Question

I cannot figure out how to plot a vector field with only 1 variable. Maybe Mathematica doesn’t support this. For example:

 r(t) = cost j + sint i

same as

 <cost, sint>

This doesn’t work:

VectorPlot[{cos t, sin t}, {t, 0, 2 Pi}] 

As a bonus how to take the derivative of a vector?

Answer 1

An easy workaround would be to use a 2D-VectorPlot with a dummy variable like this:

VectorPlot[
  {Cos[t], Sin[t]}, {t, 0, 2 \[Pi]}, {s, -1/2, 1/2},
  AspectRatio -> Automatic,
  VectorPoints -> {15, 3}, 
  FrameLabel -> {"t", None}
]

Or what probably makes more sense is to discretize the curve that you get when you follow the vector while increasing t. This is e.g. useful for Feynman-style Action-integrals in quantum mechanics.

Module[
  {t, dt = 0.1, vectors, startpoints, startpoint, vector, spv, spvs},
  vectors = Table[dt {Cos[t], Sin[t]}, {t, 0, 2 \[Pi], dt}];
  startpoints = Accumulate[vectors];
  spvs = Transpose[{startpoints, vectors}];
  Graphics[Table[Arrow[{spv[[1]], spv[[1]] + spv[[2]]}], {spv, spvs}]]
]