Proving a theorem in line integrals

[anhtu2907]

At the bottom of the picture, I couldn't understand why differentiating with respect to x gives the first integral at the right-hand side 0. Thanks for reading.

 

[HallsofIvy]

In my opinion, that is simply wrong. Rather than taking C_1 to be "any path from (a, b) to (x_1, y)" we must choose C_1 to be the vertical line from (a, b) to (a, y) then take C_2 to be the horizontal line from (a, y) to (x, y).

 

[Gianmarco]

I'm no math pro, but my guess would be that since you have the hypothesis that your integral is path independent, then
<br /> \int_{C_1} F\cdot dr = \int^{(x1,y)}_{(a,b)} F\cdot dr = f(x1,y) - f(a,b)<br />
which differentiated w.r.t. x gives 0 and w.r.t. y does not (since the point (x,y) is arbitrary, y is arbitrary but x1,a,b are fixed)