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\Delta f=\frac{\partial^{2} f}{\partial x^{2}}+\frac{\partial^{2} f}{\partial y^{2}}+\frac{\partial^{2} f}{\partial z^{2}}\ | \Delta f=\frac{\partial^{2} f}{\partial x^{2}}+\frac{\partial^{2} f}{\partial y^{2}}+\frac{\partial^{2} f}{\partial z^{2}}\ | ||
| − | <math> | + | </math> |
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| + | The basis for this conversion is, in the opinion of the authors, quite clear. The dot product of the vectors: <math> \nabla f = \Bigg[\frac{\partial f}{\partial x_{1}},\cdots,\frac{\partial f}{\partial x_{n}}\Bigg] </math> | ||
Revision as of 14:10, 6 December 2020
Coordinate Conversions for the Laplace Operator
It is most common to use the Laplace Operator $ \Delta $ in three-dimensions, as that is the dimensionality of our physical universe. Thus, the Laplace Operator is often used in 3-D Cartesian coordinates, cylindrical coordinates, and spherical coordinates.
$ \Delta f=\frac{\partial^{2} f}{\partial x^{2}}+\frac{\partial^{2} f}{\partial y^{2}}+\frac{\partial^{2} f}{\partial z^{2}}\ $
The basis for this conversion is, in the opinion of the authors, quite clear. The dot product of the vectors: $ \nabla f = \Bigg[\frac{\partial f}{\partial x_{1}},\cdots,\frac{\partial f}{\partial x_{n}}\Bigg] $
