field F and the (9) is of course for the Laplacian of the scalar S. Here, the symbol d is exterior differentiation.

2. Cartesian Coordinates

The scalar field in Cartesian coordinates is written as , ,

S=S x y z^ h (10)

and the vector field is defined to be .

F=F ex x+F ey y+F ez z (11)

The basis one-forms are also written for vector r=xex+yey+zez from (3), such that

, ,

ex=dx ey=dy ez=dz (12)

2.1. Gradient

From (6) for the scalar S we can see S dS S dx_x S dy_y S dz_z

d = = 2₂ +2₂ + 2_{2 (13)}

and by using (12) to make the gradient a vector change dq_i’s into ei’s

. S dS S e_x x S e_y y S e_z z

d = = 2₂ + 2₂ + 2₂ (14)

2.2. Divergence

From equation (7) for the given vector (11) in Cartesian coordinates, we obtain F=F ex x+F ey y+F ez z (15) F F ex x F ey y F ez z ) = ) + ) + ) ₍₁₆₎ ( ) . F ex y/ez F ey z/ex F e2 x/ey = + ^ h+ ^ h (17)

After this step we will need d F) , so we should convert e_i’s into dq_i’s from (12)

F F dy dzx F dz dxy F dx dyz

) = ^ / h+ ^ / h+ ^ / h (18)

and to calculate the exterior differentiation of (18), we proceed by d F F_x dx dy dz F_x dx dz dx x F _{dx dx dy} y F _{dy dy dz} y F dy dz dx F_y dy dx dy z F _{dz dy dz} z F dz dz dx x y z x y z x y ) / / / / / / / / / / / / / / / / 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 = + + + + + + + ^ ^ ^ ^ ^ ^ ^ ^ h h h h h h h h (19)

After canceling the repeating indices in triple wedge products dq dq dqi/ j/ k=fijk.

Now we will convert dqi’s into ei’s, because we will take their

Hodge in the final step,

d F F_x e e e F_y e e e z F _{e e e} x x y z y y z x z z x y ) / / / / / / 2 2 2 2 2 2 = + + ^ ^ ^ h h h (20) d F F_xx F_yy F_zz ) ) = 2₂ + 2₂ +2_{2 (21)}

where we have used (5) and obtained the divergence equation in Cartesian coordinates for a vector field F

F d F F_xx F_yy F_zz

$ ) )

d = = 2₂ + 2₂ +2₂

2.3. Curl

We begin by writing F from (8) with dq_i’s instead of e_i’s, so

F=F dx F dy F dzx + y + z (22)

and we take the exterior differentiation for using in

F dF # ) d = _{as follows,} . dF F_x dx dx F_x dx dy F_x dx dz y F _{dy dx} y F dy dy F_y dy dz z F _{dz dx} z F dz dy F_z dz dz x y z x y z x y z / / / / / / / / / 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 = + + + + + + + + (23)

We expect to get a vector at the end of the calculation, so we convert the differential one-forms into basis one-forms, but before eliminate the repeating dq dqi/ i=0 terms, and remaining is . dF F_y e e F_z e e F_x e e z F e e F_x e e F_y e e x y x x z x y x y y z y z x z z y z / / / / / / 2 2 2 2 2 2 2 2 2 2 2 2 = + + + ^ + + ^ ^ ^ ^ ^ h h h h h h (24)

When we apply Hodge operator on (24) and by using (4), we obtain . dF F_yxe F_z e F_x e F_z e F_x e F_y e z x y y z y x z y z x ) = -2₂ + 2₂ + 2₂ -2₂ - 2₂ +2₂ (25)

By rearranging it, we finally obtain the curl

F dF F_y F_z e F_z F_x e x F y F _e z y x x z y y x z # ) d 2₂ 2₂ 2₂ 2₂ 2 2 2 2 = = - + -+c -c a m m k (26)

S d dS

2 _{) )}

d = _{in (9). Now we take its star} dS S e_x x S e_y y S e_z z

) = 2₂ ) + 2₂ ) +2₂ ) (27)

from (4), we are led to

. dS S e e_x y z S e e_y z x S e e_z x y

) = 2₂ ^ / h+ 2₂ ^ / h+ 2₂ ^ / h (28) We convert the basis one-forms into differential one-forms, because we will take the differentiation

dS S dy dz_x S dz dx_y S dx dy_z

) = 2₂ ^ / h+ 2₂ ^ / h+ 2₂ ^ / h (29) and take the differential of it, and get

d dS _x S_x dx dy dz _x S_y dx dz dx x Sz dx dx dy y Sx dy dy dz y Sy dy dz dx y Sz dy dx dy z Sx dz dy dz z yS dz dz dx z Sz dz dx dy ) / / / / / / / / / / / / / / / / / / 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 = + + + + + + + + a a b a ^ ^ ^ ^ a ^ b a a ^ ^ ^ b ^ k k l k k h h h h h l k k l h h h h (30)

The repeating terms in wedge product of basis-one forms or differential one-forms vanishes, and the remaining is

. d dS S_x dy dz S_y dz dx z S _{dx dy} x dx y dy z dz ) / / / / / / 2 2 2 2 2 2 22 22 22 = + + a ^ a ^ b ^ k k h h l h (31) Before we take its star again, we convert the dq_i’s into e_i’s, and use (5), then get

d dS _xS2 _yS _zS 2 2 2 2 2 ) ) = 2₂ + 2₂ + 2₂ (32)

and this is already the Laplacian itself; . S d dS _xS _yS _zS 2 2 2 2 2 2 2 ) ) d = = 2₂ + 2₂ + 2₂

3. Cylindrical Coordinates

The scalar field in cylindrical coordinates is written as , ,

S=S s^ zzh ₍₃₃₎

and the vector field is defined to be .

F=F es s+F ez z+F ez z (34)

The basis one-forms are also written for vector

cos sin r=xex+yey+zez=s zex+s zey+zez from (3), such that cos sin cos sin sin cos cos sin e r ds_s e e ds ds ds e ds e r d s e s e d s s d sd e sd e dz / / s x y s x y z 2 2 1 2 2 2 2 2 1 2 & & 2 2 22 z z z z z z z z z z z z z z = = + = + = = = = - + = + = = = z z ^ ^ h h (35) 3.1. Gradient

From (6) for the scalar S we can see

S dS S ds_s S d S dz_z

d = = 2₂ +₂2_z z+ 2₂ (36)

and by using (35) to make the gradient a vector change dq_i’s into e_i’s . S dS S e_s S e_s S e_z S dS S e_s 1_s S e S e_z s z s z d 2_{2 2}2 2₂ d 2_{2 2}2 2₂ z z = = + + = = + + z z (37) 3.2. Divergence

By equation (7) for the given vector F in (34) in cylindrical coordinates, we obtain F=F es s+F ez z+F ez z (38) F F es s F e F ez z ) = ) + z) z+ ) (39) . F es /ez F ez/es F ez s/e = ^ z h+ z^ h+ ^ zh (40)

After this step we will need d F) , so we should convert e_i’s into dq_i’s from (35) ) F sF ds dz F dz ds sF ds dz ) = ^ z/ h+ z^ / h+ ^ / zh (41) d F sF_s ds d dz F_s ds dz ds s sF _{ds ds d} sF _{d d dz} F d dz ds sF d ds d z sF dz d dz F_z dz dz ds z sF dz ds d s z s z s z ) / / / / / / / / / / / / / / / / / / 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 z z _z z z z z z z z z z = + + + + + + + + z z z ^ ^ ^ ^ ^ ^ ^ ^ _^ ^ ^ ^ ^ ^ ^ h h h h h h h h h h h h h h h (42)

After canceling the repeating indices in triple wedge products dq dq dqi/ j/ k=fijk

Now we will convert dq_i’s into e_i’s, because we will take their Hodge in the final step,

3.4. Laplacian

We already obtain the part such given in (37) for

S d dS

2 _{) )}

d = _{. Now we take its star} dS S e_s s 1_s S e S e_z z

) = 2₂ ) + ₂2_z ) z+ 2₂ ) (50)

from (4), we are led to

. dS S e e_s z 1_s S e ez s S e e_z s

) = 2₂ ^ z/ h+ ₂2_z^ / h+ 2₂ ^ / zh (51) We convert the basis one-forms into differential one-forms, because we will take the differentiation

dS _{s s}S d dz 1_s S dz ds S s ds d_z ) = 2₂ ^ z/ h+ ₂2_z^ / h+ 2₂ ^ / zh

(52) and take the differential of it, and get

d dS _{s s s}S ds d dz _{s s} S ds dz ds s s zS ds ds d s sS d d dz s S d dz ds s zS d ds d s sS d d dz _s S d dz ds s zS d ds d z z z z z z 1 1 1 ) / / / / / / / / / / / / / / / / / / 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 2 2 22 22 z _z z _z z z z z z z z z z _z z = + + + + + + + + a a a b a ^ ^ ^ ^ ^ b a a b ^ ^ ^ ^ k k k l k h h h h h k l k l h h h h (53)

The repeating terms in wedge product of basis-one forms or differential one-forms vanishes, and the remaining is

. d dS _{s s s}S ds d dz s S d dz ds z s zS ds ds d 1 ) / / / / / / 22 22 22 22 22 22 z z z z z = + b + a ^ ^ a ^ l k h h k h (54)

Before we take its star again, we convert the dq_i’s into e_i’s, and use (5), then get

d dS _{s s s s}S _{s s} S _{s z s z}S d dS _{s s s s}S _s S _z S_z 1 1 1 1 1 1 2 ) ) ) 22 22 22 22 22 22 22 22 22 22 22 22 z z z z = + + = + + a a b b a a k k l l k k (55) and this is already the Laplacian itself;

. S d dS 1_{s s s s}S _s1 S _z S_z 2 2 ) ) d = = ₂2 a 2₂ k+ ₂2_zb₂2_zl+₂2 a2₂ k

4. Spherical Coordinates

d F sF_s e e_s e F e e_s e z sF s e e e s sFs e e e s F e e e s s Fz e e e 1 1 1 s s z z s z z s s s z s z z s z ) / / / / / / / / / / / / 2 2 2 2 2 2 2 2 2 2 2 2 z z = + + = + + z z z z z z z z ^ ^ ^ ^ ^ ^ ^ ^ ^ h h h h h h h h h (43) d F 1_s sF_ss 1_s F F_zz ) ) 2 ₂ 2 2 2 2 z = ^ h+ z+ (44)

where we have used (5) and obtained the divergence equation in cylindrical coordinates for a vector field F

F d F 1_s sF_ss 1_s F F_zz

$ ) )

d = = 2^ h₂ + 2_2zz+ 2₂ 3.3. Curl

We begin by writing F with dq_i’s instead of e_i’s, so

F=F ds sF ds + z z+F dzz (45)

and we take the exterior differentiation for using in

F dF # ) d = _{as follows,} . dF F_s ds ds sF_s ds d F_s ds dz F _{d ds} sF d d F d dz z F _{dz ds} z sF dz d F_z dz dz s z s z s z / / / / / / / / / 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 z z z z z z z z z = + + + + + + + + z z z ^ ^ ^ h h h (46) We expect to get a vector at the end of the calculation, so we convert the differential one-forms into basis one-forms, but before eliminate the repeating dq dqi/ i=0 terms, and remaining is dF sF_s e _se F_s e e F e _s e F s e e z F _{e e} z sF s e e s z s z s s z z s z s z / / / / / / 2 2 2 2 2 2 2 2 2 2 2 2 z z = + + + + + z z z z z z ^ ^ h h (47)

When we apply Hodge operator on (47) and by using (4), we obtain dF _s F e F_z e _s sF_s e s s Fz e Fs e s F e 1 1 1 1 s z s z s z z s ) 2₂ 2₂ 2 ₂ 2 2 2 2 2 2 z z = - + + - - + z z z z ^ h (48)

By rearranging it, we finally obtain the curl

F dF _s F F_z e F_z F_s e s Fs F e 1 1 z s s z s z # ) d 2₂ 2₂ 2₂ 2₂ 2 2 2 z = = - + -+ -z z z c c a m m k (49)

sin sin sin sin sin sin d F _{r r} F dr d d r r F dr d dr r rF dr dr d r F d d d r F d d dr rF d dr d r F d d d r F d d dr rF d dr d r r r 2 2 2 ) / / / / / / / / / / / / / / / / / / 22 22 22 2 2 2 2 2 2 2 2 2 2 2 2 i i z i z i i i i i z i i i z i i i z i z i z z i z z z z i = + + + + + + i z i z i z ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ h h h h h h h h h h h h h h h h h h (65)

After canceling the repeating indices in triple wedge products dq dq dqi/ j/ k=fijk

Now we will convert dq_i’s into e_i’s, because we will take their Hodge in the final step,

sin _sin

sin _{sin sin}

sin sin sin d F _{r r} F e_r e e r F e_r e e rF e_r e e r r r F e e e r F e e e rF e e e 1 r r r r r r r r 2 2 2 2 2 2 ) / / / / / / / / / / / / 22 2 2 2 2 22 2 2 2 2 i i i i i z i i i i i z = + + = + + i z i i z z z i i z i i z z i z ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ h h h h h h h h h h h h (66)

sin sin sin

d F _r12 _{r r F}2 r _r 1 F _r 1 F ) ) ₂2 2 2 2 2 i i i i z = + i + z ^ h ^ h (67)

where we have used (5) and obtained the divergence equation in cylindrical coordinates for a vector field F

sin sin sin F d F _{r r r F r} F r F 1 1 1 r 2 2 $ ) ) d ₂2 2 2 2 2 i i i i z = = + + i z ^ h ^ h 4.3. Curl

We begin by writing F with dq_i’s instead of e_i’s, so sin

F=F dr rF dr + i i+r iF dz z (68) and we take the exterior differentiation for using in

F dF

# )

d = in (8) as follows, and the vector field is defined to be

F=F er r+F ei i+F ez z (57)

The basis one-forms are also written for vector

sin cos sin sin cos

r=xex+yey+zez=r i zex+r i zey+r iez from (3), such that

cos sin

sin

cos cos sin

sin sin sin cos

sin

sin cos sin sin cos

sin cos e _rr r d e e e d d e e r d r e r e d r d e r r r r r rd rd d dr e e e dr dr dr e dr e / / / r x y z x y x y z r 2 1 2 2 2 1 2 2 2 1 2 & & & 2 2 2 2 2 2 i i i i z i z i i i i z z i i i i z z i i z i i i i i i i = = + + = = = = = + = = -= + + = + = = = i z z i ^ ^ ^ h h h (58) 4.1. Gradient

From (6) for the scalar S we can see

S dS S dr_r S S d

d = = 2₂ +2_2i+ ₂2_z z (59)

and by using (58) to make the gradient a vector change dq_i’s into e_i’s . sin sin S dS S e_r S e_r S _r e S dS S e_{r r}1 Se _r 1 Se r r d 2₂ 2 2 2 2 d 2₂ 2 2 2 2 i z i i i z = = + + = = + + i z i z (60) 4.2. Divergence

By equation (7) for the given vector F in (57) in spherical coordinates, we obtain F=F er r+F ei i+F ez z (61) F F er r F e F e ) = ) + i) i+ z) z (62) . F er /e F e /er F er/e0 = ^ i zh+ i^ z h+ z^ h (63)

After this step we will need d F) , so we should convert e_i’s into dq_i’s from (58)

sin sin

F r2 F dr d r F d dr rF dr d

) = i ^ i/ zh+ i i^ zK h+ z^ / ih

and take the differential of it, and get sin sin sin sin sin sin sin sin sin d dS _{r r} S_r dr d d r S dr d dr r S dr dr d r S_r d d d S _{d d dr} S _{d dr d} r S_r d d d S _{d d dr} S _{d dr d} 1 1 1 2 2 2 ) / / / / / / / / / / / / / / / / / / 22 22 22 22 22 22 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 22 22 i i z i i z i z i i i i i z i i i i z i i z i i z i z i z z i i z z z i z z i = + + + + + + + + b b a b b a b b a ^ ^ ^ ^ ^ ^ ^ ^ ^ l l l l l l k k k h h h h h h h h h (76)

The repeating terms in wedge product of basis-one forms or differential one-forms vanishes, and the remaining is

sin sin sin d dS _{r r} S_r dr d d S _{d d dr} S _{d dr d} 1 2 ) / / / / / / 22 22 2 2 2 2 22 22 i i z i i i i z z i z z i = + + b b a ^ ^ ^ l l k h h h (77)

Before we take its star again, we convert the dq_i’s into e_i’s, and use (5), then get

) sin sin sin _sin sin sin sin d dS _{r r r} S_r S S _{e e e} d dS _{r r r} S_{r r} S r S 1 1 1 1 1 r 2 2 2 2 2 2 2 2 2 ) / / ) ) 22 22 2 2 2 2 22 22 22 22 2 2 2 2 2 2 i i i i i z i z i i i i i z = + + = + + i z b a a b ^ b l k k l h l (78)

and this is already the Laplacian itself;

sin sin sin

S d dS _{r r r} S_r r S r S 1 1 1 2 2 2 2 2 2 2 2 ) ) d ₂2 2₂ 22 22 22 i i i i i z = = + b + a l k

5. Generalized Coordinates

The scalar field in generalized coordinates is written as . sin sin sin dF F dr dr_r rF_r dr d r _r F dr d F d dr rF _{d d} r F d d F d dr rF_r d d r F d d r r r / / / / / / / / / 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 i i z i i i i i i i i z z z z z i z i z z = + + + + + + + + i z i z i z ^ ^ ^ ^ ^ ^ h h h h h h (69)

We expect to get a vector at the end of the calculation, so we convert the differential one-forms into basis one-forms, but before eliminate the repeating dq dqi/ i=0 terms, and remaining is

sin

sin sin

sin sin sin

dF rF_r e _re r _r F _re e F e _re r F r e e F r e e r rF r e e r r r r r r 2 2 / / / / / / 2 2 2 2 2 2 2 2 2 2 2 2 i i i i i i z i z i = + + + + + i i z z i z i z z i z i ^ ^ ^ ^ h h h h (70)

When we apply Hodge operator on (70) and by using (4), we obtain sin sin sin sin sin sin dF _r rF_r e _r r _r F e _r F e r r F e _r F e _r rF_r e 1 1 1 1 1 1 r r r r 2 2 ) 2 ₂ 2 ₂ 2 2 2 2 2 2 2 2 i i i i i i i z i z = - -+ + -i z z i z z i i ^ ^ ^ ^ h h h h (71)

By rearranging it, we finally obtain the curl sin sin sin F dF _r F _rF e _r F r rF e _r rF_r F e 1 1 1 1 r r r # ) d 2 2 2 2 2 2 2 2 2 2 2 2 i i i z i z i = = - + - + -z i z i i z ^ ^ ^ h h h (72) 4.4. Laplacian

We already obtain the part dS such given in (59) for

S d dS

2 _{) )}

d = _{and now we take its star as} sin dS S e_r r _r1 S e _r 1 S e ) 2₂ ) ) ) 2 2 2 2 i i z = + i+ z (73)

from (4), we are led to

. sin dS S e e_r 1_r S e er _r 1 S e er ) 2₂ / / / 2 2 22 i i z = ^ i zh+ ^ z h+ ^ ih (74)

We convert the basis one-forms into differential one-forms, because we will take the differentiation

sin sin

dS r2 S d d S d dr

5.3. Curl

We begin by using (8) and writing F with dq_i’s instead of ei’s, so F F h dqi i i i 1 3 = =

/

(90)

and we take the exterior differentiation for using in

F dF # ) d = _{as follows,} dF h F_q dq dq , j i i j i j/ i 2 2 =

/

^ h^ h (91)

We expect to get a vector at the end of the calculation, so we convert the differential one-forms into basis one-forms, but before eliminate the repeating dq dqi/ i=0 terms, and remaining is dF _{h h}1 h F_q e e , j i j i j i i j/ i 2 2 =

/

^ h^ h (92)

When we apply Hodge operator on (92) and by using (4), we obtain dF _{h h}1 h F_q e , , j i i j k j i i ijk k ) =

/

2^₂ hf (93)

By rearranging it, we finally obtain the curl

F dF _{h h} h F_q e , , i j ijk i j k j i i k # ) d = =

/

f 2^₂ h (94) 5.4. Laplacian

We already obtain the part such given in (83) for

S d dS

2 _{) )}

d = _{and its star is} dS _h1 _qS e

i

i i

i

) =

/

₂2 ) (95)

from (4), we are led to . dS _h1 _qS e e , , i i j k i ijk j k ) =

/

₂2 ^f / h (96)

We convert the basis one-forms into differential one-forms, because we will take the differentiation

dS h h_{h q}S dq dq , , ijk i j k i j k i j k ) =

/

f ₂2 ^ / h (97)

and take the differential of it, and get

d dS _{q h}h h _qS dq dq dq , , ijk i i j k i j k i i j k ) =

/

f ₂2 ₂2 ^ / / h (98)

The repeating terms in wedge product of basis-one forms or differential one-forms vanishes, and the remaining is

d dS _{q h}h h _qS dq dq dq , , ijk i i j k i j k i i j k ) =

/

f ₂2 ₂2 ^ / / h (99)

Before we take its star again, we convert the dq_i’s into e_i’s, and use (5), then get

. F F ei i i 1 3 = =

/

(80)

The basis one-forms are also written for vector r q ej j j 1

3 =

=

/

from (3), such that ei _{q dq}r h dq i i i i 2 2 = = ₍₈₁₎ 5.1. Gradient

From (6) for the scalar S we can see

S dS _qSdq

i i i

d = =

/

₂2 (82)

and by using (35) to make the gradient a vector change dq_i’s into e_i’s . S dS _h1 _qSe i i i i d = =

/

₂2 (83) 5.2. Divergence

By equation (7) for the given vector F in (80) in spherical coordinates, we obtain F F ei i i 1 3 = =

/

(84) then, F F e F e e , , i i i ijk j k i j k i 1 3 ) = ) = f / =

/

^ h

/

(85)

After this step we will need d F) , so we should convert e_i’s into dq_i’s from (80)

F h h F , , dq dq , , k j i i j k j k i j k ) =

/

f ^ / h (86) d F _{q h h F} dq dq dq , , i i j k k j i ijk i j k ) =

/

₂2 ^ f h^ / / h (87)

After canceling the repeating indices in triple wedge products dq dq dqi/ j/ k=fijk

Now we will convert dq_i’s into e_i’s, because we will take their Hodge in the final step,

d F h h F e e e q h h F h h h e e e d F _{h h h}1 _q , , , , i j k k j i ijk i j k i k j i ijk i j k i j k i j k i i j k ) / / / / ) 22 22 f f = = ^ ^ ^ ^ h h h h

/

/

(88) d F _{h h h q h h F} , , i j k ijk i i j k k j i ) ) =

/

f ₂2 ^ h (89)

where we have used (5) and obtained the divergence equation in cylindrical coordinates for a vector field F

F d F _{h h h q h h F} , , i j k ijk i j k i k j i $ ) ) d = =

/

f ₂2 ^ h

derivative) operator on a function, the one-forms in this function must be in terms of dq_i’s. With the considerations just mentioned, we ensure that, we can obtain the gradient, divergence, curl and Laplacian relations for any functions in any kind of generalized coordinates.

7. References

Bachman, D. 2006. A geometric approach to differential forms. Birkhäuser, Boston, USA, 45 p.

Edwards, HM. 1994. Advanced calculus: A differential forms approach. Birkhäuser, Boston, USA, 1-21 pp.

Kreyszig, E. 2006. Advanced engineering mathematics. 9th

Edition, Wiley, Singapore A71, A84, pp.  

Morita, S. 2001. Geometry of differential forms. Am. Math. Soc., 201: 1-150. d dS _{h h h q} h h_{h q}S , , i j k ijk i j k i j k i i ) ) =

/

f ₂2 ₂2 (100)

and this is already the Laplacian itself; . S d dS _{h h h q h}h h _qS , , i j k ijk i j k i i j k i 2 _{) )} d = =

/

f ₂2 ₂2

6. Conclusion

In this study, we derive the various derivative operators on functions of fields in various coordinate systems by using the differential one-forms, Hodge star operators and the exterior derivative operators. For a classical inner product we use A B) ) to imply A B$ . Also for a classical nabla operator, we use the exterior differentiation d. If we use the Hodge operator on a function, the one-forms in this function must be in terms of e_i’s. If we use differential (or exterior