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Gradient, Divergence and Curl in Usual Coordinate Systems

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Gradient, Divergence and Curl in Usual Coordinate Systems 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 = 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 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 = 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 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 = 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 1 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 cylin- drical 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, 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 cylin- drical 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 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 cylin- drical 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 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 cylin- drical 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).
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