Laplacian Calculator

Easily calculate the Laplacian of a scalar function with our online tool. Just enter your function and variables!

Function Details

Enter the scalar function and the variables with respect to which you want to calculate the Laplacian.

Result: Laplacian

Understanding the Laplacian

The Laplacian operator, denoted as ∇² or Δ, is a fundamental concept in calculus and physics. It measures the second-order spatial derivatives of a scalar function.

Mathematically, for a function f(x, y, z), the Laplacian is defined as the divergence of the gradient of f, which expands to:

$$ \nabla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2} $$

In simpler terms, the Laplacian at a point reflects how much the average value of the function in a small neighborhood around the point deviates from the value at the point itself. It's often described as measuring the "curvature" or "concavity" of the function.

The Laplacian has wide applications across various fields, including:

  • Physics: In heat conduction, it describes the heat flow; in electromagnetism, it appears in Poisson's and Laplace's equations; in fluid dynamics, it's part of the Navier-Stokes equations.
  • Image Processing: Used for edge detection and image sharpening.
  • Mathematics: Crucial in harmonic analysis, potential theory, and the study of differential equations.

Laplacian Calculator - Quick Guide

The Laplacian Calculator is a tool designed to compute the Laplacian of a given scalar function. To use it, simply input your function in terms of variables (like x, y, z) in the 'Function' field. Then, specify the variables with respect to which you want to calculate the Laplacian, separated by commas, in the 'Variables' field. For instance, for a function f(x, y) = x² + y², and variables x, y, the calculator will output the Laplacian. The Laplacian essentially tells you about the concavity of the function at each point. It's a vital operation in many areas of science and engineering for analyzing fields and solving partial differential equations.