Definition:Laplacian/Vector Field/Cartesian 3-Space/Definition 2
Jump to navigation
Jump to search
Definition
Let $R$ be a region of Cartesian $3$-space $\R^3$.
Let $\map {\mathbf V} {x, y, z}$ be a vector field acting over $R$.
Let $\tuple {\mathbf i, \mathbf j, \mathbf k}$ be the standard ordered basis on $\R^3$.
Let $\mathbf V$ be expressed as vector-valued function:
- $\mathbf V := V_x \mathbf i + V_y \mathbf j + V_z \mathbf k$
The Laplacian on $\mathbf V$ is defined as:
- $\nabla^2 \mathbf V = \nabla^2 V_x \mathbf i + \nabla^2 V_x \mathbf j + \nabla^2 V_y \mathbf k$
where $\nabla^2 V_x$ and so on are the laplacians of $V_x$, $V_y$ and $V_z$ as scalar fields.
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 $\nabla$: $3$. The Operator $\nabla^2$ with Vector Operand: $(5.4)$