Introduction to Vector Fields: Learn It 1

  • Recognize a vector field in a plane or in space
  • Sketch a vector field from a given equation
  • Identify a conservative field and its associated potential function

Vector Fields

Vector fields provide a powerful tool for describing physical phenomena that affect regions of space. They help us model gravitational forces, electromagnetic fields, fluid flow, and atmospheric patterns—anything where a quantity has both magnitude and direction at each point in space.

How can we model the gravitational force exerted by multiple astronomical objects? How can we model the velocity of water particles on a river’s surface? Figure 1 gives visual representations of such phenomena.

Figure 1(a) shows a gravitational field exerted by two astronomical objects, such as a star and a planet or a planet and a moon. At any point in the figure, the vector associated with that point gives the net gravitational force exerted by the two objects on an object of unit mass. The vectors of largest magnitude are closest to the larger object. The larger object has greater mass, so it exerts a gravitational force of greater magnitude than the smaller object.

Figure 1(b) shows the velocity of a river at points on its surface. The vector at a given point gives the velocity of the water at that point. The small vectors on the left indicate slow-flowing water. As the water moves from left to right, it encounters rapids around a rock. The speed increases, and a whirlpool forms in part of the rapids.

Two images, labeled A and B. Image A shows the gravitational field exerted by two astronomical bodies on a small object. The earth is on the left, and the moon is on the right. The earth is surrounded by long arrows pointing towards its center arranged in concentric circles. There is a break in the circle on the right, across from the moon. The moon is surrounded by smaller arrows that curve out and to the right. Image B shows the vector velocity field of water on the surface of a river with a large rock in the middle. The arrows tend to point at the same angle as the riverbank. Where the river meets the rock, the arrows point around the rock. After the rock, the some arrows point forward, and others turn back to the rock. The water flows fastest towards the middle of the river and around the rock and slowest along the riverbank.
Figure 1. (a) The gravitational field exerted by two astronomical bodies on a small object. (b) The vector velocity field of water on the surface of a river shows the varied speeds of water. Red indicates that the magnitude of the vector is greater, so the water flows more quickly; blue indicates a lesser magnitude and a slower speed of water flow.

Each figure illustrates a vector field. Intuitively, a vector field is a map that assigns a vector to each point in a region. We study vector fields in both [latex]\mathbb{R}^2[/latex] (the plane) and [latex]\mathbb{R}^3[/latex] (three-dimensional space).

vector field in [latex]\mathbb{R}^2[/latex]

A vector field [latex]\mathbf{F}[/latex] in [latex]\mathbb{R}^2[/latex] assigns a two-dimensional vector [latex]\mathbf{F}(x,y)[/latex] to each point [latex](x, y)[/latex] in a subset [latex]D[/latex] of [latex]\mathbb{R}^2[/latex]. The subset [latex]D[/latex] is the domain of the vector field.

vector field in [latex]\mathbb{R}^3[/latex]

A vector field [latex]\mathbf{F}[/latex] in [latex]\mathbb{R}^3[/latex] assigns a three-dimensional vector [latex]\mathbf{F}(x,y,z)[/latex] to each point [latex](x, y, z)[/latex] in a subset [latex]D[/latex] of [latex]\mathbb{R}^3[/latex]. The subset [latex]D[/latex] is the domain of the vector field.

You can think of a vector field as a function whose inputs are points in space and whose outputs are vectors. At each point in the domain, the vector field “tells” you which direction to point and how strongly.

Representing Vector Fields in [latex]\mathbb{R}^2[/latex]

A vector field in [latex]\mathbb{R}^2[/latex] can be represented in two equivalent ways.

Component form uses a vector with components that are two-variable functions:

[latex]\mathbf{F}(x,y) = \langle P(x,y), Q(x,y) \rangle[/latex]

Standard unit vector form uses the unit vectors [latex]\mathbf{i}[/latex] and [latex]\mathbf{j}[/latex]:

[latex]\mathbf{F}(x,y) = P(x,y)\mathbf{i} + Q(x,y)\mathbf{j}[/latex]

Component Functions

The functions [latex]P(x,y)[/latex] and [latex]Q(x,y)[/latex] are called the component functions of the vector field [latex]\mathbf{F}[/latex].

  • [latex]P(x,y)[/latex] gives the horizontal component (in the [latex]\mathbf{i}[/latex] direction)
  • [latex]Q(x,y)[/latex] gives the vertical component (in the [latex]\mathbf{j}[/latex] direction)

A vector field is continuous if its component functions are continuous. This means there are no sudden jumps or breaks in the vector field.

Let [latex]{\bf{F}}(x,y)=(2y^2+x-4){\bf{i}}+\cos(x){\bf{j}}[/latex] be a vector field in [latex]\mathbb{R}^2[/latex]. Note that this is an example of a continuous vector field since both component functions are continuous. What vector is associated with point [latex](0, -1)[/latex]?

In physical applications:

  • Force fields like gravity or electromagnetism are represented as vector fields where each vector shows the force acting at that point
  • Velocity fields like wind or water currents show the velocity at each point in the fluid
  • The magnitude of the vector indicates the strength of the force or speed of flow
  • The direction of the vector indicates which way the force points or the fluid moves