Plotting a path in a vector field

I am trying to make a visual representation of the curve $y=x^{2}$, with $x\in\left[0,1\right]$ in the vector field defined by $\vec{F}=\left[-y,x\right]$.

However, as far as I can tell there does not seem to be any facility which allows me to do this in Mathematica (I've looked through as much of the documentation as I could find which seemed pertinent).

I know how to do each of these separately, e.g.

Plot[x^2,{x,0,1}]
VectorPlot[{-y,x},{x,-3,3},{y,-3,3}]

However, I am unsure how I can combine these two plots (or plot them both at the same time).

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You should make use of Show and appropriate options e.g. we added PlotRange, AspectRatio etc. :

Show[{ Plot[x^2, {x, -3, 3}, PlotRange -> 3, PlotStyle -> {Thick, Darker @ Green}],
VectorPlot[{-y, x}, {x, -3, 3}, {y, -3, 3}]}, AspectRatio -> 1]

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Thank you, this seems to be what I was looking for! +1 –  Shaktal Jul 13 '12 at 23:21

Instead of using Show you might prefer the paradigm of David Park's Presentations package:

<< Presentations`

Draw2D[{VectorDraw[{-y, x}, {x, -3, 3}, {y, -3, 3}],
Darker@Green, Draw[x^2, {x, -3, 3}]},
PlotRange->3, Axes->True]
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