How to compute divergence, curl, and interpret a 2D vector field

Expert intro: For a vector field in two variables, the work is mostly bookkeeping: take the right partial derivative of each component, then combine them in the divergence or curl formula. The algebra looks messy here because of the denominator, but there is a clean simplification hiding in the structure.

Detailed walkthrough

Let

$\mathbf u(x,y)=A\left[\left(1-\frac{x^2-y^2}{(x^2+y^2)^2}\right)\mathbf i-\frac{2xy}{(x^2+y^2)^2}\mathbf j\right],\qquad A>0.$

Write the components as

$u_1(x,y)=A\left(1-\frac{x^2-y^2}{(x^2+y^2)^2}\right),\qquad u_2(x,y)=-A\frac{2xy}{(x^2+y^2)^2}.$

Assume $(x,y)\neq(0,0)$.

(a) Divergence

In 2D,

$\nabla\cdot \mathbf u=\frac{\partial u_1}{\partial x}+\frac{\partial u_2}{\partial y}.$

A helpful simplification is to recognize the nonconstant part as the real and imaginary parts of a rational form related to

$\frac{1}{(x+iy)^2}.$

If you differentiate directly, the terms cancel and you get

$\nabla\cdot \mathbf u=0 \quad \text{for } (x,y)\neq(0,0).$

(b) Curl

For a planar field $\langle P,Q\rangle$, the scalar curl is

$\operatorname{curl}\mathbf u=\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}.$

Again, direct differentiation shows the mixed terms cancel, so

$\operatorname{curl}\mathbf u=0 \quad \text{for } (x,y)\neq(0,0).$

(c) Physical meaning

• Zero divergence means the field has no local source or sink behavior away from the origin.
• Zero curl means the field has no local tendency to rotate there.

So, away from the singularity at the origin, this field behaves like a potential flow: it is locally conservative and neither expands nor swirls.

If your course expects a source citation, use the textbook or notes that define divergence and curl in 2D and explain their physical meaning.

💡 Pitfall guide

Do not forget the origin. The formulas above are valid only where $(x^2+y^2)^2\neq 0$. Another common slip is mixing up the 2D curl formula with the 3D vector curl. In a planar problem, the curl is usually treated as a scalar (the $k$-component).

🔄 Real-world variant

If the constant $A$ changes, the divergence and curl stay zero everywhere the field is defined, because multiplying by a constant does not change the cancellation. If the denominator were changed, though, the field might no longer be divergence-free or curl-free; then you would need to recompute from scratch instead of relying on the pattern.

🔍 Related terms

divergence, curl, vector field