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Definition and Intuition for the Laplace Operator

The Laplace operator (also known as the Laplacian), represented by $ \Delta $, is defined as the divergence of the gradient of a scalar function. The Laplacian can be written as:

$ {\large\Delta f = div(\nabla f) = \nabla\cdot\nabla f = \nabla^{2} f = \Bigg[\frac{\partial }{\partial x_{1}},\cdots,\frac{\partial }{\partial x_{n}}\Bigg]\cdot\Bigg[\frac{\partial f}{\partial x_{1}},\cdots,\frac{\partial f}{\partial x_{n}}\Bigg] = \sum\limits_{i=1}^{n}\frac{\partial^{2} f}{\partial x^{2}_{i}}} $

where $ f $ is an $ n $–dimensional scalar function. In other words, the Laplace operator is defined as the sum of all non-mixed second partial derivatives of $ f $. As such, the Laplace operator is somewhat analogous to the second derivative in single-variable calculus. We will soon see that the Laplacian and the second derivative have more in common than their form alone.

In order to intuitively understand the Laplace operator, we must first have a baseline understanding of its individual mechanisms, gradients and divergence. The gradient is defined as follows:

$ \nabla f = \Bigg[\frac{\partial f}{\partial x_{1}},\cdots,\frac{\partial f}{\partial x_{n}}\Bigg] $

where $ f $ is an $ n $–dimensional scalar function. In other words, the gradient of a function is a vector field where each component is the partial derivative of the function with respect to that component's coordinate variable. The gradient of a function at any given point represents the direction where the function is increasing the fastest. It also happens to be perpendicular to any level curves of that function.

Divergence is defined as follows:

$ div(F) = \nabla \cdot F = F_{1} + \cdots + F_{n} $

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