View source for Laplace Operator Page 3 2020 - Rhea

[[Category:MA271Fall2020Walther]]
==Coordinate Conversions for the Laplace Operator==

It is most common to use the Laplace Operator <math>\Delta</math> 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.

<math>
\Delta f=\frac{\partial^{2} f}{\partial x^{2}}+\frac{\partial^{2} f}{\partial y^{2}}+\frac{\partial^{2} f}{\partial z^{2}}\
</math>

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}, \frac{\partial f}{\partial y},\frac{\partial f}{\partial z}\Bigg] </math> and <math> \left[\begin{array}{l}
\frac{\partial}{\partial x} \\
\frac{\partial}{\partial y} \\
\frac{\partial}{\partial z}
\end{array}\right] </math> will of course be <math> \frac{\partial^{2} f}{\partial x^{2}}+\frac{\partial^{2} f}{\partial y^{2}}+\frac{\partial^{2} f}{\partial z^{2}}\ </math>