Introduction to Vector Fields

Calculus 3 · Axiom Academy

LESSON Introduction to Vector Fields A vector field attaches a vector to every point in space — turning a plane into a landscape of arrows. Pick any point (x,y) in the plane. A vector field hands that point a vector — an arrow with a length and a direction. Do it for every point at once and the plane fills with arrows. The components P and Q give the arrow's x - and y -parts The field animated below is : at the point (x,y) the arrow is itself, so every arrow points straight away from the origin and grows longer the farther out you go. Change the rule and the whole picture changes. Take . At each point the arrow is the position vector turned a quarter-turn , so it runs along the circle through that point instead of away from the center — the field circulates. , so each arrow is tangent to its circle. At (1,0) the arrow is — pointing up, so the swirl turns counterclockwise. , which equals the circle's radius. This is the velocity field of a rigid spin — think of a turntable or a whirlpool. The faint circles are drawn only as a guide: notice every arrow rides tangent to the circle it sits on, and the arrows on the bigger circles are longer. That is made visible. 3. Fields From a Surface: the Gradient Some vector fields are born from a single scalar function f(x,y) — a height, a temperature, a potential. Its gradient field points, at every point, in the direction of steepest increase of f .

This is the written version of the interactive lesson above. See the full Calculus 3 course.