## Differentiating fields

This chapter introduces the directional derivatives at a point and the three directional derivatives in directions parallel to the Cartesian axes, known as partial derivatives. It examines gradient, divergence and curl fields, and shows how these fields can be expressed in terms of partial derivatives and how the field values are calculated. The chapter discusses a selection of physical laws and processes that depend on the gradient of a scalar field or the divergence or curl of a vector field. There is a gradient vector field associated with any scalar field. The magnitude of the gradient vector at a point is the largest directional derivative of the scalar field at that point. The direction of the gradient vector is the direction in which this largest value occurs. The flux of a vector field can be visualised by sketching field lines to show both the magnitude and direction of the field.