Laplacian

The Laplacian of a scalar field f (x, y, z) is the divergence of f and is written

2f = ∇⋅ 6ptf  
  = ∇⋅$\displaystyle \left(\vphantom{ \frac{\partial f}{\partial x},\frac{\partial f}{%
\partial y},\frac{\partial f}{\partial z}}\right.$$\displaystyle {\frac{{\partial f}}{{\partial x}}}$,$\displaystyle {\frac{{\partial f}}{{%
\partial y}}}$,$\displaystyle {\frac{{\partial f}}{{\partial z}}}$$\displaystyle \left.\vphantom{ \frac{\partial f}{\partial x},\frac{\partial f}{%
\partial y},\frac{\partial f}{\partial z}}\right)$ 6pt  
  = $\displaystyle {\frac{{\partial ^{2}f}}{{\partial x^{2}}}}$ + $\displaystyle {\frac{{\partial ^{2}f}}{{\partial y^{2}%
}}}$ + $\displaystyle {\frac{{\partial ^{2}f}}{{\partial z^{2}}}}$  

The default is that the field variables are x, y, and z, in that order. If you wish to label the field variables differently, reset the default with Set Basis Variables on the Vector Calculus submenu.

$\blacktriangleright$ Evaluate

2$\left(\vphantom{ x+y^{2}+2z^{3}}\right.$x + y2 +2z3$\left.\vphantom{ x+y^{2}+2z^{3}}\right)$ = 2 + 12z

$\left(\vphantom{ x+y^{2}+2z^{3}}\right.$x + y2 +2z3$\left.\vphantom{ x+y^{2}+2z^{3}}\right)$ = $\left(\vphantom{ 1,2y,6z^{2}}\right.$1, 2y, 6z2$\left.\vphantom{ 1,2y,6z^{2}}\right)$

∇⋅∇$\left(\vphantom{ x+y^{2}+2z^{3}}\right.$x + y2 +2z3$\left.\vphantom{ x+y^{2}+2z^{3}}\right)$ = 2 + 12z