Introduction to Laplacian

In mathematics, the Laplacian is a differential operator which appears in the divergence of a scalar function defined over an Euclidean space. It is known as an application of derivative since it uses a second-order derivative. Let’s understand how to calculate Laplacian and its applications in various fields of science. 

Understanding the Laplacian

The Laplacian is a second-order differential operator which is also known as the second order partial derivative of a scalar function. It is used in vector calculus to find the divergence of a gradient of a scalar function. It calculates the local variations or the changes with the scalar function or field. It was discovered by a French Mathematician and astronomer named Pierre-Simon de Laplace and denoted by the operator . or 2 which is pronounced as ‘nabla’.

Laplacian formula and equation

The Laplacian operator is the sum of second partial derivatives of a function. In mathematics, the Laplacian of f(x,y,z) in three-dimensions is expressed by;

$\nabla^2 f(x,y,z) = \frac{∂²f}{∂x²} + \frac{∂²f}{∂y²} +\frac{ ∂²f}{∂z²}$

Where,

• ∂/∂x is the partial derivative of f(x,y,z) with respect to x.
• ∂/∂y is the partial derivative of f(x,y,z) with respect to y.
• ∂/∂z is the partial derivative of f(x,y,z) with respect to z.

Laplacian measures the behavior of functions and fields that allows us to understand the properties such as smoothness, curvature and rate of change. It is an important equation in mathematical physics as it appears in many partial differential equations (PDEs). 

Laplace’s equation in Cylindrical Coordinates

The cylindrical coordinates are used to transform two-dimensional coordinates into three-dimensional coordinates. The radial distance, azimuthal angle, and height of the point from a particular plane are useful for finding the location of a point. To write the Laplace equation in polar or cylindrical coordinates, we use the following parameters;

$x=r \cos \theta $

$y=r\sin \theta $

$z=z$

And the transformations of Laplace equation into polar or cylindrical coordinates is;

$\nabla^2 u=u_{rr}+\frac{1}{r}u_r +\frac{1}{r^2}u_{\theta \theta}+u_{zz}$

How do you calculate Laplacian?

The Laplacian of a function can be calculated by finding the second-order partial derivative. You can also use the following steps to simplify calculations. These steps are:

1. Write the function and identify the independent variable. 
2. Calculate the partial derivative of f(x,y) with respect to x. Differentiate it again to get a second order partial derivative with respect to x. the relevant rules according to the type of function. 
3. Now use step 2 to calculate the second-order partial derivative of f(x,y) with respect to y. 
4. Find the sum of both partial derivatives of f(x,y).
5. Simplify if needed. The resulting expression will be the Laplacian of the given function.

Let’s understand the calculations of the Laplacian in the following example.

Laplacian example

Find the Laplacian of the given function.

$f(x) = x3 - xy^2+y^3$

In first step, we will calculate the partial derivative with respect to x, so, 

$\frac{\partial f}{\partial x} = \frac{\partial}{\partial x[x3 - xy^2 + y^3]$

Since the function f(x) contains an algebraic expression with an exponent, therefore, we will use the derivative power rule.

$\frac{\partial f}{\partial x} = 3x2 – y^2$

Differentiating again to get the second derivative.

$\frac{\partial^2 f}{\partial x^2}=6x{2}nbsp;

Similarly, the second partial derivative with respect to y is;

$\frac{\partial^2 f}{\partial y^2}=6y-2$

The equation of Laplacian is;

$\nabla^2 f(x,y,z) = \frac{∂²f}{∂x²} + \frac{∂²f}{∂y²}$

Substituting the values of partial derivatives.

$\nabla^2 f(x,y,z) =6x + 6y-2$

Hence the Laplacian of the given function is equal to 6x+6y-2.

Applications of Laplacian in Physics

The Laplacian operator has many applications in different fields of science. Some of its applications in physics are:

• In physics, it appears in the Laplace equation which is a second-order partial differential equation that describes the behavior of scalar fields in physical systems. This equation is;

$\nabla^2 \phi =0$

• It appears in the Heat equation which states the rate of change of the temperature. This equation is given by;

$u_t = \alpha^2 u_{xx}$

• Similar to the Heat equation, it plays an important role in the Wave equation. This equation explains the propagation of a wave which depends on the displacement. This equation is written as;

$u_{tt}=c^2 u_{xx}$

Conclusion

The Laplacian is a mathematical operator that is an application of derivative as it uses the concept of partial derivatives. It is used to analyze the behavior of functions and fields. By understanding the equation of Laplacian allows us to solve complex problems related to physics and image processing.