Algebraic and differential operators evalution rules

Evaluation of the \(\mathrm{grad}\) operator

Rule ID	symbolic expression	evaluation
[OG1]	\(\mathrm{grad}(f+g)\)	\(\mathrm{grad}(f) + \mathrm{grad}(g)\)
[OG2]	\(\mathrm{grad}(\alpha ~ f)\)	\(\alpha~\mathrm{grad}(f)\)
[OG3]	\(\mathrm{grad}(f ~ g)\)	\(f~\mathrm{grad}(g) + g~\mathrm{grad}(f)\)
[OG4]	\(\mathrm{grad}(\frac{f}{g})\)	\(-\frac{f}{g^2}\mathrm{grad}(g) + \frac{1}{g}\mathrm{grad}(f)\)
[OG5]	\(\mathrm{grad}(F+G)\)	\(\mathrm{grad}(F) + \mathrm{grad}(G)\)
[OG6]	\(\mathrm{grad}(\alpha ~ F)\)	\(\alpha~\mathrm{grad}(F)\)
[OG7]	\(\mathrm{grad}(\mathrm{dot}(F, G))\)	\(\mathrm{convect}(F, G) + \mathrm{convect}(G, F) + \mathrm{cross}(F, \mathrm{curl}(G)) - \mathrm{cross}(\mathrm{curl}(F), G)\)

Evaluation of \(\mathrm{curl}\) and \(\mathrm{rot}\) operators in 2D

Rule ID	symbolic expression	evaluation
[OC1a]	\(\mathrm{curl}(f+g)\)	\(\mathrm{curl}(f) + \mathrm{curl}(g)\)
[OC2a]	\(\mathrm{curl}(\alpha ~ f)\)	\(\alpha~\mathrm{curl}(f)\)
[OC1b]	\(\mathrm{rot}(F+G)\)	\(\mathrm{rot}(F) + \mathrm{rot}(G)\)
[OC2b]	\(\mathrm{rot}(\alpha ~ F)\)	\(\alpha~\mathrm{rot}(F)\)

Evaluation of \(\mathrm{curl}\) operator in 3D

Rule ID	symbolic expression	evaluation
[OC1]	\(\mathrm{curl}(F+G)\)	\(\mathrm{curl}(F) + \mathrm{curl}(G)\)
[OC2]	\(\mathrm{curl}(\alpha ~ F)\)	\(\alpha~\mathrm{curl}(F)\)
[OC3]	\(\mathrm{curl}(f ~ F)\)	\(f~\mathrm{curl}(F) + \mathrm{cross}(\mathrm{grad}(f), F)\)
[OC4]	\(\mathrm{curl}(\mathrm{cross}(F, G))\)	\(\mathrm{div}(G)~F - \mathrm{div}(F)~G - \mathrm{convect}(F, G) +\mathrm{convect}(G, F)\)

Evaluation of \(\mathrm{div}\) operator

Rule ID	symbolic expression	evaluation
[OD1]	\(\mathrm{div}(F+G)\)	\(\mathrm{div}(F) + \mathrm{div}(G)\)
[OD2]	\(\mathrm{div}(\alpha ~ F)\)	\(\alpha~\mathrm{div}(F)\)
[OD3]	\(\mathrm{div}(f ~ G)\)	\(f~\mathrm{div}(G) + \mathrm{dot}(G, \mathrm{grad}(f))\)
[OD4]	\(\mathrm{div}(\mathrm{cross}(F, G))\)	\(-\mathrm{dot}(F, \mathrm{curl}(G)) + \mathrm{dot}(G, \mathrm{curl}(F))\)

Evaluation of \(\mathrm{laplace}\) operator

Rule ID	symbolic expression	evaluation
[OL1]	\(\mathrm{laplace}(f+g)\)	\(\mathrm{laplace}(f) + \mathrm{laplace}(g)\)
[OL2]	\(\mathrm{laplace}(\alpha ~ f)\)	\(\alpha~\mathrm{laplace}(f)\)
[OL3]	\(\mathrm{laplace}(f~g)\)	\(f ~ \mathrm{laplace}(g) + g ~ \mathrm{laplace}(f) + 2 \mathrm{dot}(\mathrm{grad}(f), \mathrm{grad}(g))\)

Evaluation of specific combination of operators

Rule ID	symbolic expression	evaluation
[OS1]	\(\mathrm{curl}(\mathrm{grad}(f))\)	\(0\)
[OS2]	\(\mathrm{div}(\mathrm{curl}(F))\)	\(0\)
[OS3]	\(\mathrm{div}(\mathrm{cross}(\mathrm{grad}(F), \mathrm{grad}(G)))\)	\(0\)
[OS4]	\(\mathrm{curl}(\mathrm{curl}(F))\)	\(\mathrm{grad}(\mathrm{div}(F)) - \mathrm{laplace}(F)\)
[OS5]	\(\mathrm{curl}(f~\mathrm{grad}(g))\)	\(\mathrm{cross}(\mathrm{grad}(f), \mathrm{grad}(g))\)