Vector Fields

Main idea

A vector field puts an arrow at every point, like the wind arrows on a weather map: each one says which way things are pushed there and how hard. In the plane it's two ordinary functions, $\vec F(x, y) = \langle P(x, y), Q(x, y)\rangle$ , one for each component.

\vec F(x, y, z) = \langle {\htmlData{sym=0}{P}},\; Q,\; R\rangle, \qquad \text{gradient field: } \vec F = {\htmlData{sym=1}{\nabla f}}

What the symbols mean

$P$
x-component
A function of position.
$\nabla f$
Gradient field
Arrows pointing uphill on the graph of f, perpendicular to its level curves.

Explanation

A vector field is simply a rule that hands you an arrow at every point in space. Think of a weather map covered in little wind arrows: at each spot, the arrow's direction is which way the wind blows and its length is how fast. The notation $\vec F(x, y) = \langle P, Q\rangle$ just says that the arrow at $(x, y)$ goes $P$ to the right and $Q$ up, where $P$ and $Q$ are ordinary functions of position. That's all a vector field is: two (or three) functions working as a team.

So far a function gave you a number at each point, like a temperature. Now each point gets a direction too, and that's what lets us talk about forces and flows. The rest of this unit asks what happens when you move through a field: how much it helps or fights you, and how much it swirls.

Drawing one: pick points, evaluate $\vec F$ , draw a short arrow there. Patterns jump out quickly:

$\langle x, y\rangle$ points straight out from the origin, getting longer farther away. At each point the arrow is the position vector itself.
$\langle -y, x\rangle$ swirls counterclockwise around the origin. Notice it's $\langle x, y\rangle$ rotated $90°$ , so it's always at right angles to the line from the origin: it pushes you around, never in or out.
Gravity, $-\frac{GMm}{\|\vec r\|^3}\vec r$ , points toward the mass and weakens with distance.

Gradient fields come from a potential function $f$ : $\vec F = \nabla f$ . You met $\nabla f$ with partial derivatives as the direction of steepest climb, perpendicular to the level curves. Draw $\nabla f$ at every point and you get a whole field of uphill arrows. Electric fields are the negative gradient of the electric potential, which is why field lines cross equipotential lines at right angles. Not every field is a gradient, though; telling them apart is a big theme of this unit.

A common mistake. It's tempting to think the arrow at a point shows where a particle at that point will end up. It only shows the velocity (or force) right there. A particle following the field keeps changing direction as it reaches new arrows, which is why its path, a flow line, can curve or spiral even though each arrow is straight.

See it

A storm's wind field F = ⟨−0.3x − y, x − 0.3y⟩ with a balloon drifting in it. The balloon always moves with the arrow where it is, so it spirals in toward the eye, and notice the wind dies down as it gets closer: about 2.6 at the start, under 0.2 by the end. The −y, x parts make the swirl; the −0.3x, −0.3y parts pull it inward.

Worked examples

Example 1

Find the gradient field of $f(x, y) = x^2y$ and evaluate it at $(1, 3)$ .

1
Take the partial derivatives: $f_x = 2xy$ and $f_y = x^2$ , so $\nabla f = \langle 2xy, x^2\rangle$ .

Practice

Type your answer and check it. Solve them all to complete the lesson.

Problem 1+5 XP

Evaluate $\vec F = \langle y, -x\rangle$ at $(1, 2)$ .

Problem 2+5 XP

Find $\nabla f$ for $f = x^2y$ at $(1, 3)$ .

Problem 3+5 XP

Which field swirls counterclockwise around the origin?

Problem 4+5 XP

How long are the arrows of the constant field $\langle 3, 4\rangle$ ?

Problem 5+5 XP

Which way do the arrows of $\langle x, y\rangle$ point?

Problem 6+5 XP

A weather model gives the wind over a lake as $\vec F(x, y) = \langle 2 + y, -x\rangle$ m/s, with $x$ and $y$ in km. How fast is the wind at a sailboat at $(1, 3)$ , in m/s?

Problem 7+5 XP

The temperature on a metal plate is $T(x, y) = 100 - x^2 - 2y^2$ . A heat-seeking bug always crawls in the direction of $\nabla T$ . Along which unit vector does it head at $(2, 1)$ ?

Problem 8+5 XP

A particle drifts so that its velocity always equals $\vec F = \langle x, y\rangle$ : it follows a flow line. It starts at $(1, 2)$ . Where is it at $t = 1$ ?

Challenge problem+25 XP

The hardest one here. Get it, and you've got this lesson.

The field $\vec F = \nabla f$ comes from the saddle $f(x, y) = x^2 - y^2$ . A particle follows the flow lines of $\vec F$ (its velocity is always $\vec F$ ), starting at $(1, 1)$ . Where is it at $t = \ln 2$ ? Enter $x$ and $y$ .

Where it's used

Weather models, ocean currents, electric and magnetic fields, and the potential fields robots use to steer around obstacles are all vector fields.

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