Vector Fields (div, curl)

F ℝ^n → ℝ^n — divergence ( · F) measures outflow, curl ( × F) measures rotation.

The idea

A vector field assigns a vector to each point of space, as a scalar function assigns a number: $\mathbf{F}(x, y, z) = (F_1, F_2, F_3),$ where each component is itself a function of position, so the vector varies from point to point. Picture a fluid in motion, with $\mathbf{F}$ the velocity of the fluid at each point, or a force field, with $\mathbf{F}$ the force felt there.

Two derivatives of a vector field measure its local behavior: whether the flow spreads out or piles up at a point, and whether it rotates there. The first is the divergence, a scalar: $\nabla \cdot \mathbf{F} = \partial_x F_1 + \partial_y F_2 + \partial_z F_3.$ The second is the curl, a vector, which points along the axis that a paddlewheel placed in the flow would spin about: $\nabla \times \mathbf{F} = \left(\partial_y F_3 - \partial_z F_2,\ \partial_z F_1 - \partial_x F_3,\ \partial_x F_2 - \partial_y F_1\right).$ Both are assembled from the same nine partial derivatives. Divergence adds the three that measure each component's change along its own direction; curl combines the six cross terms in signed pairs.

Ways to work on it

• Walkthrough. Meet three example fields (constant, radial, rotational), then learn the divergence and curl operators.
• Practice. Compute divergence of a small 3D field, term by term.
• Hardest. Compute a curl from the component formula, then classify the field.

Not sure where to start? Take the ten-question placement test.