Laplacian

The Laplacian

The Laplacian of $f(x, y)$ is the sum of its second-order partial derivatives:

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

It measures how much $f$ differs from its average in the neighborhood of a point.

Computing Second Derivatives Numerically

Using central differences twice:

$\frac{\partial^2 f}{\partial x^2} \approx \frac{f(x+h,y) - 2f(x,y) + f(x-h,y)}{h^2}$

$\frac{\partial^2 f}{\partial y^2} \approx \frac{f(x,y+h) - 2f(x,y) + f(x,y-h)}{h^2}$

Applications

The Laplacian appears throughout physics and engineering:

• Heat equation: $\partial T / \partial t = \alpha \nabla^2 T$ (how temperature diffuses)
• Wave equation: $\partial^2 u / \partial t^2 = c^2 \nabla^2 u$
• Electrostatics: $\nabla^2 \varphi = 0$ in free space (Laplace's equation)
• Image processing: edge detection (Laplacian filter)

Interpretation

• $\nabla^2 f > 0$ at a point: $f$ is below its local average (like a bowl)
• $\nabla^2 f < 0$ at a point: $f$ is above its local average (like a hill)
• $\nabla^2 f = 0$: harmonic function — a steady-state distribution

Example

For $f(x,y) = x^2 + y^2$:

• $\partial^2 f / \partial x^2 = 2$
• $\partial^2 f / \partial y^2 = 2$
• $\nabla^2 f = 4$ everywhere

For $f(x,y) = x^2 - y^2$:

• $\nabla^2 f = 2 + (-2) = 0$ (harmonic!)

Your Task

Implement double laplacian_2d(double (*f)(double, double), double x, double y, double h) using the central difference formulas for second derivatives.