5.3 Curl and Divergence

Vector Field Analysis

Curl and divergence are two differential operators that extract specific local information from a vector field. Curl detects rotation at a point, while divergence detects expansion or contraction. Together, they form the foundation for connecting vector fields to the integral theorems (Stokes' and Divergence) you'll encounter throughout the rest of this course.

Curl of vector fields

The curl of a vector field measures the tendency of the field to rotate around a point. It's defined as:

$\text{curl } \mathbf{F} = \nabla \times \mathbf{F}$

For a 3D field $\mathbf{F}(x, y, z) = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k}$, you compute it using a determinant-style formula:

$\text{curl } \mathbf{F} = \left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}\right)\mathbf{i} + \left(\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}\right)\mathbf{j} + \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)\mathbf{k}$

A reliable way to remember this: write out the symbolic determinant

$\nabla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ P & Q & R \end{vmatrix}$

and expand along the first row. The middle component picks up a sign flip, which is the most common source of errors.

For a 2D field $\mathbf{F}(x, y) = P\mathbf{i} + Q\mathbf{j}$, the curl simplifies to a scalar (technically the $\mathbf{k}$-component):

$\text{curl } \mathbf{F} = \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}$

This scalar tells you the intensity and direction of rotation in the plane: positive means counterclockwise, negative means clockwise.

Curl as rotational measure

The physical picture: imagine placing a tiny paddle wheel at a point in the field. The curl tells you how fast that paddle wheel spins and around which axis.

• The magnitude $|\nabla \times \mathbf{F}|$ gives the rotation strength.
• The direction of the curl vector is the axis of rotation (by the right-hand rule).
• A field with $\nabla \times \mathbf{F} = \mathbf{0}$ everywhere is called irrotational (or curl-free). This is exactly the condition needed for a vector field to be conservative, meaning it has a potential function $f$ where $\mathbf{F} = \nabla f$.
• Non-zero curl at a point means line integrals of $\mathbf{F}$ are path-dependent near that point. You can't find a single potential function.

Physical examples: fluid vorticity (the curl of the velocity field) describes local spinning in a fluid. In electromagnetism, $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$ (Faraday's law) relates the curl of the electric field to a changing magnetic field.

Divergence of vector fields

The divergence of a vector field measures how much the field spreads out from (or converges toward) a point. It's defined as:

$\text{div } \mathbf{F} = \nabla \cdot \mathbf{F}$

For a 3D field $\mathbf{F}(x, y, z) = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k}$:

$\text{div } \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}$

For a 2D field $\mathbf{F}(x, y) = P\mathbf{i} + Q\mathbf{j}$:

$\text{div } \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y}$

Notice that divergence is a scalar, not a vector. You're taking each component's partial derivative with respect to its own variable and adding them up.

Divergence as flux density

Think of divergence as measuring the net "outflow" of a field per unit volume at a point.

• Positive divergence means the field is spreading outward: the point acts as a source. Example: the electric field near a positive point charge radiates outward, so $\nabla \cdot \mathbf{E} > 0$ there.
• Negative divergence means the field is converging inward: the point acts as a sink. Example: a velocity field where fluid drains toward a point.
• Zero divergence means no net creation or destruction of "field stuff" at that point. A field with $\nabla \cdot \mathbf{F} = 0$ everywhere is called solenoidal (or incompressible in fluid contexts). Magnetic fields are always solenoidal: $\nabla \cdot \mathbf{B} = 0$ is one of Maxwell's equations.

The Divergence Theorem connects this local measurement to a global one: the total divergence inside a closed region equals the net flux of $\mathbf{F}$ through the boundary surface.

Applications of curl and divergence

These operators let you classify and decompose vector fields:

• A conservative field satisfies $\nabla \times \mathbf{F} = \mathbf{0}$ (curl-free). You can find a potential function.
• A solenoidal field satisfies $\nabla \cdot \mathbf{F} = 0$ (divergence-free). You can find a vector potential $\mathbf{A}$ where $\mathbf{F} = \nabla \times \mathbf{A}$.
• The Helmholtz decomposition theorem says any sufficiently smooth vector field can be split into a curl-free part plus a divergence-free part. This is the theoretical backbone for many physics applications.

Two identities you should know cold:

1. $\nabla \cdot (\nabla \times \mathbf{F}) = 0$ — the divergence of any curl is always zero.
2. $\nabla \times (\nabla f) = \mathbf{0}$ — the curl of any gradient is always zero.

These aren't just formulas to memorize. Identity (1) means if you compute a curl and then take its divergence, you'll always get zero, which serves as a useful check on your work. Identity (2) is why conservative fields (gradients of scalar functions) are automatically irrotational.

The Laplacian $\nabla^2 f = \nabla \cdot (\nabla f)$ combines both operators on a scalar field and appears throughout physics (heat equation, wave equation, Poisson's equation). For vector fields, the vector Laplacian is:

$\nabla^2 \mathbf{F} = \nabla(\nabla \cdot \mathbf{F}) - \nabla \times (\nabla \times \mathbf{F})$

Finally, Stokes' Theorem and the Divergence Theorem tie everything together by relating curl and divergence (local, differential information) to line integrals and surface integrals (global, integral information). These theorems are the payoff for understanding curl and divergence well.