Laplacian

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1: 12.17 Physical Applications
§12.17 Physical Applications
where $k$ is a constant, and $\nabla^{2}$ is the Laplacian
2: 23.21 Physical Applications
The Laplacian operator $\nabla^{2}$1.5(ii)) is given by …
4: 1.5 Calculus of Two or More Variables
The Laplacian is given by
1.5.13 $\nabla^{2}f=\frac{{\partial}^{2}f}{{\partial x}^{2}}+\frac{{\partial}^{2}f}{{% \partial y}^{2}}=\frac{{\partial}^{2}f}{{\partial r}^{2}}+\frac{1}{r}\frac{% \partial f}{\partial r}+\frac{1}{r^{2}}\frac{{\partial}^{2}f}{{\partial\phi}^{% 2}}.$
1.5.15 $\nabla^{2}f=\frac{{\partial}^{2}f}{{\partial x}^{2}}+\frac{{\partial}^{2}f}{{% \partial y}^{2}}+\frac{{\partial}^{2}f}{{\partial z}^{2}}=\frac{{\partial}^{2}% f}{{\partial r}^{2}}+\frac{1}{r}\frac{\partial f}{\partial r}+\frac{1}{r^{2}}% \frac{{\partial}^{2}f}{{\partial\phi}^{2}}+\frac{{\partial}^{2}f}{{\partial z}% ^{2}}.$
The Laplacian is given by
1.5.17 $\nabla^{2}f=\frac{{\partial}^{2}f}{{\partial x}^{2}}+\frac{{\partial}^{2}f}{{% \partial y}^{2}}+\frac{{\partial}^{2}f}{{\partial z}^{2}}={\frac{1}{\rho^{2}}% \frac{\partial}{\partial\rho}\left(\rho^{2}\frac{\partial f}{\partial\rho}% \right)+\frac{1}{\rho^{2}{\sin}^{2}\theta}\frac{{\partial}^{2}f}{{\partial\phi% }^{2}}}+\frac{1}{\rho^{2}\sin\theta}\frac{\partial}{\partial\theta}\left(\sin% \theta\frac{\partial f}{\partial\theta}\right).$
6: 3.4 Differentiation
Laplacian
Those for the Laplacian and the biharmonic operator follow from the formulas for the partial derivatives. …
7: 18.39 Applications in the Physical Sciences
where $\nabla^{2}$ is the Laplacian (1.5.17). …By (1.5.17) the first term in (18.39.21), which is the quantum kinetic energy operator $T_{e}$, can be written in spherical coordinates $r,\theta,\phi$ as …