Gradient

The gradient of a scalar field f is a vector, written ∇f. At every point it points in the direction in which f increases fastest, and its magnitude is the rate of that increase (units of f per unit distance). For the temperature in a room, the gradient points from cool air toward the radiator; for the altitude of a landscape, it points uphill along the steepest slope.

The gradient has a deep connection to electric potential. The electric field is the negative gradient of the electric potential: E = −∇V. This one equation is why knowing the potential everywhere tells you the field everywhere: take the gradient (with a minus sign) and you are done. It also explains why electric field lines are perpendicular to equipotential surfaces — the gradient of any function is always perpendicular to its level sets, the way uphill is always perpendicular to contour lines on a topographic map.

Computationally, the gradient is just three partial derivatives bundled into a vector: ∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z). But conceptually it is one of the most useful objects in physics: it turns scalar potentials (which are easy to store and add up) into vector fields (which are what actually push things around).