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A vector field F: ℝⁿ → ℝⁿ assigns a vector to each point in space, such as F(x, y) = ⟨P(x,y), Q(x,y)⟩. Vector fields model physical phenomena: velocity (fluid flow), force (electric or gravitational), and heat flux. Visualization uses arrows indicating magnitude and direction.
From your study of vectors in 3D, you know how to represent individual vectors as arrows with magnitude and direction. A vector field takes this one step further: instead of a single arrow, you attach an arrow to every point in a region of space. Formally, a vector field F on ℝ² assigns to each point (x, y) a vector ⟨P(x,y), Q(x,y)⟩, where P and Q are real-valued functions. In ℝ³ you get a third component: F(x, y, z) = ⟨P, Q, R⟩. The result is not a single geometric object but an entire landscape of arrows.
The physical examples are the clearest way to build intuition. A velocity field assigns to each point in a fluid the velocity vector of the fluid particle at that point — the arrows show which way the water (or air) is flowing and how fast. A gravitational field assigns to each point in space the acceleration that a unit mass would experience if placed there — pointing toward Earth's center, growing stronger as you descend. An electric field assigns to each point the force per unit charge experienced by a positive test charge. In every case, the vector field is a function of position that produces a vector output, and the arrows drawn at sampled points give a qualitative picture of the whole field.
To visualize a vector field, you sample a grid of points and draw an arrow at each one, with the arrow's direction and length determined by F at that point. In practice, arrows are often normalized to a fixed length (or scaled down) to avoid clutter, showing direction more clearly than magnitude. Recognizable patterns emerge: a field like F(x, y) = ⟨−y, x⟩ produces counterclockwise rotation around the origin; F(x, y) = ⟨x, y⟩ produces outward-pointing arrows that grow with distance; F(x, y) = ⟨1, 0⟩ is a uniform horizontal flow.
Vector fields are the natural input for the two central operations of multivariable calculus that come next: line integrals and the differential operators curl and divergence. The line integral of F along a curve measures the total "work done" by the field along that path. The divergence of F measures how much the field is spreading out (or converging) at each point; the curl measures how much it is rotating. These operations extract scalar information from the vector field and are the language of Maxwell's equations, fluid mechanics, and gravitational theory — all of which you'll be equipped to read once you understand what a vector field is.