Vector Fields

A gradient attaches an arrow to each point of the plane. Loosen that idea — let the arrow be anything, not necessarily a gradient — and you have a vector field: a rule that assigns to every point (x, y) a vector

\mathbf{F}(x, y) = \big(P(x, y),\; Q(x, y)\big),

where P and Q are ordinary scalar functions, the field's two components. Picture the plane carpeted with little arrows: the wind over a map, water swirling in a basin, the pull of gravity around a planet. At a point the arrow tells you the direction and strength of the flow or force living there.

Evaluating and sketching a field

Reading a field is just plugging in a point. Take the rotation field \mathbf{F}(x, y) = (-y,\; x) — the swirl you get when stirring a cup of coffee. We sketch it by evaluating at a few points.

Step 1 — evaluate at the point. Substitute the coordinates into each component. At (1, 0):

\mathbf{F}(1, 0) = (-0,\; 1) = (0, 1).

Step 2 — read it as an arrow. Draw the vector (0, 1) with its tail at the point (1, 0): it points straight up, length 1. The field is turning the point counterclockwise.

Step 3 — repeat at more points. At (0, 1) we get (-1, 0) (pointing left); at (-1, 0) we get (0, -1) (down); at (0, -1) we get (1, 0) (right). Each arrow is perpendicular to the spoke from the origin — the hallmark of pure rotation.

Step 4 — note the magnitude. The arrow's length is \|\mathbf{F}\| = \sqrt{(-y)^2 + x^2} = \sqrt{x^2 + y^2}, the distance from the origin: the field spins faster the farther out you go, like a rigid record on a turntable.

A curve that is everywhere tangent to the arrows is a field line (a streamline). For the rotation field the field lines are circles about the origin — follow the arrows and you go round and round.

Gradient fields: when a field is a slope

A special, beautifully behaved family of fields are the gradient fields: a field \mathbf{F} is a gradient field when it is the gradient of some scalar potential f(x, y),

\mathbf{F} = \nabla f = \left( \frac{\partial f}{\partial x},\; \frac{\partial f}{\partial y} \right).

Then the arrows are the steepest-uphill directions of the surface z = f(x, y) — the field "flows downhill" off a landscape. How do we test whether a given \mathbf{F} = (P, Q) arises this way? Compare mixed partials.

Step 1 — suppose it is a gradient. If \mathbf{F} = \nabla f then P = f_x and Q = f_y.

Step 2 — differentiate across. Using partial derivatives, differentiate P by y and Q by x:

\frac{\partial P}{\partial y} = f_{xy}, \qquad \frac{\partial Q}{\partial x} = f_{yx}.

Step 3 — invoke equality of mixed partials. For a function with continuous second derivatives, f_{xy} = f_{yx} (Clairaut's theorem). Therefore

\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}.

Step 4 — apply the test. For the rotation field \mathbf{F} = (-y, x) we have P_y = -1 but Q_x = +1. Since -1 \ne 1, the swirl is not a gradient field — no landscape has it as its slope. By contrast the source field \mathbf{F} = (x, y) has P_y = 0 = Q_x, and indeed \mathbf{F} = \nabla\!\left(\tfrac12(x^2 + y^2)\right).

A vector field on a region of the plane is a map \mathbf{F}(x, y) = (P, Q) assigning a vector to each point; its magnitude is \|\mathbf{F}\| = \sqrt{P^2 + Q^2} and its field lines are the curves everywhere tangent to it.
\mathbf{F} is a gradient field (or conservative) when \mathbf{F} = \nabla f for some scalar potential f; then the arrows point steepest-uphill on the surface z = f.
A necessary condition is \frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}, from equality of mixed partials. The rotation field (-y, x) fails it; the source field (x, y) = \nabla\!\big(\tfrac12(x^2+y^2)\big) passes. (When this condition is also sufficient is the subject of conservative fields.)

Vector fields are the native language of physics.

Velocity fields. In a flowing fluid, \mathbf{F}(x, y) is the velocity of the fluid at that point. The field lines are the paths a speck of dust would trace — exactly the streamlines an engineer photographs in a wind tunnel.
Electric fields. A point charge at the origin produces the field \mathbf{E} = \dfrac{k\,(x, y)}{(x^2 + y^2)^{3/2}}, arrows radiating outward (positive charge) and weakening with the square of the distance.
Gravitational fields. The pull of a planet at the origin is \mathbf{g} = -\dfrac{GM\,(x, y)}{(x^2 + y^2)^{3/2}}, the same shape pointing inward. Both the electric and gravitational fields are gradient fields — they descend from a potential — which is exactly why "potential energy" is a thing.

Explore the carpet of arrows

Pick a field and watch its arrows fill the plane — each scaled and coloured by its magnitude (short and cool near the origin, long and warm far out). The rotation field swirls, the source field bursts outward, and the gradient field \nabla\!\big(\tfrac12(x^2+y^2)\big) = (x, y) coincides with the source — a slope made visible.