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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 = \Bigg[\frac{\partial }{\partial x_{1}},\cdots,\frac{\partial }{\partial x_{n}}\Bigg] \cdot F = F_{x_{1}} + \cdots + F_{x_{n}} $

where $ F $ is a $ n $–dimensional vector field. Divergence measures how much "stuff" is going out of a region in space. It may be helpful to think of divergence in terms of heat— consider a thin metal sheet which is very hot in some spots and very cold in others. In order for the temperature of the sheet to become uniform, the heat energy must spread from where it is originally located. If the heat transfer of the sheet was modeled in a vector field, the hot parts would have a positive divergence because the energy moves out of those areas. Likewise, the cold parts of the sheet would have a negative divergence because the energy has to move into them in order to reach a temperature equilibrium.

Now, let's apply these definitions to the Laplace operator. Since the gradient of a function measures the direction in which the function increases the fastest, it makes sense that the gradient points towards the "highest" areas of a function and away from the "lowest" areas. When we take the divergence of a gradient, we then see that the divergence would be negative in these high areas and positive in these low areas, since the vector field points towards and out of these places, respectively. Thinking more physically about it, the Laplacian is positive in local minima ("valleys") and negative in local maxima ("hills") of a function. This aligns well with the intuition for the second derivative of a single-variable function, where the second derivative is positive when the slope is increasing and it is negative when the slope is decreasing. These increasing and decreasing slopes naturally create valleys and hills in the graphs of functions, creating a parallel between the second derivative and the Laplace operator.

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