Engineering Math Practice

Curl and Divergence

The two ways to apply ∇ to a vector field — divergence (a scalar) and curl (a vector).

The setup. Both are ∇ applied to a vector field — one as a dot product, one as a cross product. Each extracts a different physical feature of the field.

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Physical meaning: positive divergence means the point acts like a faucet (field flows outward); negative means a drain. In EM this is Gauss's law territory — divergence of the E-field tells you where charge is. ∇·E = ρ/ε₀: charge is the source of the field's divergence.

div⁡F=∇⋅F=∂P∂x+∂Q∂y+∂R∂z\operatorname{div}\mathbf{F} = \nabla \cdot \mathbf{F} = \dfrac{\partial P}{\partial x} + \dfrac{\partial Q}{\partial y} + \dfrac{\partial R}{\partial z}

Divergence — measures how much the field spreads out from a point (a source) or flows in (a sink)

  • ∇⋅F\nabla \cdot \mathbf{F}Divergence — a scalar, one number at each point.

Curl measures the rotation or "swirl" of the field at a point:

curl⁡F=∇×F=∣ijk∂∂x∂∂y∂∂zPQR∣\operatorname{curl}\mathbf{F} = \nabla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\[4pt] \dfrac{\partial}{\partial x} & \dfrac{\partial}{\partial y} & \dfrac{\partial}{\partial z} \\[8pt] P & Q & R \end{vmatrix}

Curl, as a determinant

  • ∇×F\nabla \times \mathbf{F}Curl — a vector, with its own i, j, k components.

Physical meaning: curl points along the axis a tiny paddlewheel placed in the field would spin around, and its magnitude is how fast. Zero curl means no swirl (irrotational).

∇×F=⟨∂R∂y−∂Q∂z,    ∂P∂z−∂R∂x,    ∂Q∂x−∂P∂y⟩\nabla \times \mathbf{F} = \left\langle \dfrac{\partial R}{\partial y} - \dfrac{\partial Q}{\partial z},\;\; \dfrac{\partial P}{\partial z} - \dfrac{\partial R}{\partial x},\;\; \dfrac{\partial Q}{\partial x} - \dfrac{\partial P}{\partial y} \right\rangle

Curl, expanding the determinant

  • irrotational\text{irrotational}Zero curl — no swirl anywhere in the field.

The two facts that tie it together. A conservative field has zero curl: if F = ∇f, then ∇×F = 0. So curl is the test for conservative — swirl-free is a necessary condition. (This is why the ∂P/∂y = ∂Q/∂x check from the vector-fields section worked: it's exactly the 2D curl being zero.)

A curl field never has sources — this is always true, for any field F, not just conservative ones.

∇⋅(∇×F)=0\nabla \cdot \left(\nabla \times \mathbf{F}\right) = 0

Divergence of a curl is always zero

Worked example

F=⟨xy, yz, xz⟩,find ∇⋅F and ∇×F\mathbf{F} = \langle xy,\ yz,\ xz \rangle, \quad \text{find } \nabla \cdot \mathbf{F} \text{ and } \nabla \times \mathbf{F}
  1. ∇⋅F=∂∂x(xy)+∂∂y(yz)+∂∂z(xz)=y+z+x\nabla \cdot \mathbf{F} = \dfrac{\partial}{\partial x}(xy) + \dfrac{\partial}{\partial y}(yz) + \dfrac{\partial}{\partial z}(xz) = y + z + x

    Divergence: sum the three matching partials.

  2. ∂R∂y−∂Q∂z=∂(xz)∂y−∂(yz)∂z=0−y=−y\dfrac{\partial R}{\partial y} - \dfrac{\partial Q}{\partial z} = \dfrac{\partial(xz)}{\partial y} - \dfrac{\partial(yz)}{\partial z} = 0 - y = -y

    Curl, i-component.

  3. ∂P∂z−∂R∂x=∂(xy)∂z−∂(xz)∂x=0−z=−z\dfrac{\partial P}{\partial z} - \dfrac{\partial R}{\partial x} = \dfrac{\partial(xy)}{\partial z} - \dfrac{\partial(xz)}{\partial x} = 0 - z = -z

    Curl, j-component.

  4. ∂Q∂x−∂P∂y=∂(yz)∂x−∂(xy)∂y=0−x=−x\dfrac{\partial Q}{\partial x} - \dfrac{\partial P}{\partial y} = \dfrac{\partial(yz)}{\partial x} - \dfrac{\partial(xy)}{\partial y} = 0 - x = -x

    Curl, k-component.

  5. ∇×F=⟨−y, −z, −x⟩\nabla \times \mathbf{F} = \langle -y,\ -z,\ -x \rangle

    Assemble the three components. Since the curl isn't zero, F is not conservative.

Answer:

∇⋅F=x+y+z,∇×F=⟨−y, −z, −x⟩\nabla \cdot \mathbf{F} = x+y+z, \quad \nabla \times \mathbf{F} = \langle -y,\ -z,\ -x \rangle

How to solve one of these

  1. 1.

    Identify P, Q, R. The three component functions of the field.

  2. 2.

    Divergence: sum the three matching partials. ∂P/∂x+∂Q/∂y+∂R/∂z\partial P/\partial x + \partial Q/\partial y + \partial R/\partial z. Result is a scalar.

  3. 3.

    Curl: set up the ijk determinant and expand, or use the component formula. Result is a vector.

  4. 4.

    Watch the cross-partials in curl. Each component is (partial of the later variable) minus (partial of the earlier). Sign errors live here.

  5. 5.

    Interpret. Zero curl means possibly conservative; the sign of divergence means a source or a sink.

∇×F=∣ijk∂∂x∂∂y∂∂zPQR∣\nabla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\[4pt] \dfrac{\partial}{\partial x} & \dfrac{\partial}{\partial y} & \dfrac{\partial}{\partial z} \\[8pt] P & Q & R \end{vmatrix}

Divergence (∇·F) is a scalar measuring source/sink; curl (∇×F) is a vector measuring rotation. Curl zero means possibly conservative; divergence of a curl is always zero.

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