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

Graphical or geometric meaning

In horizontal planes, $\langle-y,x\rangle$ circulates counterclockwise, matching upward curl by the right-hand rule. The $z$ component grows with height, creating vertical outflow and divergence one.

Common mistakes and why they fail

Verification and reasonableness checks

• Confirm output type: scalar divergence, vector curl.
• Use a determinant mnemonic only after fixing coordinate order.
• Check identities such as $\nabla\cdot(\nabla\times\mathbf F)=0$ under $C^2$ hypotheses.

Local operators answer different questions

For a vector field in three dimensions, divergence $\nabla\cdot\mathbf F$ measures local source strength, while curl $\nabla\times\mathbf F$ measures local rotation with orientation. One is scalar and the other vector, so units and output type catch many errors. Compute partial derivatives component by component and preserve the determinant signs in curl. Under sufficient smoothness, $\nabla\cdot(\nabla\times\mathbf F)=0$ and $\nabla\times(\nabla\phi)=\mathbf0$; these identities are checks, not substitutes for hypotheses on a domain. A zero divergence field need not have zero curl, and a zero curl field need not be globally conservative when the domain has holes. Interpret the operator before invoking an integral theorem.

Practice

1. Find divergence of $\langle x,y,z\rangle$.
2. Find curl of $\langle x,y,z\rangle$.
3. What is two-dimensional scalar curl of $\langle-y,x\rangle$?

Connections and next steps

• Vector Fields
• Green's Theorem
• Divergence Theorem