Does Mathematica 11 have spherical coordinate unit vectors?

Question

I have a spherical vector wave for an electric field:

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

which points in the ϕ direction. In mathematics, I would simply append a unit vector $\hat{\phi}$ to the end of the equation. Likewise for other possible components pointing in the θ and r directions. Does Mathematica 11 have such unit vectors?

Answer 1

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.

Answer 2

You need to be careful in using MMA's coordinate transform function. MMA's formulas are for converting a single point vector from one coordinate system to another. From what I am reading, you are more interested in the electric field vector {er, eth, ephi} which is similar to a velocity vector {vr, vtheta, vphi} and the like. None of the components of those vectors are angles as some components of spherical and cylindrical coordinates. The point conversions MMA uses going from Cylindrical and Spherical to Cartesian and back are.

(* Point Conversions *)
CylToCart[{ρ_,ϕ_,z_}]:={ρ Cos[ϕ],ρ Sin[ϕ],z};
SphToCart[{r_,θ_,ϕ_}]:={r Sin[θ] Cos[ϕ],r Sin[θ] Sin[ϕ],r Cos[θ]};
CartToCyl[{x_,y_,z_}]:={Sqrt[x^2+y^2],ArcTan[x,y],z};
CartToSph[{x_,y_,z_}]:={Sqrt[x^2+y^2+z^2],ArcTan[z,Sqrt[x^2+y^2]],ArcTan[x,y]};

Now we can convert E field, velocity, force, etc. vectors from Cartesian to Spherical and back.

(* Vector Component Conversons *)
(* Spherical *)
arSphToCart[ar_]:=SphToCart[{ar,θ,ϕ}];
aθSphToCart[aθ_]:=SphToCart[{aθ,θ+π/2,ϕ}];
aϕSphToCart[aϕ_]:=SphToCart[{aϕ,π/2,ϕ+π/2}];
aSphToCart[{ar_,aθ_,aϕ_}]:=arSphToCart[ar]+aθSphToCart[aθ]+aϕSphToCart[aϕ];
aCartToSph[{ax_,ay_,az_}]:={Sin[θ] (ax Cos[ϕ]+ay Sin[ϕ])+az Cos[θ],Cos[θ] (ax Cos[ϕ]+ay Sin[ϕ])-az Sin[θ],ay Cos[ϕ]-ax Sin[ϕ]};

And Cylindrical to Cartesian and back.

(* Cylindrical *)
aρCylToCart[aρ_]:=CylToCart[{aρ,ϕ,0}];
aϕCylToCart[aϕ_]:=CylToCart[{aϕ,ϕ+π/2,0}];
azCylToCart[az_]:=CylToCart[{0,0,az}];
aCylToCart[{aρ_,aϕ_,az_}]:=aρCylToCart[aρ]+aϕCylToCart[aϕ]+azCylToCart[az];
aCartToCyl[{ax_,ay_,az_}]:={ax Cos[ϕ]+ay Sin[ϕ],ay Cos[ϕ]-ax Sin[ϕ],az};

I do not know of any way to tell MMA what coordinate system a particular vector is using. As long as you are consistent, you can perform dot and cross products without conversion. For operations that include derivatives such as Div and Grad, you probably need to convert to Cartesian, operate, and then convert back.
As long as you are consistent, and you know what coordinate system a vector is using, you can use {0,0,1} for a spherical unit vector in the phi direction.