Definition:Laplacian/Scalar Field/Cartesian 3-Space
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Definition
Let $R$ be a region of Cartesian $3$-space $\R^3$.
Let $\map U {x, y, z}$ be a scalar field acting over $R$.
The Laplacian of $U$ is defined as:
- $\nabla^2 U := \dfrac {\partial^2 U} {\partial x^2} + \dfrac {\partial^2 U} {\partial y^2} + \dfrac {\partial^2 U} {\partial z^2}$
where $\nabla$ denotes the del operator.
Also known as
The Laplacian is also known as the Laplace operator, Laplace's operator or Laplace-Beltrami operator.
The last name is usually used in the context of submanifolds in Euclidean space and on (pseudo-)Riemannian manifolds.
Also see
- Results about the Laplacian can be found here.
Sources
- 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text {V}$: Further Applications of the Operator $\nabla$: $1$. The Operator $\operatorname {div} \grad$: $(5.1)$, $(5.2)$
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 22$: The Laplacian: $22.32$