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VectorCalculus

 Laplacian
 compute the Laplacian of a function from R^n to R, or of a vector field

 Calling Sequence Laplacian(f, c) Laplacian(F)

Parameters

 f - algebraic expression c - (optional) list(name) or name[name, name, ...]; specify the list of variable names or coordinate system indexed by coordinate names F - (optional) Vector or Vector-valued procedure; specify a vector field

Description

 • The Laplacian(f, c) command computes the Laplacian of the function f.  This is a synonym for (Del . Del)(f) or Del . Del(f).
 • If c is a list of names, the Laplacian is taken in the current default coordinate system by using the names in c as the coordinate names.  If the number of given names is not compatible with this coordinate system, an error is raised.
 If c is a name indexed by other names, the Laplacian is computed in this coordinate system by using the indices as the coordinate names.  If the number of names is not compatible with the coordinate system, an error is raised.
 If c is not specified, the current default coordinates are used. The default coordinates must be indexed by coordinate names, otherwise an error is raised.
 • The Laplacian(F) command, where F is either a vector field or a Vector-valued procedure (which is interpreted as a vector field), computes the Laplacian of the vector field as follows:
 – If the coordinate system of F is cartesian, then map the algebraic Laplacian onto the component functions.
 – Otherwise, if F is a 3-D vector field, use the formula $\mathrm{Del}\left(\mathrm{Del}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}F\right)-\mathrm{Del}&x\left(\mathrm{Del}&xF\right)$.
 – Otherwise, map F to cartesian coordinates, apply the algebraic Laplacian to the component functions, and then map the result back to the original coordinate system of F.
 • The Laplacian() command returns the differential form of the Laplacian operator in the current coordinate system.  For more information, see SetCoordinates.

Examples

 > $\mathrm{with}\left(\mathrm{VectorCalculus}\right):$
 > $\mathrm{Laplacian}\left({x}^{2}+{y}^{2}+{z}^{2},\left[x,y,z\right]\right)$
 ${6}$ (1)
 > $\mathrm{Laplacian}\left(f\left(r,\mathrm{θ}\right),{'\mathrm{polar}'}_{r,\mathrm{θ}}\right)$
 $\frac{\frac{{\partial }}{{\partial }{r}}{}{f}{}\left({r}{,}{\mathrm{θ}}\right){+}{r}{}\left(\frac{{{\partial }}^{{2}}}{{\partial }{{r}}^{{2}}}{}{f}{}\left({r}{,}{\mathrm{θ}}\right)\right){+}\frac{\frac{{{\partial }}^{{2}}}{{\partial }{{\mathrm{θ}}}^{{2}}}{}{f}{}\left({r}{,}{\mathrm{θ}}\right)}{{r}}}{{r}}$ (2)
 > $\mathrm{SetCoordinates}\left({'\mathrm{cylindrical}'}_{r,\mathrm{θ},z}\right)$
 ${{\mathrm{cylindrical}}}_{{r}{,}{\mathrm{θ}}{,}{z}}$ (3)
 > $\mathrm{Laplacian}\left(f\left(r,\mathrm{θ},z\right)\right)$
 $\frac{\frac{{\partial }}{{\partial }{r}}{}{f}{}\left({r}{,}{\mathrm{θ}}{,}{z}\right){+}{r}{}\left(\frac{{{\partial }}^{{2}}}{{\partial }{{r}}^{{2}}}{}{f}{}\left({r}{,}{\mathrm{θ}}{,}{z}\right)\right){+}\frac{\frac{{{\partial }}^{{2}}}{{\partial }{{\mathrm{θ}}}^{{2}}}{}{f}{}\left({r}{,}{\mathrm{θ}}{,}{z}\right)}{{r}}{+}{r}{}\left(\frac{{{\partial }}^{{2}}}{{\partial }{{z}}^{{2}}}{}{f}{}\left({r}{,}{\mathrm{θ}}{,}{z}\right)\right)}{{r}}$ (4)
 > $\mathrm{Del}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{Del}\left(f\left(r,\mathrm{θ},z\right)\right)$
 $\frac{\frac{{\partial }}{{\partial }{r}}{}{f}{}\left({r}{,}{\mathrm{θ}}{,}{z}\right){+}{r}{}\left(\frac{{{\partial }}^{{2}}}{{\partial }{{r}}^{{2}}}{}{f}{}\left({r}{,}{\mathrm{θ}}{,}{z}\right)\right){+}\frac{\frac{{{\partial }}^{{2}}}{{\partial }{{\mathrm{θ}}}^{{2}}}{}{f}{}\left({r}{,}{\mathrm{θ}}{,}{z}\right)}{{r}}{+}{r}{}\left(\frac{{{\partial }}^{{2}}}{{\partial }{{z}}^{{2}}}{}{f}{}\left({r}{,}{\mathrm{θ}}{,}{z}\right)\right)}{{r}}$ (5)
 > $\mathrm{Laplacian}\left(\right)$
 $\frac{\frac{{\partial }}{{\partial }{r}}{}\left({r}{}\left(\frac{{\partial }}{{\partial }{r}}{}{\mathrm{SF}}{}\left({r}{,}{\mathrm{θ}}{,}{z}\right)\right)\right){+}\frac{{\partial }}{{\partial }{\mathrm{θ}}}{}\left(\frac{\frac{{\partial }}{{\partial }{\mathrm{θ}}}{}{\mathrm{SF}}{}\left({r}{,}{\mathrm{θ}}{,}{z}\right)}{{r}}\right){+}\frac{{\partial }}{{\partial }{z}}{}\left({r}{}\left(\frac{{\partial }}{{\partial }{z}}{}{\mathrm{SF}}{}\left({r}{,}{\mathrm{θ}}{,}{z}\right)\right)\right)}{{r}}$ (6)
 > $F≔\mathrm{VectorField}\left(⟨{r}^{3},\frac{z}{\mathrm{θ}},\sqrt{z}⟩\right)$
 ${F}{≔}\left({{r}}^{{3}}\right){\stackrel{{_}}{{e}}}_{{r}}{+}\left(\frac{{z}}{{\mathrm{θ}}}\right){\stackrel{{_}}{{e}}}_{{\mathrm{θ}}}{+}\left(\sqrt{{z}}\right){\stackrel{{_}}{{e}}}_{{z}}$ (7)
 > $\mathrm{simplify}\left(\mathrm{Laplacian}\left(F\right)\right)$
 $\frac{{2}{}\left({4}{}{{r}}^{{3}}{}{{\mathrm{θ}}}^{{2}}{+}{z}\right)}{{{\mathrm{θ}}}^{{2}}{}{{r}}^{{2}}}{\stackrel{{_}}{{e}}}_{{r}}{-}\frac{{z}{}\left({{\mathrm{θ}}}^{{2}}{-}{2}\right)}{{{r}}^{{2}}{}{{\mathrm{θ}}}^{{3}}}{\stackrel{{_}}{{e}}}_{{\mathrm{θ}}}{-}\frac{{1}}{{4}{}{{z}}^{{3}{/}{2}}}{\stackrel{{_}}{{e}}}_{{z}}$ (8)