What is the difference between the Laplacian and the gradient?

The gradient $\nabla f$ is a vector of first partial derivatives that points in the direction of steepest increase. The Laplacian $\nabla^2 f$ is a scalar formed by taking the divergence of that gradient — it sums the second partial derivatives and tells you about curvature, not direction.

Laplacian — Definition, Formula & Examples

The Laplacian of a scalar function is the sum of its unmixed second partial derivatives. It measures how much the function's value at a point differs from the average value in a small surrounding neighborhood.

For a twice-differentiable scalar function $f : \mathbb{R}^n \to \mathbb{R}$ , the Laplacian is defined as $\nabla^2 f = \sum_{i=1}^{n} \dfrac{\partial^2 f}{\partial x_i^2}$ . Equivalently, it is the divergence of the gradient: $\nabla^2 f = \nabla \cdot (\nabla f)$ .

Key Formula

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

Where:

$\nabla^2 f$ = Laplacian of the scalar function f
$\frac{\partial^2 f}{\partial x^2}$ = Second partial derivative of f with respect to x
$\frac{\partial^2 f}{\partial y^2}$ = Second partial derivative of f with respect to y
$\frac{\partial^2 f}{\partial z^2}$ = Second partial derivative of f with respect to z

How It Works

To compute the Laplacian, take the second partial derivative of your function with respect to each independent variable, then add the results together. In two dimensions with variables

x

and

y

, this means computing

f_{xx} + f_{yy}

. In three dimensions, you add

f_{zz}

as well. A positive Laplacian at a point means the function value there is below the local average (like a valley bottom in a bowl), while a negative Laplacian means it is above the local average (like a hilltop). When the Laplacian equals zero everywhere, the function is called harmonic — it satisfies Laplace's equation

\nabla^2 f = 0

, one of the most important partial differential equations in physics.

Worked Example

Problem: Find the Laplacian of

f(x, y, z) = x^2 + 3y^2 - 2z^2 + xy

Step 1: Compute the second partial derivative with respect to x.

f_x = 2x + y \quad \Rightarrow \quad f_{xx} = 2

Step 2: Compute the second partial derivative with respect to y.

f_y = 6y + x \quad \Rightarrow \quad f_{yy} = 6

Step 3: Compute the second partial derivative with respect to z.

f_z = -4z \quad \Rightarrow \quad f_{zz} = -4

Step 4: Sum all three second partial derivatives.

\nabla^2 f = 2 + 6 + (-4) = 4

Answer:

\nabla^2 f = 4

. Since the Laplacian is a positive constant everywhere, the function is subharmonic — its value at any point is below the average over surrounding spheres.

Another Example

Problem: Show that

f(x, y) = e^x \sin y

is harmonic (i.e., its Laplacian is zero).

Step 1: Find the second partial derivative with respect to x.

f_x = e^x \sin y \quad \Rightarrow \quad f_{xx} = e^x \sin y

Step 2: Find the second partial derivative with respect to y.

f_y = e^x \cos y \quad \Rightarrow \quad f_{yy} = -e^x \sin y

Step 3: Add the results.

\nabla^2 f = e^x \sin y + (-e^x \sin y) = 0

Answer:

\nabla^2 f = 0

everywhere, confirming that

f

is harmonic and satisfies Laplace's equation.

Why It Matters

The Laplacian is central to courses in multivariable calculus, partial differential equations, and mathematical physics. It appears in the heat equation, wave equation, and Schrödinger equation, governing phenomena from temperature diffusion to quantum mechanics. Engineers, physicists, and data scientists working with image processing or mesh smoothing rely on discrete versions of the Laplacian every day.

Common Mistakes

Mistake: Including mixed partial derivatives like

\frac{\partial^2 f}{\partial x \, \partial y}

in the sum.

Correction: The Laplacian only sums unmixed (pure) second partial derivatives — each variable differentiated twice. Cross-partials are not included.

Mistake: Confusing

\nabla^2 f

(scalar Laplacian) with

\nabla f

(gradient vector) or with the Hessian matrix.

Correction: The Laplacian is the trace of the Hessian matrix, not the full matrix. It produces a single scalar value, not a vector or matrix.