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Divergence and curl

The divergence of a vector field measures how much the field "flows out" from a point, while the curl measures how much it swirls around a point. Both are built from partial derivatives of the components.

∇⋅F⃗=∂Fx∂x+∂Fy∂y+∂Fz∂z\nabla\cdot\vec F = \frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial y}+\frac{\partial F_z}{\partial z}divergence
∇×(∇f)=0⃗\nabla\times(\nabla f) = \vec 0the curl of a gradient field is always zero

Symbols

F⃗\vec Fvector field

Example

∇⋅(x,y,z)=1+1+1=3\nabla\cdot(x,y,z) = 1+1+1 = 3.

A gradient field is always irrotational: ∇×(∇f)=0⃗\nabla\times(\nabla f)=\vec 0.
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Part of Multivariable Calculus: Vector calculus.