Why do we take the dot product with the normal vector when we do Stokes ...
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Same here, you got vector field which is a vector field on the xy plane, when you find the curl for this vector field, you will get the magnitude of the curl in the xy plane but pointing at k hat direction. So in order to get that magnitude of curl, you need to dot it with the normal of surface which in this case is normal to the xy plane.
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