Laplacian

One-Sentence Idea / 一句话

The Laplacian is the multidimensional second derivative: it measures how a value compares with the surrounding average trend.

Laplacian 是二阶导数的多维推广，用来衡量一点相对于四周平均趋势是偏高还是偏低。

Core Form

对三维 scalar field $T(x,y,z)$：

$${\nabla }^{2}T=T_{xx}+T_{yy}+T_{zz}.$$

从 vector calculus 看：

$${\nabla }^{2}T=\nabla \cdot (\nabla T).$$

它先取 gradient，再测量该 gradient field 的 divergence。

Why It Matters

• Heat Equation：$T_t=\alpha {\nabla }^{2}T$。
• 正 Laplacian 通常表示该点低于周围平均趋势；负值表示高于周围平均趋势。
• 在 diffusion、waves、potential theory 等 PDE 中反复出现。

Connections

• Partial Derivatives
• Second Derivative
• Heat Equation
• Electric Potential
• 02 - But What Is a Partial Differential Equation > Knowledge Point 12: The Laplacian Generalizes Spatial Curvature

Search and Source Mentions

• Main source: 02 - But What Is a Partial Differential Equation