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Mathematica does all its computations in an orthonormal basis. You simply need to specify what coordinate system you're working in. So for your example, you just multiply by {0,0,1}:

e[r_, θ_, ϕ_, t_] := (Sin[θ]/r)[Cos[r - t] - Sin[r - t]/r] {0, 0, 1} 

Apprently this is a pure wave in vacuum, as the divergence is zero:

Div[e[r, θ, ϕ, t], {r, θ, ϕ}, "Spherical"] (* 0 *) 

Similarly, a pure Coloumb electric field would be

 col[r_, θ_, ϕ_] := {1/r^2,0,0} Div[col[r, θ, ϕ], {r, θ, ϕ}, "Spherical"] (* 0 *) 

I suggest you look at the the tutorials tutorial/VectorAnalysis and tutorial/ChangingCoordinateSystems and the functions linked therefrom for more.

Mathematica does all its computations in an orthonormal basis. You simply need to specify what coordinate system you're working in. So for your example, you just multiply by {0, 0, 1}:

e[r_, θ_, ϕ_, t_] := (Sin[θ]/r)[Cos[r - t] - Sin[r - t]/r] {0, 0, 1} 

Apparently this is a pure wave in vacuum, as the divergence is zero:

Div[e[r, θ, ϕ, t], {r, θ, ϕ}, "Spherical"] (* 0 *) 

Similarly, a pure Coulomb electric field would be

 col[r_, θ_, ϕ_] := {1/r^2, 0, 0} Div[col[r, θ, ϕ], {r, θ, ϕ}, "Spherical"] (* 0 *) 

I suggest you look at the the tutorials tutorial/VectorAnalysis and tutorial/ChangingCoordinateSystems and the functions linked therefrom for more.

Mathematica does all its computations in an orthonormal basis. You simply need to specify what coordinate system you're working in. So for your example, you just multiply by {0,0,1}:

e[r_, θ_, ϕ_, t_] := (Sin[θ]/r)[Cos[r - t] - Sin[r - t]/r] {0, 0, 1} 

Apprently this is a pure wave in vacuum, as the divergence is zero:

Div[e[r, θ, ϕ, t], {r, θ, ϕ}, "Spherical"] (* 0 *) 

Similarly, a pure Coloumb electric field would be

 col[r_, θ_, ϕ_] := {1/r^2,0,0} Div[col[r, θ, ϕ], {r, θ, ϕ}, "Spherical"] (* 0 *) 

I suggest you look at the the tutorials tutorial/VectorAnalysis and tutorial/ChangingCoordinateSystems and the functions linked therefrom for me.

Mathematica does all its computations in an orthonormal basis. You simply need to specify what coordinate system you're working in. So for your example, you just multiply by {0,0,1}:

e[r_, θ_, ϕ_, t_] := (Sin[θ]/r)[Cos[r - t] - Sin[r - t]/r] {0, 0, 1} 

Apprently this is a pure wave in vacuum, as the divergence is zero:

Div[e[r, θ, ϕ, t], {r, θ, ϕ}, "Spherical"] (* 0 *) 

Similarly, a pure Coloumb electric field would be

 col[r_, θ_, ϕ_] := {1/r^2,0,0} Div[col[r, θ, ϕ], {r, θ, ϕ}, "Spherical"] (* 0 *) 

I suggest you look at the the tutorials tutorial/VectorAnalysis and tutorial/ChangingCoordinateSystems and the functions linked therefrom for more.

Mathematica does all its computations in an orthonormal basis. You simply need to specify what coordinate system you're working in. So for you example, you just multiply by {0,0,1}

e[r_, \[Theta]_, \[Phi]_, t_] := (Sin[\[Theta]]/r)[Cos[r - t] - Sin[r - t]/r] {0, 0, 1} 

Apprently this is a pure wave in vacuum, as the divergence is zero:

Div[e[r, \[Theta], \[CurlyPhi], t], {r, \[Theta], \[CurlyPhi]}, "Spherical"] (* 0 *) 

Similarly, a pure Coloumb electric field would be

 col[r_,t_,p_] := {1/r^2,0,0} Div[col[r,t,p], {r, t, p}, "Spherical"] (* 0 *) 

I suggest you look at the the tutorials tutorial/VectorAnalysis and tutorial/ChangingCoordinateSystems and the functions linked therefrom for me.

Mathematica does all its computations in an orthonormal basis. You simply need to specify what coordinate system you're working in. So for your example, you just multiply by {0,0,1}:

e[r_, θ_, ϕ_, t_] := (Sin[θ]/r)[Cos[r - t] - Sin[r - t]/r] {0, 0, 1} 

Apprently this is a pure wave in vacuum, as the divergence is zero:

Div[e[r, θ, ϕ, t], {r, θ, ϕ}, "Spherical"] (* 0 *) 

Similarly, a pure Coloumb electric field would be

 col[r_, θ_, ϕ_] := {1/r^2,0,0} Div[col[r, θ, ϕ], {r, θ, ϕ}, "Spherical"] (* 0 *) 

I suggest you look at the the tutorials tutorial/VectorAnalysis and tutorial/ChangingCoordinateSystems and the functions linked therefrom for me.