Vector Fields

Quick answer

A vector field F assigns a vector to each point of the plane or of space, such as the velocity of a fluid or the force of gravity. In the plane it has two component functions, F = (P, Q). Drawing arrows at a grid of points shows the pattern: rotation, flow outward, flow along a slope. The gradient of any function f is a vector field, perpendicular to the level curves of f. A flow line is a curve whose velocity at each moment equals the field there.

What you'll learn

Evaluate and sketch a vector field in the plane
Recognize rotational, radial and gradient fields
Compute the gradient field of a function
Verify that a curve is a flow line of a field

An arrow at every point

A weather map shows wind as arrows scattered across a country, each one giving the wind’s direction and speed at its location. That is a vector field: a function that takes a point and returns a vector. In the plane,

\mathbf{F}(x, y) = P(x, y)\,\mathbf{i} + Q(x, y)\,\mathbf{j} = \big(P(x, y),\ Q(x, y)\big)

with two ordinary functions $P$ and $Q$ as its components. In space a field has three components, $\mathbf{F} = (P, Q, R)$ , each a function of $x$ , $y$ and $z$ . Velocity fields describe fluids, force fields describe gravity and electricity, and gradient fields describe how a function changes.

To draw a field, evaluate it at a grid of points and draw each vector as an arrow there. The arrows below are all scaled by one factor, so their lengths compare correctly even though they are shorter than the true vectors.

The rotation field (−y, x)

Why fields are the right language

A single vector describes one velocity or one force. Most of physics needs a vector at every point at once: the velocity of every drop of a river, the pull of the Sun at every position a planet might occupy. Calculus with fields answers questions that single vectors cannot. How much work does the wind do on a kite flying along a path? How much water crosses a net each second? Does a fluid swirl, and does it spread out? A vector field packages the whole situation into one object, so these questions become integrals and derivatives of the field. The next lessons build exactly those tools: line integrals, then Green’s theorem.

Worked examples

The radial field (x, y)

xy = 1
xy = −1

The gradient field of xy with its level curves

Flow lines

Drop a cork into a fluid with velocity field $\mathbf{F}$ . Its path $\mathbf{r}(t)$ has velocity equal to the field wherever it is:

\mathbf{r}'(t) = \mathbf{F}\big(\mathbf{r}(t)\big)

Such a curve is a flow line, or streamline. In a picture of the field, flow lines are the curves that the arrows are tangent to.

Common mistakes

Practice problems

Evaluate $\mathbf{F}(x, y) = (x - y,\ x)$ at $(2, -1)$ and find the length of the vector.

Answer

$(3, 2)$ , length $\sqrt{13}$

Full solution

$P = 2 - (-1) = 3$ and $Q = 2$ . The length is $\sqrt{9 + 4} = \sqrt{13}$ .
Find the gradient field of $f(x, y) = x^2 + y^2$ and describe it.

Answer

$\nabla f = (2x,\ 2y)$ , the radial field doubled

Full solution

The arrows point straight away from the origin, twice as long as the position vector. They cross the circular level curves $x^2 + y^2 = k$ at right angles.
Find the gradient field of $f(x, y) = e^{xy}$ .

Answer

$\nabla f = \big(y e^{xy},\ x e^{xy}\big)$

Full solution

By the chain rule, $f_x = e^{xy} \cdot y$ and $f_y = e^{xy} \cdot x$ .
Find the gradient field of $f(x, y, z) = xyz$ .

Answer

$\nabla f = (yz,\ xz,\ xy)$

Full solution

Each partial derivative treats the other two variables as constants.
Which field has every arrow pointing straight toward the origin: $(y, x)$ , $(-x, -y)$ or $(-y, x)$ ?

Answer

$(-x, -y)$

Full solution

$(-x, -y)$ is the negative of the position vector, so it points from each point back to the origin. $(y, x)$ is the gradient of $xy$ and $(-y, x)$ circles the origin.
Show that $\mathbf{r}(t) = (e^t,\ e^t)$ is a flow line of $\mathbf{F}(x, y) = (x,\ y)$ and say which point it passes through at $t = 0$ .

Answer

$\mathbf{r}'(t) = (e^t, e^t) = \mathbf{F}(\mathbf{r}(t))$ ; it passes through $(1, 1)$ .

Full solution

The field at $(e^t, e^t)$ is $(e^t, e^t)$ , which equals the velocity. The cork moves out along the line $y = x$ , faster and faster.
Decide whether $\mathbf{F} = (y,\ x)$ and $\mathbf{G} = (y,\ -x)$ can be gradient fields, using the test $\tfrac{\partial P}{\partial y} = \tfrac{\partial Q}{\partial x}$ .

Answer

$\mathbf{F}$ passes, and it is $\nabla(xy)$ ; $\mathbf{G}$ fails.

Full solution

For $\mathbf{F}$ , both partials equal $1$ . For $\mathbf{G}$ , $\tfrac{\partial P}{\partial y} = 1$ but $\tfrac{\partial Q}{\partial x} = -1$ , so no function has gradient $\mathbf{G}$ .
A student draws $\mathbf{F}(0, 2) = (-2, 0)$ for the rotation field as an arrow from the origin to $(-2, 0)$ . What should the picture show?

Answer

An arrow placed at $(0, 2)$ , pointing left.

Full solution

A vector field attaches each vector to its own point. At $(0, 2)$ , on top of the circle of radius $2$ , the counterclockwise flow heads left.

Frequently asked questions

What is a vector field?

A function that assigns a vector to each point of a region. In the plane, F(x, y) = P(x, y) i + Q(x, y) j, with two component functions P and Q.

How do you sketch a vector field?

Evaluate F at a grid of points and draw each vector as an arrow starting at, or centered on, its point. Scaling all arrows by the same factor keeps the picture readable while preserving relative lengths.

What is a gradient field?

The vector field ∇f of a function f. Its arrows point in the direction f increases fastest and cross the level curves of f at right angles.

What is a flow line?

A curve r(t) whose velocity is the field at every point: r′(t) = F(r(t)). A particle carried by a fluid with velocity field F travels along a flow line.

Is every vector field a gradient field?

No. A gradient field in the plane must satisfy ∂P/∂y = ∂Q/∂x, and the rotation field (−y, x) fails that test. Fields that are gradients are called conservative.

What to learn next

Line IntegralsIntegrate along a curve: find the mass of a wire with ∫f ds, and the work a force field does along a path with ∫F · dr, using a parametrization of the curve.

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