Vector Fields

Some quantities in the world are perfectly described by a single number at every point — temperature in a room, say, or air pressure. But others need more: the wind at a spot isn't just "20 kilometres an hour," it also blows in a particular direction; gravity at a spot doesn't just pull "with a certain strength," it pulls toward the centre of the Earth. Whenever a quantity carries both a size and a direction at every point of space, one number per point isn't enough — you need a whole arrow at every point. That is a vector field, and you have been reading them your whole life without knowing the name: the arrows on a weather map, the streamlines painted onto a river-flow diagram, even the delicate curved lines that leap into view when iron filings are sprinkled near a magnet.

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.

Now contrast it with the source field \mathbf{F}(x,y) = (x, y), sketched the very same way — plug in points first, look for the pattern second. At (1,0) we get (1,0): pointing away from the origin, the same direction as the point itself. At (0,1) we get (0,1); at (2,0) we get (2,0) — twice as long as the arrow at (1,0). Every arrow points straight outward from the origin, growing longer the farther out you go — the pattern a splash makes bursting outward, rather than a swirl going round. Two fields, evaluated with exactly the same recipe, sketch two completely different pictures: one curls, the other bursts.

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).

Vector fields are the native language of physics.

Three more worked examples

Reading a field well takes practice at three different jobs: plugging in a point, spotting a gradient, and measuring speed. Here is one worked example of each.

Example A — evaluate and draw. Let \mathbf{F}(x, y) = (x^2, xy).

At (2, 1): \mathbf{F}(2,1) = (2^2, 2\cdot 1) = (4, 2) — an arrow pointing up and to the right, longer sideways than up.

At (-1, 3): \mathbf{F}(-1,3) = ((-1)^2, (-1)\cdot 3) = (1, -3) — short to the right, long downward.

At (0, 2): \mathbf{F}(0,2) = (0^2, 0\cdot 2) = (0, 0) — no arrow at all. The whole y-axis is still for this field, since both components carry a factor of x.

Example B — spot the gradient. Let f(x, y) = x^2y - \tfrac{1}{3}y^3. Its gradient field is built one partial derivative at a time:

f_x = 2xy, \qquad f_y = x^2 - y^2,

so \nabla f = (2xy,\; x^2 - y^2) is the gradient field of f — every arrow of this field points in the steepest-uphill direction of the surface z = f(x,y).

Example C — magnitude as speed. A river's velocity field is measured as \mathbf{F}(x, y) = (3, 2y) (metres per second): a steady rightward current of 3, plus a sideways drift that grows with y. At the point (4, -1), the speed of the water is the magnitude of the arrow there:

\|\mathbf{F}(4,-1)\| = \sqrt{3^2 + (2\cdot(-1))^2} = \sqrt{9+4} = \sqrt{13} \approx 3.6 \text{ m/s}.

Notice the point's x-coordinate never entered the calculation — this particular field only changes with y, so the current is exactly as fast anywhere along a horizontal line.

A weather map bristling with wind barbs is a vector field frozen at one instant: each barb's direction shows which way the wind blows there, and the little flags on its tail count the speed — more feathers, faster wind. Follow the arrows with your eye and you can trace out field lines just like the ones you sketched by hand: they spiral into a low-pressure system (counterclockwise in the Northern Hemisphere) and spiral out of a high-pressure system, the very same curling and bursting patterns as the rotation and source fields above — just borrowed from the sky. Seen from a weather satellite, a hurricane is nothing more than a rotation field with real wind roaring through it.

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.

Fields fill space, not just the plane

A vector field pins an arrow to every point of space, not just a flat page. Here is a three-dimensional field — a vortex that swirls around the vertical axis while drifting upward, like a rising whirlwind. Drag to rotate it and look down the axis to see the swirl, or from the side to see the upward climb.

See it explained