Gradient, Divergence and Curl in Usual Coordinate Systems Albert - PowerPoint PPT Presentation

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 co- ordinates. 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 i = ∇ i φ . (1) If x → V i ( x ) is a vector density field, its divergence is the scalar density defined by i V i . D = ∇ (2) If x → F i ( x ) is a form field, its curl (or rotational ) is the vector density defined by R i = ε ijk ∇ j F k . (3)

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 i = ∇ i φ = ∂ i φ , (4) for the divergence of a vector density, i V i = ∂ i V i , D = ∇ (5) and for the curl of a form, R i = ε ijk ∇ j F k = ε ijk ∂ j F k (6) [this equation is only valid for spaces without torsion; the general formula is R i = ε ijk ∇ j F k = ε ijk ( ∂ j F k − 1 2 S jk ℓ V ℓ ) ]. 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 cylin- drical coordinates (or for whatever coordinate system).

Cartesian Spherical Cylindrical G x = ∂ x φ G r = ∂ r φ G r = ∂ r φ Gradient G y = ∂ y φ G θ = ∂ θ φ G ϕ = ∂ ϕ φ G z = ∂ z φ G ϕ = ∂ ϕ φ G z = ∂ z φ D D D Divergence = = = ∂ x V x + ∂ y V y + ∂ z V z ∂ r V r + ∂ θ V θ + ∂ ϕ V ϕ ∂ r V r + ∂ ϕ V ϕ + ∂ z V z R x = ∂ y F z − ∂ z F y R r = ∂ θ F R r = ∂ ϕ F z − ∂ z F ϕ − ∂ ϕ F ϕ θ R y = ∂ z F x − ∂ x F z R θ = ∂ ϕ F r − ∂ r F R ϕ = ∂ z F r − ∂ r F z Curl ϕ R z = ∂ x F y − ∂ y F x R ϕ = ∂ r F R z = ∂ r F θ − ∂ θ F r ϕ − ∂ ϕ F 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 i = ∇ i φ , (7) from equation 2 we can write the definition of the divergence of a (true) vector field, i V i , D = ∇ (8) and from equation 3 we can write the definition of the curl of a form as a true vector, R i = ε ijk ∇ j F k , (9) or a true form, R ℓ = g ℓ i ε ijk ∇ j F k . (10)

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 D = 1 D = ∂ i V i g D = ∂ i ( g V i ) g ∂ i ( g V i ) . ⇐ ⇒ ⇐ ⇒ (11) 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 r 2 sin θ and r , this leads to the results of the coordinates, respectively the values 1 , following table. D = ∂ V x ∂ x + ∂ V y ∂ y + ∂ V z Divergence, Cartesian coordinates : (12) ∂ z ∂ ( r 2 V r ) ∂ ( sin θ V θ ) + ∂ V ϕ D = 1 1 + Divergence, Spherical coordinates : ∂ ϕ (13) r 2 ∂ r sin θ ∂ θ ∂ ( rV r ) ∂ ϕ + ∂ V z + ∂ V ϕ D = 1 Divergence, Cylindrical coordinates : (14) r ∂ r ∂ z Replacing the components on the natural basis by the components on the normed basis (see main text) gives D = ∂ � ∂ x + ∂ � V x ∂ y + ∂ � V y V z Divergence, Cartesian coordinates : (15) ∂ z ∂ ( r 2 � ∂ ( sin θ � ∂ � V r ) V θ ) V ϕ D = 1 1 1 + + Divergence, Spherical coordinates : (16) r 2 ∂ r r sin θ ∂ θ r sin θ ∂ ϕ ∂ ( r � ∂ � ∂ ϕ + ∂ � V r ) V ϕ V z D = 1 + 1 Divergence, Cylindrical coordinates : (17) r ∂ r r ∂ z

� � � r ∂ r r ∂ z ∂ ϕ 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, R i = 1 R ℓ = 1 R i = ε ijk ∂ j F k g R i = ε ijk ∂ j F k g ε ijk ∂ j F k g g ℓ i ε ijk ∂ j F 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 , r 2 sin θ and r , this leads to the results of the following table. R x = ∂ y F z − ∂ z F y Curl, Cartesian coordinates : R y = ∂ z F x − ∂ x F z (19) R z = ∂ x F y − ∂ y F x 1 R r = r 2 sin θ ( ∂ θ F ϕ − ∂ ϕ F θ ) 1 Curl, Spherical coordinates : R θ = sin θ ( ∂ ϕ F r − ∂ r F ϕ ) (20) R ϕ = sin θ ( ∂ r F θ − ∂ θ F r ) R r = 1 r ( ∂ ϕ F z − ∂ z F ϕ ) R ϕ = r ( ∂ z F r − ∂ r F z ) Curl, Cylindrical coordinates : (21) R z = 1 r ( ∂ r F ϕ − ∂ ϕ F r )

Replacing the components on the natural basis by the components on the normed basis (see main text) gives R x = ∂ y � � F z − ∂ z � F y R y = ∂ z � � F x − ∂ x � Curl, Cartesian coordinates : F z (22) R z = ∂ x � � F y − ∂ y � F x � � ∂ ( sin θ � − ∂ � ϕ ) 1 F F θ � R r = r sin θ ∂ θ ∂ ϕ � � ∂ � ∂ ϕ − ∂ ( r � ϕ ) R θ = 1 1 F r F � Curl, Spherical coordinates : (23) r sin θ ∂ r � � ∂ ( r � − ∂ � θ ) R ϕ = 1 F F r � r ∂ r ∂ θ � � ∂ � ∂ ϕ − ∂ ( r � R r = 1 F z F ϕ ) � r ∂ z R ϕ = ∂ � ∂ z − ∂ � F r F z � Curl, Cylindrical coordinates : (24) ∂ r � � ∂ ( r � − ∂ � F ϕ ) F r R z = 1 � 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: