conservative field

Some force fields are economical in a precise way: the work they do moving you from one place to another depends only on where you start and where you finish, never on the scenic route in between. Hike from base camp to summit by any trail you like — for a conservative field the total work is the same. Such a field 'conserves' a quantity (an energy) that can be assigned to each location, and that is what gives it its name.

A vector field F is conservative exactly when it is the gradient of some scalar field: F = grad phi for a potential phi. Several conditions turn out to be equivalent on a nice (simply connected) region: F is a gradient; the line integral of F is path-independent; the line integral around every closed loop is zero; and curl F = 0 everywhere (the field is irrotational). The chain that links them is the gradient theorem, which says integral over C of grad phi dot dr = phi(end) - phi(start). In two dimensions the practical test is the cross-partial condition partial P/partial y = partial Q/partial x.

Conservative fields are why energy conservation works. Gravity and the electrostatic field are conservative, so we can speak of gravitational and electric potential energy as a well-defined function of position; the kinetic energy a falling object gains depends only on the height dropped. The whole machinery of potentials, voltage, and energy diagrams rests on conservativeness, and recognizing that a field is conservative collapses a hard line integral into a simple subtraction of two potential values.