The properties of the Laplace differential operator, denoted $\Delta$ or $\nabla^2$, and defined as the divergence of the gradient. For Laplace equation, see (harmonic-functions)

Laplacian Operator is a derivative operator which is used to find edges in an image.

Mathematically, the Laplacian operator is defined as: $$\nabla^2 \equiv \frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}$$

The Laplacian is a scalar operator. If it is applied to a scalar field, it generates a scalar field.

If the scalar Laplacian operator is applied to a vector field, it acts on each component in turn and generates a vector field.

**Note that** the operator $~\nabla ^2~$ is commonly written as $~\Delta~$ by mathematicians.

**Differences with other operators:**

The major difference between Laplacian and other operators like Prewitt, Sobel, Robinson and Kirsch is that these all are first order derivative masks but Laplacian is a second order derivative mask. Another difference between Laplacian and other operators is that unlike other operators Laplacian didn’t take out edges in any particular direction but it take out edges in following classification.

$1.\quad$ Inward Edges $\qquad$ $2.\quad$ Outward Edges

**Applications:**

The Laplacian is extremely important in mechanics, electromagnetics, wave theory, and quantum mechanics, and appears in

$1.\quad$ Laplace's equation $$\nabla^2 \phi=0$$

$2.\quad$ The Helmholtz differential equation$$\nabla^2 \phi+k^2 \phi=0$$

$3.\quad $ The wave equation$$\nabla^2 \phi=\frac{1}{c^2}\frac{\partial^2 \phi}{\partial t^2}$$

$4.\quad$ The Schrödinger equation$$i \hbar \frac{\partial }{\partial t}\psi(x,y,z,t)=\left[-\frac{\hbar^2}{2m}\nabla^2+V(x)\right]\psi(x,y,z,t)$$

**Laplacian, Various Coordinates :**

$1.\quad$ In rectangular coordinates $~(x,y,z)~$: $$\nabla^2\equiv \nabla \cdot \nabla \equiv \frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}$$

$2.\quad$ In cylindrical polar coordinates $~(r,\theta,z)~$:$$\nabla^2\equiv \frac{\partial^2}{\partial r^2}+\frac{1}{r}\frac{\partial}{\partial r}+\frac{1}{r^2}\frac{\partial^2}{\partial \theta^2}+\frac{\partial^2}{\partial z^2}$$

$3.\quad$ In spherical polar coordinates $~(r,\theta,\phi)~$:$$\nabla^2\equiv \frac{\partial^2}{\partial r^2}+\frac{2}{r}\frac{\partial}{\partial r}+\frac{1}{r^2}\frac{\partial^2}{\partial \theta^2}+\frac{\cos \theta}{r^2\sin \theta}\frac{\partial}{\partial \theta}+\frac{1}{r^2\sin^2\theta}\frac{\partial^2}{\partial \phi^2}$$

**Note:** This tag is also for questions concerning the properties, such as self-adjointness, compactness of inverse, and spectral structure of the Laplace differential operator. The operator, denoted $\Delta$ or $\nabla^2$, is defined as the divergence of the gradient, and generalized to the Laplace-Beltrami and Laplace-deRham operators.

**References:**

https://en.wikipedia.org/wiki/Laplace_operator

http://mathworld.wolfram.com/Laplacian.html

https://nptel.ac.in/courses/115101005/downloads/lectures-doc/Lecture-5.pdf