6.1 Vector Fields

Big idea

Everything integrated so far attached a number to a point: a density, a temperature, a height above the plane. This chapter changes the cargo. A vector field attaches a whole vector to each point, so at every location you carry a direction and a magnitude rather than a single value. The picture is a plane or a region of space bristling with arrows, one arrow per point, and the formula for the field is the rule that produces the arrow from the coordinates.

The reason to build such an object is that most of the quantities physics cares about are of exactly this type. The velocity of a fluid at a point in a river, the force a magnet exerts at a point in the room, the gravitational pull at a point above a planet: each is a vector that depends on where you are standing. Once you have the field, the natural questions are how much a particle gains while moving through it, how much of it crosses a surface, and whether the field is doing anything rotational. Those three questions are the rest of the chapter.

You have already met one supplier of vector fields. The gradient of a function of two or three variables is a vector at each point, so every differentiable scalar function generates a field automatically. Fields that arise this way are the well behaved ones, and recognizing them on sight is worth real effort later, because integrals of gradient fields collapse to a subtraction of two numbers.

For now the work is descriptive. Learn to go from a formula to a picture and from a picture back to a formula, and learn the two or three families of fields that show up everywhere: fields that point straight out from a center, fields that swirl around one, and fields that are constant. Almost every example in the chapter is built from those pieces.

Decoder

A vector field on a region is a rule that assigns exactly one vector to each point of the region. Writing that vector in components turns the single rule into two ordinary functions of two variables in the plane, or three functions of three variables in space.

The phrase carrying the content is “to each point”. The input of a vector field is a location, not a time and not a parameter along a curve. When you evaluate $\mathbf{F}(2, -1)$ you are asking what arrow sits at the point $(2,-1)$, and you draw that arrow with its tail at that point. Drawing it from the origin instead is the standard first mistake, and it destroys the picture: the whole content of a field is which arrow sits where.

A field is only as smooth as its component functions. If $P$ and $Q$ are continuous, the arrows turn gradually as you move, which is what makes the sketches look like flows rather than noise. Where a denominator vanishes the field is undefined, and that single missing point can change the answers to questions in later sections dramatically.

A common source of confusion is scale. A picture of a field shortens every arrow by the same factor so the drawing stays readable, so arrow lengths in a sketch tell you relative magnitude, not absolute magnitude. Read the formula, not the ink, when you need the actual size.

Definitions and results

Vector field in the plane. A vector field on a region $D$ of the plane is a function $$ \mathbf{F}(x,y) = P(x,y)\,\mathbf{i} + Q(x,y)\,\mathbf{j} $$ where $P$ and $Q$ are scalar functions on $D$, called the component functions.

Vector field in space. On a region of $\mathbf{R}^3$ the same definition reads $\mathbf{F}(x,y,z) = P\,\mathbf{i} + Q\,\mathbf{j} + R\,\mathbf{k}$, with three component functions. The angle-bracket notation $\langle P, Q, R\rangle$ means the same thing.

Sketching a field. Choose a grid of sample points. At each one, evaluate the field and draw the resulting vector with its tail at that point, shortening all of them by one common factor. Look for points where the field is zero, curves along which the direction is constant, and the general trend of the magnitude.

Radial fields. A radial field points along the line from the origin to the point. With $\mathbf{r} = \langle x, y\rangle$, the field $\mathbf{F} = \mathbf{r}$ points outward with magnitude equal to the distance from the origin, and $\mathbf{F} = -\mathbf{r}$ points inward. Dividing by $\|\mathbf{r}\|$ produces a unit radial field, and dividing by $\|\mathbf{r}\|^3$ produces an inverse square field, the shape of gravity and of the electric field of a point charge.

Rotational fields. The field $\mathbf{F} = \langle -y, x\rangle$ is perpendicular to the position vector at every point, since $\langle -y, x\rangle \cdot \langle x, y\rangle = -xy + xy = 0$. Its magnitude is $\sqrt{x^2+y^2}$, so it turns counterclockwise and speeds up away from the origin, like a rigid disk spinning. Reversing the sign reverses the spin.

Gradient fields. If $f$ is a differentiable scalar function, then $\nabla f$ is a vector field, called the gradient field of $f$. A field $\mathbf{F}$ that equals $\nabla f$ for some $f$ is called conservative, and $f$ is a potential function for it.

Gradients and level curves. At each point the gradient is perpendicular to the level curve of $f$ through that point and points toward larger values of $f$. So the arrows of a gradient field cross the contour lines of $f$ at right angles, which is often the fastest way to sketch such a field.

Unit fields and normalization. Dividing a nowhere-zero field by its own magnitude produces a field of unit vectors carrying the direction information alone. This is how a flow direction is separated from a flow speed.

Worked examples

A shear field

Describe $\mathbf{F}(x,y) = \langle y, 0 \rangle$.

Every arrow is horizontal, since the second component is always zero. On the $x$-axis the field is the zero vector. Above the axis, $y > 0$ gives arrows pointing in the positive $x$ direction, longer the higher you go; below the axis they point the other way. The picture is a flow that slides to the right on top, to the left underneath, and stands still along the middle line. This is the standard model of a shearing fluid, and the fact that neighboring layers slide past each other is exactly what the curl will measure later.

A radial field and its potential

Take $\mathbf{F}(x,y) = \langle x, y\rangle$. Its magnitude at $(x,y)$ is $\sqrt{x^2+y^2}$, so the arrows grow linearly with distance from the origin and each one points directly away from it.

This field is a gradient field. Let $f(x,y) = \tfrac{1}{2}\left(x^2+y^2\right)$. Then $$ \nabla f = \left\langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right\rangle = \langle x, y \rangle = \mathbf{F} $$ The level curves of $f$ are circles centered at the origin, and the arrows of $\mathbf{F}$ point straight out along radii, which are indeed perpendicular to those circles. The two descriptions agree.

Checking a gradient against a contour

Let $f(x,y) = x^2 y$, so $\nabla f = \langle 2xy, \; x^2 \rangle$. At the point $(1,2)$ the gradient is $\langle 4, 1\rangle$.

Test the perpendicularity claim there. The level curve through $(1,2)$ is $x^2 y = 2$. Differentiating implicitly, $2xy + x^2 \frac{dy}{dx} = 0$, so at $(1,2)$ we get $4 + \frac{dy}{dx} = 0$ and the tangent direction is $\langle 1, -4\rangle$. The dot product with the gradient is $4(1) + 1(-4) = 0$, so the gradient is perpendicular to the contour, as promised.

A field in space

Consider $\mathbf{F}(x,y,z) = \langle -y, \; x, \; 2 \rangle$.

The first two components are the planar rotational field and the third is a constant upward push. So in any horizontal plane the arrows swirl counterclockwise about the $z$-axis, and every arrow also leans upward by the same fixed amount. A particle carried by this field spirals up a helix. Its magnitude is $\sqrt{x^2+y^2+4}$, never zero, and smallest on the $z$-axis itself where the swirl dies out and only the upward push remains.

Practice

These problems ask you to evaluate a field at given points, match formulas to pictures, identify radial and rotational pieces, and produce the gradient field of a given function.

Quiz

Five items on evaluating and describing vector fields, recognizing radial and rotational types, and building gradient fields.