Curl and Divergence

The central idea

For a $C^1$ field $\mathbf F=\langle P,Q,R\rangle$ on an open subset of $\mathbb R^3$, divergence is the scalar $\nabla\cdot\mathbf F=P_x+Q_y+R_z$, and curl is the vector $\nabla\times\mathbf F=\langle R_y-Q_z,P_z-R_x,Q_x-P_y\rangle$. In two dimensions, scalar curl is $Q_x-P_y$.

Definitions, hypotheses, and notation

Divergence and curl are coordinate differential operators, so smoothness is required for identities involving mixed partial derivatives. Units also differ: if $\mathbf F$ has units $U$, both operators have units $U/$length, but curl additionally carries an oriented axis. A visual field plot can suggest behavior but does not replace derivatives.

Zero divergence does not mean the field is zero; it means no first-order net source. Zero curl does not mean no global circulation unless domain topology supports a potential theorem. Local differential statements and global integral conclusions must be connected through the appropriate hypotheses.

Conceptual meaning

Divergence measures net outward source strength per unit volume. Curl measures local rotation, with direction set by the right-hand rule and magnitude related to circulation density. They have different output dimensions and cannot be interchanged.

A dependable method and decision rule

1. Write field components and coordinate order explicitly.
2. For divergence, differentiate each component with respect to its matching coordinate and add.
3. For curl, use the component formula with signs checked.
4. Evaluate at a point only after symbolic differentiation.
5. Interpret sign and direction in the field's units and domain.

Fully worked example

Common mistakes and why they fail