vector field
Now imagine that at every point of the room there is not a number but a little arrow: the wind has a speed and a direction at each location, water in a river flows this way here and that way there. A vector field attaches a vector to every point of space — a quantity that carries both magnitude and direction and that varies smoothly from place to place. Wind velocity, the flow of a fluid, the gravitational force on a test mass, the electric and magnetic fields: all are vector fields.
In coordinates a vector field is written F(x, y, z) = (P, Q, R), three ordinary scalar fields P, Q, R giving the components along the x, y, z axes. You can draw it as a forest of arrows, or trace its field lines — curves that are everywhere tangent to the arrows, like streamlines threading a flow. Two derived quantities measure how the arrows behave: the divergence div F asks whether the field is spreading out of a point (a source) or piling into it (a sink), and the curl, written curl F or nabla cross F, asks how much the field swirls or circulates around a point. Together with the gradient they are built from the single del operator nabla.
Vector fields are the central object of this whole field of study. Volume I introduced them; here they become the language of fluid dynamics (the velocity field of a flow), electromagnetism (Maxwell's equations are statements about the divergence and curl of E and B), and gravitation. The great integral theorems — Green's, Stokes', and the divergence theorem — are precisely the rules for converting local information about a vector field (its divergence or curl) into global information (total flux or circulation), and vice versa.
The field F(x, y) = (-y, x) is a uniform rotation: at each point the arrow is perpendicular to the radius, so the whole plane spins counterclockwise about the origin, and its curl is the constant 2.
An arrow at every point; here they all swirl, which is exactly what a nonzero curl detects.
The components (P, Q, R) depend on the coordinate system you chose, but the field itself — the arrow at each point — does not; physically meaningful quantities like div F and curl F are coordinate-independent even though their formulas look different in polar or spherical coordinates.