Greens theorem direction of line integral

[SpartanG345]

My course notes said that in greens theorem

where the closed line integral of F.r = the double integral (...)dxdy

the curve c is taken once anti-clockwise, why does it matter which way you take the line integral? Does it matter at all?

Thanks

 

[ben.zhang98]

Green theorem tells that we can transform a closed line integral of curl to a surface integral. As we know, the curl of a vector will still be a vector, so the line integral of curl will have its direction, and it points to the direction of the surface. So if you change the circular as a clockwise one, the direction will change, and you will obtain a minus.

 

[HallsofIvy]

Note that in Stoke's theorem, which generalizes Green's theorem to the boundary of a surface in three dimensions, the direction of the path integral dependes upon the direction the normal to the surface points. For example, if you surface is a sphere, whether the normal is pointing inward or outward. As ben.zhang98 said, changing the the direction changes the sign.