This theorem asserts that if a vector field is independent of path and is continuous on an open, connected domain, then the field is conservative.
I highlighted the part of the proof I am struggling with. I don't understand how the first integral regarding C1 is independent of x and can thus equals zero. How can this be true if there is a change of x from A to x1, what does it mean for x1 to be "constant?"
Thank you.
When computing $\partial f/\partial x$, you're investigating how the value $f(x,y)$ changes as you move the point $(x,y)$ right/left along the horizontal line in the picture (i.e., you vary the value of $x$ while keeping $y$ constant). But the point $(x_1,y)$ doesn't move.