Question about conservative vector field and a statement in Wolfram Mathworld

Question

In this reference https://mathworld.wolfram.com/LineIntegral.html in expression (7) we have that if $F(.)$ is a conservative vector field then "a Cartesian path can therefore be chosen between starting and ending point to give

$$ \int_{(a,b,c)}^{(x,y,z)} F_1(x)dx+F_2(y)dy+F_3(z)dz=\int_{(a,b,c)}^{(x,b,c)} F(x)dx+ \int_{(x,b,c)}^{(x,y,c)}F(y)dy+ \int_{(x,y,c)}^{(x,y,z)}F(z)dz $$

My questions are:

1. which "ending point" is meant here, what are $(x,y,z)$ at the upper bound?

2. what is the name of this result, so that I can search more, for the intuition and formal proof? (Of course happy to get some ideas on those directly here!)

Thanks!

Answer 1

1. $(x, y, z)$ is an arbitrary point in three-dimensional space
2. This is a straightforward consequence of that fact that the line integral of a conservative vector field is path-independent, so we can calculate along any path we wish. You could consider this as the definition of a conservative vector field, so it doesn't require a proof. (If you want to use one of the other equivalent definitions, showing path independence is not difficult. See Wikipedia.)