Laplacian Calculator
Calculate the Laplacian operator (nabla squared) of a scalar function.
Scalar Function f(x, y, z)
Use: x, y, z, ^, sin, cos, tan, exp, log, sqrt, pi
Point of Evaluation
Laplacian Formula
nabla^2 f = d^2f/dx^2 + d^2f/dy^2 + d^2f/dz^2
Also known as: div(grad f) = nabla dot nabla f
Laplacian nabla^2 f
6.000000
at point (1, 1, 1)
ff(x,y,z)
3.000000
|G||nabla f|
3.464102
xxd^2f/dx^2
2.000000
yyd^2f/dy^2
2.000000
Second Derivatives
d^2f/dx^22.000000
d^2f/dy^22.000000
d^2f/dz^22.000000
Harmonic Function
No - nabla^2 f != 0
About the Laplacian
Physical Interpretation
The Laplacian measures how much a function at a point differs from its average in a small neighborhood. It appears in heat diffusion, wave propagation, and electrostatics.
Applications
- Heat and diffusion equations
- Wave equations in physics
- Image processing (edge detection)
- Quantum mechanics (Schrodinger equation)