Plotting a 3D vector field on 2D plane

Question

I want to visualize vector field that is of the form $(v_1(x,y), v_2(x,y), v_3(x,y))$, i.e. there is a three component vector that depends on only two coordinates $x$ and $y$ in the real plane. There are Mathematica functions VectorPlot[] and VectorPlot3D[] but both of them assume that the dimensionality of the space is the same as the number of vector components. In the latest version of Mathematica there also seems to be something called SliceVectorPlot3D[], but I don't have this in my Mathematica so I would prefer to use some other way. Furthermore, if it matters at all, the vectors are normalized so that I only care about the direction of the vector at a given point on the plane.

How do I do this?

EDIT: Here's an example of something I would like to get:

Answer 1

You can just do this with graphics primitives:

v = Flatten[
   Table[ { x , y  , { Sin[x ] , Cos[ y] , 2} } , {x, -Pi, 
     Pi, .5} , {y, -Pi, Pi, .5}], 1];
Graphics3D[{Red, 
    Arrow[Tube[{{#[[1]], #[[2]], 
        0}, {#[[1]] + #[[3, 1]], #[[2]] + #[[3, 2]], #[[3, 
          3]]}}]]} & /@ v, Axes -> True]

Answer 2

Another take, using ListPointPlot3D:

Define vector field on the x-y plane:

f[x_, y_] = {.3 Sin[x], .3 Cos[y], 1}

Define grid on the x-y plane where the vector tails will be placed:

data = Flatten[Table[{x, y, 0}, {x, -Pi, Pi, Pi/4}, {y, -Pi, Pi, Pi/4}], 1]

Plot with ListPointPlot3D using a replacement rule to convert points to vectors:

ListPointPlot3D[data] /. Point[a_] :> (Arrow[Tube[{#, # + f @@ Most@#}, .05]] & /@ a)

Easily extendable to arbitrary surfaces (just add some z[x,y] instead of 0 in the data specification.

z[x_, y_] = .3 Sin[x] + .3 Cos[y];
data = Flatten[Table[{x, y, z[x, y]}, {x, -Pi, Pi, Pi/4}, {y, -Pi, Pi, Pi/4}], 1];
ListPointPlot3D[data] /. {Point[a_] :> (Arrow[Tube[{#, # + f @@ Most@#}]] & /@ a)};
ListPlot3D[data];
Show[{%, %%}, PlotRangePadding -> {.3, .3, {.1, .9}}]

Answer 3

@J.M. is right. Define each vector as a function (eg v1[x_,y_]:=x+y) and then use ParametricPlot3D[] parametrized for x and y.

Example:

v1[x_, y_] := Sin[x]*y;
v2[x_, y_] := Cos[x]*y;
v3[x_, y_] := x;
ParametricPlot3D[{v1[x, y], v2[x, y], v3[x, y]},
 {x, -2 \[Pi], 2 \[Pi]}, {y, -2 \[Pi], 2 \[Pi]}]