Use the fundamental theorem of line integrals to calculate r c ~f â·d~r exactly, if ~f = 2x 1/3~i+e y/7 ~j, and c is the quarter of the unit circle in the first quadrant, traced counterclockwise from (1,0) to (0,1).

Assuming the vector field is , we want to find a scalar function such that . We require




Integrating both sides of the first PDE with respect to gives



Differentiating with respect to gives






Such a scalar function exists, so the fundamental theorem holds and any integral along any path is determined exactly by its endpoints. In this case,

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