Conservative Vector Fields
Let's say you have a function
TFAE means "The Following Are Equivalent". So, the following things are either all true or all false.
Independence of Path
If the entire phrase of part five is true, including continuous first partials and simply connected, then the integral is independent of path. This means that if you want to take the path integral from A to B, you can go along a straight line, a curve, split it up into two straight lines, it doesn't matter. Any possible path from two points has the same path integral. This is only true when it is conservative. That is crucial to understand. This is something specific to conservative vector fields.