# Smooth vector field along an immersion. [closed]

I need to show that every smooth vector field along an immersion has local smooth extensions.

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## closed as too localized by Will Jagy, Ryan Budney, Deane Yang, Charles Siegel, Willie WongMay 9 '11 at 3:13

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Your question is too vague and ill-formed. I suggest reading MO for a while before making your first post, to get a sense for what is expected from question-askers. Also, reading the FAQ helps if you have not yet. – Ryan Budney May 9 '11 at 0:25

@r0b0t Isn't the tangent vector field a function from the image set into $R^2$? If so, then it is not even continuous at the two interior points where the "ends" approach the image, since the tangent vectors at those points are not the limits of the tangent vectors along the approaching ends. – Dick Palais May 9 '11 at 0:29
You could consider the ring $C^\infty(\mathbb R^2)$ modulo the ideal of smooth functions that vanish on the image of your set. You could then define a "vector field" to be a derivation of this algebra. In the case of embedded submanifolds, this definition also agrees with the various others. – Theo Johnson-Freyd May 25 '11 at 2:04