Sketching F(x,y) = <-y, x>

Calculus 3 · Axiom Academy

Evaluate the field at a grid of points, draw each arrow, and read off the pattern. Sketch the two-dimensional vector field by hand. Evaluate at a grid of sample points, draw the arrow that assigns to each point, and describe the pattern the arrows reveal. Nice work — you sketched from scratch and identified it as a rotation field. Sketch by sampling: to draw any field, evaluate at a grid of points and draw each vector starting at its point. It rotates: points 90° counterclockwise from , so the arrows circle the origin. Perpendicular everywhere: , so each arrow is at a right angle to the radius. Length grows with r : , zero at the origin and larger farther out. Flow lines are circles: a particle carried by this field travels in a circle centered at the origin. Rotation fields like this model spinning fluids (vortices) and angular velocity — the same sampling recipe sketches any vector field you meet in multivariable calculus.

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