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Vector Field — Definition, Formula & Examples

A vector field is a function that assigns a vector to every point in a region of space. You can picture it as a collection of arrows, one at each point, showing a direction and magnitude — like a wind map showing speed and direction at every location.

A vector field on a subset D⊆RnD \subseteq \mathbb{R}^n is a function F:D→Rn\mathbf{F}: D \to \mathbb{R}^n that maps each point x∈D\mathbf{x} \in D to a vector F(x)\mathbf{F}(\mathbf{x}). In two dimensions, F(x,y)=⟨P(x,y), Q(x,y)⟩\mathbf{F}(x,y) = \langle P(x,y),\, Q(x,y) \rangle; in three dimensions, F(x,y,z)=⟨P, Q, R⟩\mathbf{F}(x,y,z) = \langle P,\, Q,\, R \rangle, where PP, QQ, and RR are scalar-valued component functions.

Key Formula

F(x,y)=⟨P(x,y),  Q(x,y)⟩\mathbf{F}(x, y) = \langle P(x, y),\; Q(x, y) \rangle
Where:
  • F\mathbf{F} = The vector field
  • (x,y)(x, y) = A point in the domain
  • P,QP, Q = Scalar-valued component functions giving the x- and y-components of each vector

How It Works

To evaluate a vector field, plug a specific point into the component functions to get the vector at that location. In a plot, you draw this vector as an arrow based at the point. The gradient of a scalar function ff produces a vector field ∇f\nabla f, meaning every scalar field gives rise to a vector field. Operations like divergence and curl take a vector field as input and measure how the vectors spread out or rotate around each point.

Worked Example

Problem: Given the vector field F(x,y)=⟨2x, −y⟩\mathbf{F}(x,y) = \langle 2x,\, -y \rangle, find the vector at the point (3,4)(3, 4).
Substitute into P: The first component is P(x,y)=2xP(x,y) = 2x. At (3,4)(3,4):
P(3,4)=2(3)=6P(3,4) = 2(3) = 6
Substitute into Q: The second component is Q(x,y)=−yQ(x,y) = -y. At (3,4)(3,4):
Q(3,4)=−4Q(3,4) = -4
Write the result: Combine the components into a vector.
F(3,4)=⟨6, −4⟩\mathbf{F}(3,4) = \langle 6,\, -4 \rangle
Answer: The vector at (3,4)(3,4) is ⟨6,−4⟩\langle 6, -4 \rangle, an arrow pointing right and downward with magnitude 36+16=52=213\sqrt{36+16} = \sqrt{52} = 2\sqrt{13}.

Why It Matters

Vector fields model fluid flow, electromagnetic forces, and gravitational fields in physics and engineering. In multivariable calculus, they are the central objects in line integrals, surface integrals, and the major theorems (Green's, Stokes', Divergence) that connect local behavior to global quantities.

Common Mistakes

Mistake: Confusing a vector field with a scalar field.
Correction: A scalar field assigns a single number to each point (like temperature), while a vector field assigns a vector (like wind velocity). The gradient converts a scalar field into a vector field, not the other way around.