What Makes a Vector Field Constant?
A vector field assigns a vector to each point in its domain. Its arrows may change from point to point: their directions, lengths, or both may vary. A constant vector field is the special case in which the assigned vector never changes. Consequently, its diagram has the same arrow repeated at every point of the domain.
The domain matters. A component formula might involve a coordinate, but that coordinate may be fixed on the domain under consideration. To decide whether a field is constant, compare its values at points in its domain—not just the symbols appearing in its formula.
For example, if \(D=\mathbb R^2\) and \(\mathbf F(x,y)=\langle 3,-2\rangle\), the same vector is assigned at every point. But when the domain is restricted, coordinate dependence in a formula is not by itself enough to show that the field varies. The test is always whether the assigned values actually differ on that domain.
Reading a Constant Field from Its Components
In two dimensions, write a field in component form as \(\mathbf F(x,y)=\langle P(x,y),Q(x,y)\rangle\). It is constant on \(D\) precisely when there are fixed numbers \(a\) and \(b\) such that \(P(x,y)=a\) and \(Q(x,y)=b\) for every \((x,y)\in D\). In three dimensions, the same test applies to all three components.
If the domain is all of \(\mathbb R^2\) or all of \(\mathbb R^3\), a formula with constant components is an immediate way to recognize a constant field. More generally, coordinate-independent components are a sufficient test on such a domain. For a restricted domain, however, evaluate the components on that domain. A coordinate that appears in a component formula may have only one value there.
Once the fixed vector \(\mathbf c\) is known, sketching the field amounts to drawing a copy of that vector based at each chosen point. For instance, if \(\mathbf c=\langle 3,-2\rangle\), each arrow points three units in the positive \(x\)-direction and two units in the negative \(y\)-direction, relative to its own base point. All arrows have the same direction and length.
Worked Example: A Constant Field with Two Nonzero Components
Let \(\mathbf F:\mathbb R^2\to\mathbb R^2\) be given by \(\mathbf F(x,y)=\langle 3,-2\rangle\). Find the field values at three points and describe a sketch.
At \((0,0)\), the component formula gives \(\mathbf F(0,0)=\langle 3,-2\rangle\).
At \((1,2)\), the formula has no \(x\)- or \(y\)-dependent terms, so \(\mathbf F(1,2)=\langle 3,-2\rangle\).
At \((-2,1)\), it likewise gives \(\mathbf F(-2,1)=\langle 3,-2\rangle\).
Thus all three sample arrows have the same displacement: three units right and two units down. Their lengths are all \(\sqrt{3^2+(-2)^2}=\sqrt{13}\). To sketch the field, place arrows with that displacement at the chosen grid points. Check: substituting each point into the original formula returns \(\langle 3,-2\rangle\), confirming that the sampled arrows agree.
Why the Arrows Are Equal and Parallel
An arrow based at a point \(\mathbf p\) with vector value \(\mathbf c\) ends at \(\mathbf p+\mathbf c\). Its displacement from base to tip is therefore \(\mathbf c\), regardless of the base point. If the field is constant, every arrow has this same displacement. That is why the arrows are equal in length and direction, and therefore parallel when \(\mathbf c\ne\mathbf 0\).
The length and direction statements follow directly from the definition of a vector arrow. Its length is the magnitude of its displacement, \(\|\mathbf c\|\), which is unchanged from point to point. If \(\mathbf c\ne\mathbf 0\), its direction is also unchanged. Conversely, if the arrows at every point represent the same displacement, then \(\mathbf F\) takes the same vector value everywhere, so the field is constant.
Read the component values, or evaluate them on the specified domain, to identify a possible fixed vector \(\mathbf c\).
Verify that every point in the domain receives that same vector, rather than checking only a few sample points.
At each selected point, draw an arrow whose displacement is \(\mathbf c\). Use the same scale for every arrow.
Worked Sketches in Two and Three Dimensions
Worked Example: Sketching a Two-Dimensional Field
Consider \(\mathbf F(x,y)=\langle -1,2\rangle\) on \(\mathbb R^2\). Describe the arrows based at \((0,0)\), \((2,0)\), and \((0,1)\).
At \((0,0)\), the value is \(\langle -1,2\rangle\), so the arrow ends at \((0-1,0+2)=(-1,2)\).
At \((2,0)\), the value is again \(\langle -1,2\rangle\), so the arrow ends at \((2-1,0+2)=(1,2)\).
At \((0,1)\), the value is still \(\langle -1,2\rangle\), so the arrow ends at \((0-1,1+2)=(-1,3)\).
Each arrow points one unit left and two units up, and each has length \(\sqrt{(-1)^2+2^2}=\sqrt{5}\). The arrows are translated copies of one another. Check: subtracting each base point from its tip gives \(\langle -1,2\rangle\) in all three cases.
Worked Example: A Three-Dimensional Constant Field
Let \(\mathbf F(x,y,z)=\langle 2,1,-2\rangle\) on \(\mathbb R^3\). Find the arrows based at \((0,0,0)\) and \((1,-1,3)\), and state what a sketch should show.
At the origin, the value is \(\langle 2,1,-2\rangle\), so the arrow ends at \((0+2,0+1,0-2)=(2,1,-2)\).
At \((1,-1,3)\), the value is also \(\langle 2,1,-2\rangle\), so the arrow ends at \((1+2,-1+1,3-2)=(3,0,1)\).
The common length is \(\sqrt{2^2+1^2+(-2)^2}=\sqrt{9}=3\). In a three-dimensional sketch, both arrows represent the same displacement: two units in the positive \(x\)-direction, one in the positive \(y\)-direction, and two in the negative \(z\)-direction. Check: subtracting the first base from its tip gives \((2,1,-2)\); subtracting the second gives \((3-1,0-(-1),1-3)=(2,1,-2)\) as required.
In a perspective drawing, equal three-dimensional arrows may not look exactly equal on the page because of the chosen projection. The mathematical requirement is that their three-dimensional displacements agree, not that every drawn segment has identical apparent length on the screen.
Constant on a Restricted Domain
A field can be constant on a domain even when its formula contains a coordinate. This is not an exception to the definition: the definition asks whether the values are the same at every point of the stated domain. Restricting the domain can fix a coordinate or otherwise make a component take one value throughout.
Worked Example: A Coordinate-Dependent Formula That Is Constant on Its Domain
Let \(D=\{(x,0):x\in\mathbb R\}\), the \(x\)-axis, and define \(\mathbf F(x,y)=\langle y+2,5\rangle\). Determine whether \(\mathbf F\) is constant on \(D\).
An arbitrary point of \(D\) has the form \((x,0)\). Substituting \(y=0\) gives \(\mathbf F(x,0)=\langle 0+2,5\rangle=\langle 2,5\rangle\).
This value does not depend on \(x\). Therefore every point in \(D\) receives the same vector \(\langle 2,5\rangle\), and the field is constant on \(D\). Its arrows, drawn at points on the \(x\)-axis, all have displacement two units right and five units up. Check: for any two points \((x_1,0)\) and \((x_2,0)\) in \(D\), both field values equal \(\langle 2,5\rangle\).
A Common Mistake: Checking the Formula Instead of the Domain
A tempting shortcut is to say, “The formula contains a variable, so the field cannot be constant.” That conclusion can be wrong if the domain restricts the variable. In the previous example, the first component contains \(y\), but \(y=0\) at every point of \(D\).
For a concrete incorrect calculation, consider the same field on \(D=\{(x,0):x\in\mathbb R\}\). The mistaken reasoning is: “The first component is \(y+2\), which depends on \(y\); therefore the field is not constant on \(D\).” This does not test values at points in \(D\). The correction is to substitute the domain condition first: \(\mathbf F(x,0)=\langle 2,5\rangle\) for every \(x\). The field is constant on that domain.
It is also important to distinguish being constant on a restricted domain from being constant on a larger one. For the same formula on all of \(\mathbb R^2\), \(\mathbf F(0,0)=\langle 2,5\rangle\), while \(\mathbf F(0,1)=\langle 3,5\rangle\). These values differ, so the field is not constant on all of \(\mathbb R^2\). The domain must always be part of the claim.
Why Uniform-Arrow Sketches Matter
Recognizing a constant field gives a simple visual reference: every vector is a translated copy of the same fixed vector. This helps distinguish uniform behavior from fields in which arrows change as the point moves. It also makes drawing efficient: compute the fixed vector once, choose representative points in the domain, and repeat that displacement at each one.
When preparing a sketch, keep these checks in mind:
- Use the domain stated for the field; do not silently assume a larger one.
- Check every component, since a change in any component changes the vector.
- Use one consistent scale so equal vector lengths are represented consistently.
- Place each arrow at its assigned point; equal arrows do not all begin at the origin.
- If the fixed vector is zero, represent the field as zero vectors rather than arrows with a nonzero direction.
The essential test is simple: on the domain in question, does every point receive the same vector? If so, the field is constant and its arrows all represent the same displacement.
Check Your Understanding
Use the domain in each question when deciding whether a field is constant.
- Is \(\mathbf F(x,y)=\langle 4,1\rangle\) constant on \(\mathbb R^2\)? What displacement should each arrow represent?
- For \(D=\{(x,2):x\in\mathbb R\}\), determine whether \(\mathbf F(x,y)=\langle y,-3\rangle\) is constant on \(D\). If it is, give its fixed value.
- For the same formula \(\mathbf F(x,y)=\langle y,-3\rangle\), is the field constant on all of \(\mathbb R^2\)? Justify your answer by comparing two values.
- What are the length and displacement of the arrows in the field \(\mathbf F(x,y,z)=\langle 1,2,2\rangle\)?
- Why is it insufficient to conclude that a field is nonconstant merely because its formula contains a coordinate?