Laplacian: Difference between revisions
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imported>Gemma E. Mason (Notation and definition in Cartesian, spherical and cylindrical co-ordinates.) |
imported>Milton Beychok m (Added a {{subpages}} template to top of page to create a proper Citizendium article. Also bolded '''Laplacian''' in first sentence.) |
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The Laplacian is a differential operator of the form<br /> | {{subpages}} | ||
The''' Laplacian''' is a differential operator of the form<br /> | |||
<math>\sum_{i}\frac{\partial^{2}}{\partial x_{i}^{2}}</math><br /> | <math>\sum_{i}\frac{\partial^{2}}{\partial x_{i}^{2}}</math><br /> | ||
where <math>x_{i}</math> are Cartesian | where <math>x_{i}</math> are [[Cartesian coordinates]]. The Laplacian is usually denoted by the symbol <math>\Delta</math> or written as the gradient squared <math>\nabla^{2}</math>. | ||
In cylindrical | In [[cylindrical coordinates]], the Laplacian takes the form<br /> | ||
<math>\frac{1}{\rho}\frac{\partial}{\partial \rho}\bigl(\rho\frac{\partial}{\partial\rho}\bigr)+\frac{1}{\rho^{2}}\frac{\partial^{2}}{\partial\phi^{2}}+\frac{\partial^{2}}{\partial z^{2}}</math><br /> | <math>\frac{1}{\rho}\frac{\partial}{\partial \rho}\bigl(\rho\frac{\partial}{\partial\rho}\bigr)+\frac{1}{\rho^{2}}\frac{\partial^{2}}{\partial\phi^{2}}+\frac{\partial^{2}}{\partial z^{2}}</math><br /> | ||
In spherical | In [[spherical coordinates]], the Laplacian is<br/> | ||
<math>\frac{1}{\rho^{2}}\frac{\partial}{\partial \rho}\bigl(\rho^{2}\frac{\partial}{\partial\rho}\bigr)+\frac{1}{\rho^{2}\mathrm{sin}\theta}\frac{\partial}{\partial\theta}\bigl(\mathrm{sin}\theta\frac{\partial}{\partial\theta}\bigr)+\frac{1}{\rho^{2}\mathrm{sin}^{2}\theta}\frac{\partial^{2}}{\partial\phi^{2}}</math> | <math>\frac{1}{\rho^{2}}\frac{\partial}{\partial \rho}\bigl(\rho^{2}\frac{\partial}{\partial\rho}\bigr)+\frac{1}{\rho^{2}\mathrm{sin}\theta}\frac{\partial}{\partial\theta}\bigl(\mathrm{sin}\theta\frac{\partial}{\partial\theta}\bigr)+\frac{1}{\rho^{2}\mathrm{sin}^{2}\theta}\frac{\partial^{2}}{\partial\phi^{2}}</math> | ||
Latest revision as of 22:40, 3 September 2010
The Laplacian is a differential operator of the form
where are Cartesian coordinates. The Laplacian is usually denoted by the symbol or written as the gradient squared .
In cylindrical coordinates, the Laplacian takes the form
In spherical coordinates, the Laplacian is
