The Divergence(F) command computes the divergence of the vector field F.  This is a synonym for $\mathrm{Del}·F$ or DotProduct(Del, F).

If F is a Vector valued procedure, the default coordinate system is used, and it must be indexed by the coordinate names.  Otherwise, F must be a Vector with the vectorfield attribute set, and it must have a coordinate system attribute that is indexed by the coordinate names.

If F is a procedure, the returned object is a procedure. Otherwise, the returned object is an expression.

The Divergence() command returns the differential form of the divergence operator in the current coordinate system.  For more information, see SetCoordinates.

$\mathrm{with}\left(\mathrm{VectorCalculus}\right)\:$

$\mathrm{SetCoordinates}\left('\mathrm{cartesian}'\left[x,y,z\right]\right)$

${\mathrm{cartesian}}_{x,y,z}$

$F≔\mathrm{VectorField}\left(⟨x^2\,y^2\,z^2⟩\right)$

$F≔\left(x^2\right){\stackrel{\_}{e}}_x\+\left(y^2\right){\stackrel{\_}{e}}_y\+\left(z^2\right){\stackrel{\_}{e}}_z$

$\mathrm{Divergence}\left(F\right)$

$2x+2y+2z$

$\mathrm{Divergence}\left(\right)$

$\frac{\partial }{\partial x}{\mathrm{VF}}_1\left(x\,y\,z\right)+\frac{\partial }{\partial y}{\mathrm{VF}}_2\left(x\,y\,z\right)+\frac{\partial }{\partial z}{\mathrm{VF}}_3\left(x\,y\,z\right)$

$\mathrm{Del}·F$

$2x+2y+2z$

$\nabla ·F$

$2x+2y+2z$

$\mathrm{DotProduct}\left(\mathrm{Del}\,F\right)$

$2x+2y+2z$

$\mathrm{Divergence}\left(\left(x\,y\,z\right)\mapsto \mathrm{PositionVector}\left(\left[\sin \left(x\right)\,\cos \left(y\right)\,\tan \left(z\right)\right]\right)\right)$

$\left(x\,y\,z\right)\mapsto \cos \left(x\right)-\sin \left(y\right)+1+{\tan \left(z\right)}^2$

$\mathrm{SetCoordinates}\left('\mathrm{polar}'\left[r,\mathrm{\theta }\right]\right)$

${\mathrm{polar}}_{r,\mathrm{\theta }}$

$\mathrm{Divergence}\left(\left(r\,\mathrm{\theta }\right)\mapsto ⟨f\left(r\,\mathrm{\theta }\right)\,g\left(r\,\mathrm{\theta }\right)⟩\right)$

$\left(r\,\mathrm{\θ}\right)\→\frac{f\left(r\,\mathrm{\θ}\right)\+r\left(\frac{\partial }{\partial r}f\left(r\,\mathrm{\θ}\right)\right)\+\frac{\partial }{\partial \mathrm{\θ}}g\left(r\,\mathrm{\θ}\right)}{r}$