The Divergence(F) calling sequence computes the divergence of the vector field $F$.  This calling sequence is equivalent to $\mathrm{Del}·F$ and DotProduct(Del, F).

If $F$ is a Vector-valued procedure, the default coordinate system is used. The default coordinate system 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(c) calling sequence returns the differential form of the divergence operator in the coordinate system specified by $c$, which can be given as:

* an indexed name, e.g., ${\mathrm{spherical}}_{r,\mathrm{\phi },\mathrm{\theta }}$

* a name, e.g., spherical; default coordinate names will be used

* a list of names, e.g., $\left[r\,\mathrm{\phi }\,\mathrm{\theta }\right]$; the current coordinate system will be used, with these as the coordinate names

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

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

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

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

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

$2x+2y+2z$

$\mathrm{Del}·F$

$2x+2y+2z$

$\nabla ·F$

$2x+2y+2z$

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

$2x+2y+2z$

$\mathrm{Divergence}\left(\left(x\,y\,z\right)\mapsto ⟨\sin \left(x\right)\,\cos \left(y\right)\,\tan \left(z\right)⟩\right)$

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

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

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

$\mathrm{SetCoordinates}\left(\mathrm{cylindrical}\left[r,\mathrm{\theta },z\right]\right)\:$

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

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

$\mathrm{Divergence}\left(\left[s\,\mathrm{\phi }\,w\right]\right)$

$\frac{\frac{\partial }{\partial s}\left(s{\mathrm{VF}}_1\left(s\,\mathrm{\phi }\,w\right)\right)+\frac{\partial }{\partial \mathrm{\phi }}{\mathrm{VF}}_2\left(s\,\mathrm{\phi }\,w\right)+\frac{\partial }{\partial w}\left(s{\mathrm{VF}}_3\left(s\,\mathrm{\phi }\,w\right)\right)}{s}$

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

$\frac{\frac{\partial }{\partial r}\left(r^2\sin \left(\mathrm{\phi }\right){\mathrm{VF}}_1\left(r\,\mathrm{\phi }\,\mathrm{\theta }\right)\right)+\frac{\partial }{\partial \mathrm{\phi }}\left(r\sin \left(\mathrm{\phi }\right){\mathrm{VF}}_2\left(r\,\mathrm{\phi }\,\mathrm{\theta }\right)\right)+\frac{\partial }{\partial \mathrm{\theta }}\left(r{\mathrm{VF}}_3\left(r\,\mathrm{\phi }\,\mathrm{\theta }\right)\right)}{r^2\sin \left(\mathrm{\phi }\right)}$

$\mathrm{Divergence}\left(\mathrm{spherical}\left[\mathrm{\alpha },\mathrm{\psi },\mathrm{\gamma }\right]\right)$

$\frac{\frac{\partial }{\partial \mathrm{\alpha }}\left({\mathrm{\alpha }}^2\sin \left(\mathrm{\psi }\right){\mathrm{VF}}_1\left(\mathrm{\alpha }\,\mathrm{\psi }\,\mathrm{\gamma }\right)\right)+\frac{\partial }{\partial \mathrm{\psi }}\left(\mathrm{\alpha }\sin \left(\mathrm{\psi }\right){\mathrm{VF}}_2\left(\mathrm{\alpha }\,\mathrm{\psi }\,\mathrm{\gamma }\right)\right)+\frac{\partial }{\partial \mathrm{\gamma }}\left(\mathrm{\alpha }{\mathrm{VF}}_3\left(\mathrm{\alpha }\,\mathrm{\psi }\,\mathrm{\gamma }\right)\right)}{{\mathrm{\alpha }}^2\sin \left(\mathrm{\psi }\right)}$

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

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

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

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