📂Mathematical Physics

Laplacian of a Scalar Function in the Three-Dimensional Cartesian Coordinate System

Definition

The Laplacian of a 3D scalar function $f=f(x,y,z)$ is the divergence of its gradient $f$ and is denoted by $\nabla^{2}$.

$$ \nabla ^{2} f := \nabla \cdot(\nabla f)= \frac{ \partial^{2} f}{ \partial x^{2} }+\frac{ \partial^{2} f}{ \partial y^{2}}+\frac{ \partial^{2} f}{ \partial z^{2}} $$

Explanation

The name Laplacian comes from the French mathematician Laplace. The notation $\nabla^{2}$ is used for convenience. In mathematics (theory of partial differential equations), the notation $\Delta$ is more commonly used. In a nutshell, the Laplacian is an extension of the second-order derivative. If the gradient is an extension of the first-order derivative into three dimensions, then the Laplacian is an extension of the second-order derivative into three dimensions. You might have learned the following content in high school calculus.

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While the first-order derivative simply provides information on whether function $f$ is increasing or decreasing, the second-order derivative gives information on how it is increasing or decreasing. The formula for the Laplacian of $f$, as shown above, is nothing more than the formula for divergence with an additional differentiation.

Derivation

There’s really nothing to derive.

$$ \begin{align*} \nabla \cdot (\nabla f) &= \nabla \cdot \left( \frac{ \partial f}{ \partial x },\frac{ \partial f}{ \partial y},\frac{ \partial f}{ \partial z} \right) \\ &= \frac{ \partial ^{2} f }{ \partial x^{2} }+\frac{ \partial ^{2} f }{ \partial y^{2} } + \frac{ \partial ^{2}f }{ \partial z^{2} } \end{align*} $$

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See Also

• Del operator $\nabla$
• Gradient $\nabla f$
• Divergence $\nabla \cdot \mathbf{F}$
• Curl $\nabla \times \mathbf{F}$
• Laplacian $\nabla^{2} f$