1 Laplace Operator
The Laplace Operator (Laplacian) \(\Delta\) is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. In a Cartesian coordinate system the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. Informally, the laplacian \(\Delta f(p)\) of a function \(f\) at a point \(p\) measures by how much the average value of \(f\) over small balls centered at \(p\) deviates from \(f(p)\).
The Laplacian occurs in many differential equations describing physical phenomena including:
- [[poisson-equation]] - describing electric and gravitational potentials;
- [[diffusion-equation]] - describing heat and fluid flow;
- [[wave-equation]] - describing wave propagation; and
- [[schrodinger-equation]] - describing the wave function in quantum mechanics.
The Laplace operator (Laplacian) is a second-order differential operator in the \(n\)-dimensional Euclidean space, defined as the Divergence \((\nabla \cdot)\) of the Gradient \((\nabla f)\). This if \(f\) is a twice-differentiable real function, then the Laplacian is the real-valued function defined by \[ \Delta f = \nabla^2f = \nabla \cdot \nabla f, \] where the latter notations derive from formally writing \[ \nabla=\left( \frac{\partial}{\partial x_{1}}, \dots, \frac{\partial}{\partial x_{n}} \right). \]