Gradient, Divergence and Curl in Usual Coordinate Systems

Albert Tarantola

September 15, 2004

Here we analyze the 3-D Euclidean space, using Cartesian, spherical or cylindrical coordinates. The words scalar, vector, and tensor mean “true” scalars, vectors and tensors, respectively. The scalar densities, vector densities and tensor densities (see main text) are named explicitly.

1 Deﬁnitions

If x φ(x) is a scalar field, its gradient is the form defined by → G = φ . (1) i ∇i If x Vi(x) is a vector density field, its divergence is the scalar density defined by → D = Vi . (2) ∇i If x F (x) is a form field, its curl (or rotational) is the vector density defined by → i Ri = εi jk F . (3) ∇j k

2 Properties

These deﬁnitions are such that we can replace everywhere true (“covariant”) derivatives by partial derivatives. This gives, for the gradient of a density,

G = φ = ∂ φ , (4) i ∇i i for the divergence of a vector density,

D = Vi = ∂ Vi , (5) ∇i i and for the curl of a form, Ri = εi jk F = εi jk∂ F (6) ∇j k j k

1 Gradient, Divergence and Curl in Usual Coordinate Systems

Albert Tarantola

September 15, 2004

1 Deﬁnitions

2 Properties

These deﬁnitions are such that we can replace everywhere true (“covariant”) derivatives by partial derivatives. This gives, for the gradient of a density,

G = φ = ∂ φ , (4) i ∇i i for the divergence of a vector density,

D = Vi = ∂ Vi , (5) ∇i i and for the curl of a form, Ri = εi jk F = εi jk∂ F (6) ∇j k j k

1 Gradient, Divergence and Curl in Usual Coordinate Systems

Albert Tarantola

September 15, 2004

1 Deﬁnitions

2 Properties

These deﬁnitions are such that we can replace everywhere true (“covariant”) derivatives by partial derivatives. This gives, for the gradient of a density,

G = φ = ∂ φ , (4) i ∇i i for the divergence of a vector density,

D = Vi = ∂ Vi , (5) ∇i i and for the curl of a form, Ri = εi jk F = εi jk∂ F (6) ∇j k j k i i jk [this equation is only valid for spaces without torsion; the general formula is R = ε jFk = i jk 1 ! ∇ ε (∂jFk 2 S jk V!) ]. These−equations lead to particularly simple expressions. For instance, the following table shows that the explicit expressions have the same form for Cartesian, spherical and cylindrical coordinates (or for whatever coordinate system).

Cartesian Spherical Cylindrical

Gx = ∂xφ Gr = ∂rφ Gr = ∂rφ Gradient Gy = ∂yφ Gθ = ∂θφ Gϕ = ∂ϕφ Gz = ∂zφ Gϕ = ∂ϕφ Gz = ∂zφ D D D Divergence = = = x y z r θ ϕ r ϕ z ∂xV + ∂yV + ∂zV ∂rV + ∂θV + ∂ϕV ∂rV + ∂ϕV + ∂zV x r r R = ∂yFz ∂zFy R = ∂θ Fϕ ∂ϕFθ R = ∂ϕFz ∂zFϕ y − θ − ϕ − Curl R = ∂zFx ∂xFz R = ∂ϕFr ∂rFϕ R = ∂zFr ∂rFz z − ϕ − z − R = ∂ F ∂ F R = ∂ Fθ ∂θ F R = ∂ Fϕ ∂ϕF x y − y x r − r r − r

3 Remarks

Although we have only defined the gradient of a true scalar, the divergence of a vector density, and the curl of a form, the definitions can be immediately be extended by “putting bars on” and “taking bars off” (see main text). As an example, from equation 1, we can immediately write the definition of the gradient of a scalar density, G = φ , (7) i ∇i from equation 2 we can write the definition of the divergence of a (true) vector field,

D = Vi , (8) ∇i and from equation 3 we can write the deﬁnition of the curl of a form as a true vector,

Ri = εi jk F , (9) ∇j k 2 2 Properties

These deﬁnitions are such that we can replace everywhere true (“covariant”) derivatives by partial derivatives. This gives, for the gradient of a density,

G = φ = ∂ φ , (4) i ∇i i for the divergence of a vector density,

Cartesian Spherical Cylindrical

3 Remarks

D = Vi , (8) ∇i and from equation 3 we can write the deﬁnition of the curl of a form as a true vector,

Ri = εi jk F , (9) ∇j k 2 i i jk [this equation is only valid for spaces without torsion; the general formula is R = ε jFk = i jk 1 ! ∇ ε (∂jFk 2 S jk V!) ]. These−equations lead to particularly simple expressions. For instance, the following table shows that the explicit expressions have the same form for Cartesian, spherical and cylindrical coordinates (or for whatever coordinate system).

Cartesian Spherical Cylindrical

3 Remarks

D = Vi , (8) ∇i and from equation 3 we can write the deﬁnition of the curl of a form as a true vector,

Ri = εi jk F , (9) ∇j k or a true form, R = g εi jk F . (10) ! !i ∇j k Although equation 8 seems well adapted to the practical computation of the divergence of a true vector, it is better to use 5 instead. For we have successively

1 D = ∂ Vi g D = ∂ (g Vi) D = ∂ (g Vi) . (11) i ⇐⇒ i ⇐⇒ g i

This last expression provides directly compact expressions for the divergence of a vector. For instance, as the fundamental density g takes, in Cartesian, spherical and cylindrical

2 i i jk [this equation is only valid for spaces without torsion; the general formula is R = ε jFk = i jk 1 ! ∇ ε (∂jFk 2 S jk V!) ]. These−equations lead to particularly simple expressions. For instance, the following table shows that the explicit expressions have the same form for Cartesian, spherical and cylindrical coordinates (or for whatever coordinate system).

Cartesian Spherical Cylindrical

3 Remarks

D = Vi , (8) ∇i and from equation 3 we can write the deﬁnition of the curl of a form as a true vector,

Ri = εi jk F , (9) ∇j k or a true form, R = g εi jk F . (10) ! !i ∇j k

Although equation 8 seems well adapted to the practical computation of the divergence of a true vector, it is better to use 5 instead. For we have successively

1 D = ∂ Vi g D = ∂ (g Vi) D = ∂ (g Vi) . (11) i ⇐⇒ i ⇐⇒ g i

This last expression provides directly compact expressions for the divergence of a vector. For instance, as the fundamental density g takes, in Cartesian, spherical and cylindrical coordinates, respectively the values 1 , r2 sinθ and r , this leads to the results of the following table.

∂Vx ∂Vy ∂Vz Divergence, Cartesian coordinates : D = + + (15) ∂x ∂y ∂z ! ! ! 1 ∂(r2Vr) 1 ∂(sinθ Vθ) 1 ∂Vϕ Divergence, Spherical coordinates : D = + + (16) r2 ∂r r sinθ ∂θ r sinθ ∂ϕ 1 ∂(rVr!) 1 ∂Vϕ ∂Vz ! ! Divergence, Cylindrical coordinates : D = + + (17) r ∂r r ∂ϕ ∂z ! ! ! These are the formulas given in elementary texts (not using tensor concepts). Similarly, although 10 seems well adapted to a practical computation of the curl, it is better to go back to equation 6. We have, successively,

1 1 Ri = εi jk∂ F g Ri = εi jk∂ F Ri = εi jk∂ F R = g εi jk∂ F . j k ⇐⇒ j k ⇐⇒ g j k ⇐⇒ ! g !i j k (18) This last expression provides directly compact expressions for the curl. For instance, as the fundamental density g takes, in Cartesian, spherical and cylindrical coordinates, respectively the values 1 , r2 sinθ and r , this leads to the results of the following table.

3 coordinates, respectively the values 1 , r2 sinθ and r , this leads to the results of the following table.

∂Vx ∂Vy ∂Vz Divergence, Cartesian coordinates : D = + + (12) ∂x ∂y ∂z 1 ∂(r2Vr) 1 ∂(sinθ Vθ) ∂Vϕ Divergence, Spherical coordinates : D = + + (13) r2 ∂r sinθ ∂θ ∂ϕ 1 ∂(rVr) ∂Vϕ ∂Vz Divergence, Cylindrical coordinates : D = + + (14) r ∂r ∂ϕ ∂z Replacing the components on the natural basis by the components on the normed basis (see main text) gives ∂Vx ∂Vy ∂Vz Divergence, Cartesian coordinates : D = + + (15) ∂x ∂y ∂z ! ! ! 1 ∂(r2Vr) 1 ∂(sinθ Vθ) 1 ∂Vϕ Divergence, Spherical coordinates : D = + + (16) r2 ∂r r sinθ ∂θ r sinθ ∂ϕ 1 ∂(rVr!) 1 ∂Vϕ ∂Vz ! ! Divergence, Cylindrical coordinates : D = + + (17) r ∂r r ∂ϕ ∂z These are the formulas given in elementary texts (not!using tensor! concepts).! Similarly, although 10 seems well adapted to a practical computation of the curl, it is better to go back to equation 6. We have, successively, 1 1 Ri = εi jk∂ F g Ri = εi jk∂ F Ri = εi jk∂ F R = g εi jk∂ F . j k ⇐⇒ j k ⇐⇒ g j k ⇐⇒ ! g !i j k (18) This last expression provides directly compact expressions for the curl. For instance, as the fundamental density g takes, in Cartesian, spherical and cylindrical coordinates, respectively the values 1 , r2 sinθ and r , this leads to the results of the following table.

R = ∂ F ∂ F x y z − z y Curl, Cartesian coordinates : R = ∂ F ∂ F (19) y z x − x z R = ∂ F ∂ F z x y − y x 1 R = (∂θ Fϕ ∂ϕFθ) r r2 sinθ − 1 Curl, Spherical coordinates : Rθ = (∂ϕF ∂ Fϕ) (20) sinθ r − r Rϕ = sinθ (∂ Fθ ∂θ F ) r − r 1 R = (∂ϕF ∂ Fϕ) r r z − z Curl, Cylindrical coordinates : Rϕ = r(∂ F ∂ F ) (21) z r − r z 1 R = (∂ Fϕ ∂ϕF ) z r r − r 3 coordinates, respectively the values 1 , r2 sinθ and r , this leads to the results of the following table.

Replacing the components on the natural basis by the components on the normed basis (see main text) gives

R = ∂ F ∂ F x y z − z y Curl, Cartesian coordinates : Ry = ∂zFx ∂xFz (22) ! ! − ! Rz = ∂xFy ∂yFx ! ! − ! ! ! ! 1 ∂(sinθFϕ) ∂Fθ Rr = r sinθ " ∂θ − ∂ϕ # ! ! ! 1 1 ∂Fr ∂(rFϕ) Curl, Spherical coordinates : Rθ = (23) r "sinθ ∂ϕ − ∂r # ! ! ! 1 ∂(rFθ) ∂Fr Rϕ = r " ∂r − ∂θ # ! ! ! 1 ∂Fz ∂(rFϕ) Rr = r " ∂ϕ − ∂z # ! ! ! ∂Fr ∂Fz Curl, Cylindrical coordinates : Rϕ = (24) ∂z − ∂r ! ! ! 1 ∂(rFϕ) ∂Fr Rz = r " ∂r − ∂ϕ # ! ! ! These are the formulas given in elementary texts (not using tensor concepts). Comment: I should remember not to put this back in a table, as it is not very readable:

Curl R = ∂ F ∂ F x y z − z y Cartesian R = ∂ F ∂ F y z x − x z R! = ∂ F! ∂ F! z x y − y x ! 1 ∂!sinθFϕ ! ∂Fθ Rr = !r sinθ ! ∂θ −! ∂ϕ 1 1 ∂Fr ! ∂rFϕ! Spherical Rθ = $ % ! r sinθ ∂ϕ − ∂r 1 ∂rFθ! ∂Fϕ ! Rϕ =$ ( ) % ! r ∂r − ∂θ ∂Fz ! ∂rFϕ! Rr = ∂ϕ ∂z ! ∂F − ∂F Cylindrical Rϕ = ! r !z ∂z − ∂r ! 1 ∂rFϕ ∂Fr Rz = ! ! ! r ∂r − ∂ϕ $ ! ! % ! Comment: What follows is not very interesting and should be suppresed. From 7 we can write g G = (g φ) , (25) i ∇i

! 4 !