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Contours are used to visualize vector fields. And I wonder whether it's possible to prove such a result that the contours of a continuous vector field is a continuous and smooth curve.

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A continuous vector field does not entail smooth contours (level sets). –  anon Feb 22 '12 at 4:20
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I've heard of contours of scalar fields, and streamlines of vector fields, but what is the contour of a vector field? –  Rahul Feb 22 '12 at 4:29
    
@RahulNarain yeah, but a conservative vector field always corresponds to a scalar field. –  Strin Feb 22 '12 at 12:08

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