Curl, Divergence, Gradient and Laplacian, respectively return the curl, the divergence, the gradient and the Laplacian of a given vectorial or scalar function. When the command's name is prefixed by $\%$, an unevaluated representation for these operations is returned.

The %Curl, %Divergence, %Gradient and %Laplacian are the inert forms of Curl, Divergence, Gradient and Laplacian, that is: they represent the same mathematical operations while holding the operations unperformed. To activate the operations use value.

Curl, Divergence and Gradient check their arguments (for consistency) before sending the task to Nabla. So, if $\mathrm{A\_}$ is a vector, then Gradient(A_) will interrupt the computation with an error message, as well as Divergence(A) and Curl(A) when $A$ is a scalar (not a vector). All these differential operations are realized just w.r.t the geometrical coordinates $\left\{\mathrm{\phi }\,r\,\mathrm{\rho }\,\mathrm{\theta }\,x\,y\,z\right\}$. Therefore, if $A$ does not depend on these global geometrical coordinates, these commands (as well as Nabla) return 0.

For the conventions about the geometrical coordinates and vectors see ?conventions

$\mathrm{with}\left(\mathrm{Physics}\left[\mathrm{Vectors}\right]\right)$

$\left[\mathrm{\&x}\,\mathrm{`+`}\,\mathrm{`.`}\,\mathrm{Assume}\,\mathrm{ChangeBasis}\,\mathrm{ChangeCoordinates}\,\mathrm{CompactDisplay}\,\mathrm{Component}\,\mathrm{Curl}\,\mathrm{DirectionalDiff}\,\mathrm{Divergence}\,\mathrm{Gradient}\,\mathrm{Identify}\,\mathrm{Laplacian}\,\nabla \,\mathrm{Norm}\,\mathrm{ParametrizeCurve}\,\mathrm{ParametrizeSurface}\,\mathrm{ParametrizeVolume}\,\mathrm{Setup}\,\mathrm{Simplify}\,\mathrm{`^`}\,\mathrm{diff}\,\mathrm{int}\right]$

$\mathrm{Setup}\left(\mathrm{mathematicalnotation}=\mathrm{true}\right)$

$\left[\mathrm{mathematicalnotation}=\mathrm{true}\right]$

$\mathrm{Gradient}\left(f\left(x\,y\,z\right)\right)$

$\left(\frac{\partial }{\partial x}f\left(x\,y\,z\right)\right)\stackrel{\wedge }{i}+\left(\frac{\partial }{\partial y}f\left(x\,y\,z\right)\right)\stackrel{\wedge }{j}+\left(\frac{\partial }{\partial z}f\left(x\,y\,z\right)\right)\stackrel{\wedge }{k}$

$\mathrm{Divergence}\left(\mathrm{Gradient}\left(f\left(x\,y\,z\right)\right)\right)$

$\frac{{\partial }^2}{\partial x^2}f\left(x\,y\,z\right)+\frac{{\partial }^2}{\partial y^2}f\left(x\,y\,z\right)+\frac{{\partial }^2}{\partial z^2}f\left(x\,y\,z\right)$

$\mathrm{Laplacian}\left(f\left(x\,y\,z\right)\right)$

$\frac{{\partial }^2}{\partial x^2}f\left(x\,y\,z\right)+\frac{{\partial }^2}{\partial y^2}f\left(x\,y\,z\right)+\frac{{\partial }^2}{\partial z^2}f\left(x\,y\,z\right)$

$\mathrm{Laplacian}\left(\mathrm{A\_}\left(r\,\mathrm{\theta }\,\mathrm{\phi }\right)\right)$

$\frac{\left(\frac{{\partial }^2}{\partial r^2}\overrightarrow{A}\left(r\,\mathrm{\theta }\,\mathrm{\phi }\right)\right)r^2+2\left(\frac{\partial }{\partial r}\overrightarrow{A}\left(r\,\mathrm{\theta }\,\mathrm{\phi }\right)\right)r+\left(\frac{\partial }{\partial \mathrm{\theta }}\overrightarrow{A}\left(r\,\mathrm{\theta }\,\mathrm{\phi }\right)\right)\cot \left(\mathrm{\theta }\right)+\frac{{\partial }^2}{\partial {\mathrm{\theta }}^2}\overrightarrow{A}\left(r\,\mathrm{\theta }\,\mathrm{\phi }\right)+\left(\frac{{\partial }^2}{\partial {\mathrm{\phi }}^2}\overrightarrow{A}\left(r\,\mathrm{\theta }\,\mathrm{\phi }\right)\right){\csc \left(\mathrm{\theta }\right)}^2}{r^2}$

$\mathrm{eq}≔\mathrm{Gradient}\left(A\left(r\,\mathrm{\theta }\,\mathrm{\phi }\right)\right)$

$\mathrm{eq}≔\left(\frac{\partial }{\partial r}A\left(r\,\mathrm{\theta }\,\mathrm{\phi }\right)\right)\stackrel{\wedge }{r}+\frac{\left(\frac{\partial }{\partial \mathrm{\theta }}A\left(r\,\mathrm{\theta }\,\mathrm{\phi }\right)\right)\stackrel{\wedge }{\mathrm{\theta }}}{r}+\frac{\left(\frac{\partial }{\partial \mathrm{\phi }}A\left(r\,\mathrm{\theta }\,\mathrm{\phi }\right)\right)\stackrel{\wedge }{\mathrm{\phi }}}{r\sin \left(\mathrm{\theta }\right)}$

$\mathrm{Curl}\left(\mathrm{eq}\right)$

$0$

$\mathrm{\%Gradient}\left(A\left(r\,\mathrm{\theta }\,\mathrm{\phi }\right)\right)$

$\nabla A\left(r\,\mathrm{\theta }\,\mathrm{\phi }\right)$

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

$0$

$\mathrm{eq}≔\mathrm{Curl}\left(\mathrm{V\_}\left(r\,\mathrm{\theta }\,\mathrm{\phi }\right)\right)$

$\mathrm{eq}≔\nabla \times \overrightarrow{V}\left(r\,\mathrm{\theta }\,\mathrm{\phi }\right)$

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

$0$

$V≔\mathrm{\_r}+\mathrm{f1}\left(\mathrm{\theta }\,\mathrm{\phi }\right)\mathrm{\_\θ}+\mathrm{f2}\left(\mathrm{\theta }\,\mathrm{\phi }\right)\mathrm{\_\φ}$

$V≔\stackrel{\wedge }{r}+\mathrm{f1}\left(\mathrm{\theta }\,\mathrm{\phi }\right)\stackrel{\wedge }{\mathrm{\theta }}+\mathrm{f2}\left(\mathrm{\theta }\,\mathrm{\phi }\right)\stackrel{\wedge }{\mathrm{\phi }}$

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

$\frac{2}{r}+\frac{\left(\frac{\partial }{\partial \mathrm{\theta }}\mathrm{f1}\left(\mathrm{\theta }\,\mathrm{\phi }\right)\right)\sin \left(\mathrm{\theta }\right)+\mathrm{f1}\left(\mathrm{\theta }\,\mathrm{\phi }\right)\cos \left(\mathrm{\theta }\right)}{r\sin \left(\mathrm{\theta }\right)}+\frac{\frac{\partial }{\partial \mathrm{\phi }}\mathrm{f2}\left(\mathrm{\theta }\,\mathrm{\phi }\right)}{r\sin \left(\mathrm{\theta }\right)}$

$\mathrm{Curl}\left(V\right)$

$\frac{\left(\left(\frac{\partial }{\partial \mathrm{\theta }}\mathrm{f2}\left(\mathrm{\theta }\,\mathrm{\phi }\right)\right)\sin \left(\mathrm{\theta }\right)+\mathrm{f2}\left(\mathrm{\theta }\,\mathrm{\phi }\right)\cos \left(\mathrm{\theta }\right)-\frac{\partial }{\partial \mathrm{\phi }}\mathrm{f1}\left(\mathrm{\theta }\,\mathrm{\phi }\right)\right)\stackrel{\wedge }{r}}{r\sin \left(\mathrm{\theta }\right)}-\frac{\mathrm{f2}\left(\mathrm{\theta }\,\mathrm{\phi }\right)\stackrel{\wedge }{\mathrm{\theta }}}{r}+\frac{\mathrm{f1}\left(\mathrm{\theta }\,\mathrm{\phi }\right)\stackrel{\wedge }{\mathrm{\phi }}}{r}$