# Conservative Vector Fields

## Determining conservativeness

# TFAE

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.

# Fundamental Theorem of Line Integrals

It turns out $\int_{g(t)} ydx + xdy$ is actually very easy to solve. The equation above means that the form above is exact. $(\dfrac{\partial}{\partial y} y = \dfrac{\partial}{\partial x} x = 1)$. In this equation, $f = xy$, $\nabla f = f_x dx + f_ydy = ydx + xdy$ So, we plug in f(t) at the end points. $$f(\pi) = 0 * 1 = 0. f(0) = 0 * 1 = 0. f(\pi) - f(0) = \boxed{0}$$