Divergence

It is computed with the operator \(\nabla=\langle\partial_x,\partial_y,\partial_z\rangle\) as a formal dot product with \(\mathbf F\): each component is differentiated with respect to its own variable and the three results are added. So the divergence of a field in \(\mathbb R^3\) is a number at each point, not a vector, and in the plane one simply adds \(P_x+Q_y\).

The physical reading is that of a fluid with velocity \(\mathbf F\). If \(\operatorname{div}\mathbf F(P)>0\), more fluid leaves a small neighborhood of \(P\) than enters it and \(P\) acts as a source; if negative, as a sink; if zero, what enters is what leaves and the field is called incompressible (or solenoidal) at \(P\). Guiding example: \(\mathbf F=\langle x,y,z\rangle\) spreads away from the origin in every direction and has divergence \(3\) everywhere.

Divergence does not detect swirling: the field \(\langle -y,x,0\rangle\) circulates around the \(z\)-axis yet its divergence is \(0\). Nor should it be confused with the gradient, which goes from scalars to vectors; here the opposite happens. It is linear, and for a scalar \(f\) times a field, \(\operatorname{div}(f\mathbf F)=\nabla f\cdot\mathbf F+f\operatorname{div}\mathbf F\).

Together with the curl, it completes the two basic derivatives of a field. Later, the divergence theorem equates the triple integral of \(\operatorname{div}\mathbf F\) with the flux through the boundary, and Green's theorem in flux form is its planar version.