96 - Path Parametrization and Independence

When we go along the same curve in integration we add up infinitely small changes along that curve. When we walk the reverse path we just invert these, decreases become increases and vice versa.

Path Independence of Scalar Fields

For a scalar field we only have values, no direction, so the integrals will be the same no matter what. Because of this we say that scalar fields are path independent, meaning you travel the same distance getting to a point no matter the path you took.

Path Dependence of Vector Fields

When we have a vector field we can get to some point by a multitude of paths, but the back path may not be the same, so the distance we traveled depends on the path we took. Because of this we say that vector fields are path dependent.

Conservative Vector Fields

In the case that a vector field has the same distance no matter the path taken, we say that it is a conservative vector field, meaning that any integral of such field is independent of the path taken.

The following are equivalent:

is a conservative vector field
is the gradient of a scalar field
Closed loop line integrals of are always zero
Integrals of are path-independent
is irrotational, meaning it has zero curl everywhere

97 - Closed-Curve Integrals and the Fundamental Theorem of Line Integrals