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Vector Fields as Assignments of Vectors to Points

Understand a vector field as a rule from points to vectors, and tell its point inputs apart from its vector outputs.

Beginner 10 min read

What You'll Learn

  • Define a vector field on a region in two or three dimensions.
  • Distinguish the point supplied to a field from the vector it returns.
  • Read component formulas as rules for constructing vector outputs.
  • Identify the input and output spaces in planar and spatial examples.
  • Recognize and correct common input-output notation errors.

Introduction

A scalar field assigns a number to each point in a region. A vector field makes a different kind of assignment: it gives a vector to each point. The point tells us where the rule is being applied; the vector is the result of applying it there.

This distinction is important because a point and a vector may both be written using ordered coordinates. For example, \((x,y)\) can specify a point in the plane, while \(\langle a,b\rangle\) can specify a vector with two components. Their coordinate notation may look similar, but they play different roles in a vector field.

Definition: A Rule from Points to Vectors

Definition: Let \(D\) be a region in \(\mathbb{R}^n\), where \(n=2\) or \(n=3\). A vector field on \(D\) is a function that assigns to each point of \(D\) exactly one vector in \(\mathbb{R}^n\). In notation, it is a function \(\mathbf{F}:D\to\mathbb{R}^n\). The input is a point in \(D\); the output \(\mathbf{F}(\mathbf{p})\) is a vector.

In the plane, a field is commonly written using two component functions:

$$ \mathbf{F}(x,y) = \langle P(x,y),Q(x,y)\rangle. $$

Here \((x,y)\) gives the coordinates of the input point, and \(P(x,y)\) and \(Q(x,y)\) are the two components of the output vector. In three-dimensional space, a field is commonly written with three components:

$$ \mathbf{F}(x,y,z) = \langle P(x,y,z),Q(x,y,z),R(x,y,z)\rangle. $$

The output must have the same number of components as the space in which the field is defined: two for a field on a planar region and three for a field on a spatial region. In both cases, the notation describes an assignment: each allowed input point has one vector output.

Inputs and Outputs Have Different Roles

It can help to label the two roles explicitly. In a planar field, the input might be the point \(\mathbf{p}=(x,y)\). The field then returns a vector \(\mathbf{v}=\mathbf{F}(\mathbf{p})\). The coordinates \(x\) and \(y\) are used by the component formulas to produce the components of \(\mathbf{v}\).

$$ \underbrace{(x,y)}_{\text{input point}} \quad\xrightarrow{\ \mathbf{F}\ }\quad \underbrace{\langle P(x,y),Q(x,y)\rangle}_{\text{output vector}}. $$

The point and vector may each be represented by two numbers, but those numbers do not have the same meaning. The input coordinates locate the point at which the assignment is made. The output components specify the vector assigned there. For a field in three dimensions, the same distinction applies to triples.

This is the vector-valued counterpart of a scalar field. A scalar field has a number as its output, whereas a vector field has a vector as its output. The input in either case is a point.

Worked Example: A Planar Field with Two Components

Worked Example: A Planar Field with Two Components

Consider the rule \(\mathbf{F}:\mathbb{R}^2\to\mathbb{R}^2\) given by

$$ \mathbf{F}(x,y)=\langle x+y,x-y\rangle. $$

The input is a point with two coordinates, \((x,y)\). The two component formulas are \(P(x,y)=x+y\) and \(Q(x,y)=x-y\), so the output is a vector with two components.

For the input point \((2,1)\), calculate each output component:

$$ P(2,1)=2+1=3, \qquad Q(2,1)=2-1=1. $$

Therefore the vector assigned to that point is

$$ \mathbf{F}(2,1)=\langle 3,1\rangle. $$

Check: the input \((2,1)\) is a point in \(\mathbb{R}^2\), and the result \(\langle 3,1\rangle\) is a two-component vector in \(\mathbb{R}^2\). Substituting \(x=2\) and \(y=1\) into the stated component formulas gives precisely that output.

Worked Example: A Different Planar Assignment

Worked Example: A Different Planar Assignment

Let \(\mathbf{G}:\mathbb{R}^2\to\mathbb{R}^2\) be defined by

$$ \mathbf{G}(x,y)=\langle x^2,2y\rangle. $$

The input is again a point in the plane. The first output component is the square of its first coordinate, and the second output component is twice its second coordinate. Using the input point \((-1,2)\), compute the components separately:

$$ (-1)^2=1, \qquad 2(2)=4. $$

Thus the assigned vector is

$$ \mathbf{G}(-1,2)=\langle 1,4\rangle. $$

Check: inserting the input coordinates into the original rule gives \(\langle (-1)^2,2(2)\rangle=\langle 1,4\rangle\). The input has two coordinates, and the output has two vector components, as required for a field from \(\mathbb{R}^2\) to \(\mathbb{R}^2\).

Worked Example: A Field in Three Dimensions

Worked Example: A Field in Three Dimensions

Define a field \(\mathbf{H}:\mathbb{R}^3\to\mathbb{R}^3\) by

$$ \mathbf{H}(x,y,z)=\langle x+z,2y,x-z\rangle. $$

The input point has three coordinates, \((x,y,z)\). The output vector has three components: \(x+z\), \(2y\), and \(x-z\). At the input point \((1,-1,2)\), compute them in order:

$$ x+z=1+2=3, \qquad 2y=2(-1)=-2, \qquad x-z=1-2=-1. $$

Consequently,

$$ \mathbf{H}(1,-1,2)=\langle 3,-2,-1\rangle. $$

Check: substitution into the three original component formulas yields \(\langle 1+2,2(-1),1-2\rangle=\langle 3,-2,-1\rangle\). Both the input and the output have three coordinates, but the input is a point and the output is a vector.

Why Component Formulas Define a Vector Field

Component notation gives a direct way to specify an assignment. Suppose \(D\) is a region in \(\mathbb{R}^2\), and suppose \(P\) and \(Q\) are real-valued functions defined at every point of \(D\). For each \((x,y)\in D\), the values \(P(x,y)\) and \(Q(x,y)\) form an ordered pair. That pair determines one vector in \(\mathbb{R}^2\). Therefore, the rule

$$ \mathbf{F}(x,y)=\langle P(x,y),Q(x,y)\rangle $$

assigns a vector to every point of \(D\). The same reasoning applies in three dimensions: three component functions defined at every point produce one ordered triple, and hence one vector in \(\mathbb{R}^3\).

Why the definition works: A vector in \(\mathbb{R}^n\) is specified by its \(n\) components. If each component formula returns exactly one real number for each allowed input point, the ordered list of those values specifies exactly one output vector. This gives a well-defined assignment from points to vectors.

The requirements matter. The inputs must be points in the stated region, and the component formulas must produce real numbers there. If a component cannot be assigned a value at some proposed input, the rule does not define a vector output at that input. In the examples above, the stated formulas produce real components for every point in their stated regions.

A Common Input-Output Mistake

A frequent error is to copy the input coordinates as though they were automatically the output vector. For the field

$$ \mathbf{F}(x,y)=\langle x+y,x-y\rangle, $$

and input \((2,1)\), it would be wrong to report \(\mathbf{F}(2,1)=\langle 2,1\rangle\). That answer simply repeats the coordinates of the input and does not apply the field's component formulas.

The correction is to substitute the input coordinates into both components:

$$ \mathbf{F}(2,1) = \langle 2+1,2-1\rangle = \langle 3,1\rangle. $$

Another version of the same confusion is to report only \(3\), the value of the first component, as the full output. A planar vector field requires two output components. Here the complete output is \(\langle 3,1\rangle\).

Takeaway: Read \(\mathbf{F}(\text{point})\) as “the vector assigned to this point.” The coordinates inside the parentheses are the input; the components produced by the rule are the output.

Check Your Understanding

For each question, keep track of which quantities are input points and which are output vectors.

  1. In the definition \(\mathbf{F}:D\to\mathbb{R}^2\), what kind of object is an input, and what kind of object is an output?
  2. For \(\mathbf{A}(x,y)=\langle x-y,x+y\rangle\), identify the two component functions and state the dimension of the output vector.
  3. For \(\mathbf{B}(x,y,z)=\langle y,z,x\rangle\), how many coordinates does an input point have, and how many components does the output have?
  4. Explain why writing the input point \((2,1)\) as the output of a field requires checking the field's component formulas first.
  5. What must be true of the component functions at an allowed input point for their formula to assign a vector there?

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