---
tag: 05RCT1522R
kind: statement
role: proposition
name: Laplacian as a sum of second partial derivatives
aliases: [cartesian-formula-for-the-laplacian]
msc: [35J05]
uses: [0XQ2ZVN1PZ, 02NSMQVQWK, 0NS2QYVC4Y, 02RSXVM1ZN, 0SCSQWXRA4]
sources:
  - url: https://en.wikipedia.org/wiki/Laplace_operator
    locator: Definition
  - url: https://en.wikipedia.org/wiki/Laplace_operator
    locator: N dimensions
  - url: https://en.wikipedia.org/wiki/Laplace_operator
    locator: Two dimensions
---

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, and let $f \colon U \to \mathbb{R}$ be twice differentiable on $U$.
Then
$$\Delta f = \frac{\partial^2 f}{\partial x_1^2} + \frac{\partial^2 f}{\partial x_2^2} + \dots + \frac{\partial^2 f}{\partial x_n^2}.$$

## Remarks

Proof: $\nabla f = (\partial f / \partial x_1, \dots, \partial f / \partial x_n)$, and the [[divergence|02NSMQVQWK]] adds the derivative of the $i$-th component with respect to $x_i$.
So $\Delta f$ is the trace of the [[Hessian matrix|0SCSQWXRA4]] of $f$. The trace does not change under a rotation of coordinates, so neither does $\Delta f$.
Examples: $\Delta (x^2 + y^2) = 4$, $\Delta (x^2 - y^2) = 0$ and $\Delta \|x\|^2 = 2n$ on $\mathbb{R}^n$.
In polar coordinates in the plane, $\Delta f = f_{rr} + \tfrac{1}{r} f_r + \tfrac{1}{r^2} f_{\theta\theta}$ for $r > 0$.

## Explanations

### L1 human

In Cartesian coordinates, the Laplacian adds up the second derivatives of $f$ in the $n$ coordinate directions.
Only the unmixed second partial derivatives appear, not mixed ones like $f_{xy}$.