Vector fields

Wind velocity over a map, the gravitational pull at each point near the Earth, the flow of water in a pipe: each of these assigns a vector to every point of a region. Objects like this are called vector fields, and they are the main characters of this unit. Before you can integrate along curves and across surfaces, you need to be comfortable reading, sketching and computing with them.

What a vector field is

A function $f(x, y)$ from the last units takes a point and returns a number. A vector field takes a point and returns a vector.

Evaluating a vector field is just evaluating each component. If $\mathbf{F}(x, y) = \langle x^2 - y,\ 3xy \rangle$, then $\mathbf{F}(1, 2) = \langle 1 - 2,\ 6 \rangle = \langle -1, 6 \rangle$.

Drawing a vector field

To picture $\mathbf{F}$, pick a grid of sample points and draw the vector $\mathbf{F}(x, y)$ as an arrow located at $(x, y)$. Real vectors are often too long to fit, so plots scale every arrow down by the same factor. The directions and the relative lengths are what you read from the picture.

Take the rotation field $\mathbf{F}(x, y) = \langle -y, x \rangle$. At $(1, 0)$ it points straight up, $\langle 0, 1 \rangle$; at $(0, 1)$ it points left, $\langle -1, 0 \rangle$. In general $\mathbf{F}(x, y) \cdot \langle x, y \rangle = -xy + yx = 0$, so each arrow is perpendicular to the position vector, and $|\mathbf{F}(x, y)| = \sqrt{x^2 + y^2}$, so arrows get longer as you move away from the origin.

Compare the radial field $\mathbf{G}(x, y) = \langle x, y \rangle$, where every arrow points directly away from the origin.

A quick way to sketch any field by hand: find where it is zero, check what it does on the axes, then check a few diagonal points. You rarely need more than a dozen arrows to recognize the pattern.

Gradient fields

You have already met one very important family of vector fields. If $f$ is a differentiable function of two or three variables, its gradient

$$\nabla f = \langle f_x, f_y \rangle \qquad \text{or} \qquad \nabla f = \langle f_x, f_y, f_z \rangle$$

is a vector field, called a gradient field. At each point it points in the direction of fastest increase of $f$ and is perpendicular to the level curve (or level surface) through that point.

The name comes from physics: when a force field is conservative, mechanical energy is conserved. In the next two lessons you'll see why conservative fields make line integrals dramatically easier, and how to test whether a field is conservative.

Fields in space and inverse-square fields

Three-dimensional fields work the same way, although they are harder to draw. The most famous one is the gravitational field of a mass $M$ at the origin. Write $\mathbf{r} = \langle x, y, z \rangle$ and $r = |\mathbf{r}|$. Newton's law says

$$\mathbf{F}(x, y, z) = -\frac{GM}{r^3}\,\mathbf{r},$$

which points toward the origin and has magnitude $\dfrac{GM}{r^2}$: an inverse-square field. The electric field of a point charge has the same shape. These fields reappear at the end of the unit, where the divergence theorem explains Gauss's law.

Flow lines

If $\mathbf{F}$ is the velocity field of a fluid, a particle dropped into the fluid moves along a curve $\mathbf{r}(t)$ whose velocity always matches the field: $\mathbf{r}'(t) = \mathbf{F}(\mathbf{r}(t))$. Such a curve is a flow line (or streamline). Geometrically, flow lines are curves that are tangent to the arrows at every point. For $\langle -y, x \rangle$ the flow lines are circles centered at the origin; for $\langle x, y \rangle$ they are rays leaving the origin.

Practice

Practice 1

Let $\mathbf{F}(x, y) = \langle x - y,\ xy \rangle$. Find $\mathbf{F}(3, 2)$.

Your answer

Enter a point like (2, -3)

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Practice 2

Let $\mathbf{F}(x, y, z) = \langle yz,\ xz,\ xy \rangle$. Find $\mathbf{F}(1, 2, 3)$.

Your answer

Enter a point like (2, -3)

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Practice 3

For the rotation field $\mathbf{F}(x, y) = \langle -y, x \rangle$, find $|\mathbf{F}(3, 4)|$.

Your answer

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

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Practice 4

Let $f(x, y) = x^2 y + y^3$. Find the gradient field $\nabla f$ at the point $(1, 2)$.

Your answer

Enter a point like (2, -3)

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Practice 5

Which formula matches this vector field?

A$\mathbf{F}(x, y) = \langle x, y \rangle$B$\mathbf{F}(x, y) = \langle -y, x \rangle$C$\mathbf{F}(x, y) = \langle y, -x \rangle$D$\mathbf{F}(x, y) = \langle -x, -y \rangle$

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Practice 6

Let $\mathbf{F}(x, y) = \dfrac{\langle x, y \rangle}{x^2 + y^2}$. Find $|\mathbf{F}(3, 4)|$.

Your answer

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

CheckShow hint

Practice 7

The flow lines of $\mathbf{F}(x, y) = \langle x, 2y \rangle$ satisfy $x'(t) = x$ and $y'(t) = 2y$. The flow line through $(1, 1)$ lies on a curve $y = g(x)$. Find $g(x)$.

Your answer

Enter an expression, e.g. 3x^2 - 2x + 1

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Practice 8

The conservative field $\mathbf{F}$ has potential function $f(x, y) = xe^y$. Find $\mathbf{F}(2, 0)$.

Your answer

Enter a point like (2, -3)

CheckShow hint