Vector Fields & Scalar Fields

A Vector at Every Point

Think of a room with a fan running. At every point in the room, the air has some velocity - a speed and a direction. That assignment of a vector to every point is a vector field.

Outside, the wind does it too: at each location on a weather map, there's an arrow showing which way the air is moving and how fast. The field doesn't live at just one point. It lives everywhere.

Formally, a vector field in the plane assigns to each point $(x,y)$ a vector:

It turns out that a vector field is just a pair of ordinary scalar functions dressed up as a vector. But treating them as a unit reveals structure you'd never see looking at the components separately. As Kaplan puts it: "Each vector field is equivalent to a triple of scalar functions" (in 3D). Let's see this in action.

Below are four classic fields. Switch between them and watch how the arrows change - their length and color both encode the magnitude. Or type your own formulas.

Expansion ($\mathbf{v} = x\mathbf{i} + y\mathbf{j}$): every arrow points away from the origin, and the farther out you go, the longer the arrow. This is an expanding field - like an explosion in slow motion. At the origin itself, $\mathbf{v} = \mathbf{0}$.

Rotation ($\mathbf{v} = -y\mathbf{i} + x\mathbf{j}$): every arrow is perpendicular to the line from the origin. This is pure rotation - like water circling a drain, but without moving inward. These are the velocity vectors of a rigid body rotating about the $z$-axis.

Dipole: the electric field from two infinite charged wires at $(\pm 1, 0)$. Notice the saddle point between them and how the field lines splay outward from each wire.

The Gravitational Field

Newton figured out something remarkable: every mass in the universe pulls on every other mass, and the force weakens as the square of the distance.

Place a big mass $M$ at the origin. A small mass $m$ at position $\mathbf{r}$ feels a force:

The minus sign points the force inward - toward the attracting mass. The $1/r^2$ means: double the distance, quarter the force. This is an inverse-square law, and it shows up everywhere in physics - gravity, electrostatics, radiation intensity.

We'd expect the arrows to get shorter as we move away from the origin. Let's check. Hover over the plot to read the magnitude at any point.

What happens at the origin itself? The force blows up - $|\mathbf{F}| \to \infty$ as $r \to 0$. This is a singularity. The field is undefined there. In physics, this signals that we need a more refined model: a real mass isn't a point, and once you're inside the object, the field changes character.

Scalar Fields and Level Curves

Now let's flip the script. Instead of assigning a vector to each point, we assign a plain number.

Temperature in a room - at every point there's a single number (degrees). Altitude on a map - every location has an elevation. Air pressure on a weather map. Voltage in a circuit. These are all scalar fields: a function $f(x,y)$ (or $f(x,y,z)$ in 3D) that returns a single value.

The interesting question is: how do we visualize a scalar field? We can't draw arrows, but we can draw level curves - the contours where $f = c$ for various constants $c$. Think topographic maps, where each contour line is a fixed elevation.

Here's the key geometric insight: the closer the level curves are packed, the faster the function is changing. This idea leads directly to the gradient - which we'll meet in §3.3.

For $f = xy$, the level curves $xy = c$ are hyperbolas. For $c > 0$ they live in quadrants I and III; for $c < 0$ in quadrants II and IV. For $c = 0$, the level "curve" is the coordinate axes themselves.

For $f = x^2 + y^2$, the level curves are circles centered at the origin. The function is the squared distance from the origin - a perfect bowl shape.

The Bridge: Scalar Fields and Vector Fields

Here's the beautiful connection. Starting from a scalar field $f$, we can build a vector field - its gradient, $\nabla f$. And starting from a vector field $\mathbf{v}$, we can extract a scalar field - its magnitude $|\mathbf{v}|$, or (later) its divergence or curl.

Below, the left panel shows the contour plot of $f(x,y) = x^2 - y^2$. The right panel shows the gradient arrows of the same function. Watch how the gradient arrows are perpendicular to the level curves and point "uphill."

We'll explore gradients fully in §3.3. But notice the pattern already - the two worlds really are connected.

Now here's a deeper question: not every vector field comes from a scalar field this way. Figuring out which ones do - and what that means physically - is one of the big questions of this chapter.

Look at the rotation field $\mathbf{v} = \langle -y, x \rangle$. Could this be the gradient of some function? Think about it: if it were, then following the gradient would always increase $f$. But following this field takes you in circles - you'd return to where you started with the same value of $f$. So this field is not a gradient. We'll make this precise with the concept of "curl" later in the chapter.