Laplacian on Scalar Field is Divergence of Gradient
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Theorem
Let $\R^n$ denote the real Cartesian space of $n$ dimensions.
Let $U$ be a scalar field over $\R^n$.
Let $\nabla^2 U$ denote the laplacian on $U$.
Then:
- $\nabla^2 U = \operatorname {div} \grad U$
where:
- $\operatorname {div}$ denotes the divergence operator
- $\grad$ denotes the gradient operator.
Proof
From Divergence Operator on Vector Space is Dot Product of Del Operator and definition of the gradient operator:
\(\ds \operatorname {div} \mathbf V\) | \(=\) | \(\ds \nabla \cdot \mathbf V\) | ||||||||||||
\(\ds \grad \mathbf U\) | \(=\) | \(\ds \nabla U\) |
where $\nabla$ denotes the del operator.
Let $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ be the standard ordered basis on $\R^n$.
Hence:
\(\ds \nabla^2 U\) | \(:=\) | \(\ds \nabla \cdot \paren {\nabla U}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\sum_{k \mathop = 1}^n \mathbf e_k \dfrac \partial {\partial x_k} } \cdot \paren {\sum_{k \mathop = 1}^n \dfrac {\partial U} {\partial x_k} \mathbf e_k}\) | Definition of Del Operator | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 1}^n \dfrac {\partial^2 U} {\partial {x_k}^2}\) |
$\blacksquare$
Also presented as
In Cartesian $3$-space $\R^3$, where:
- $U$ is defined as $\map U {x, y, z}$
- $\tuple {\mathbf i, \mathbf j, \mathbf k}$ is the standard ordered basis on $\R^3$.
this is usually presented as:
\(\ds \nabla^2 U\) | \(:=\) | \(\ds \nabla \cdot \paren {\nabla U}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\mathbf i \dfrac \partial {\partial x} + \mathbf j \dfrac \partial {\partial y} + \mathbf k \dfrac \partial {\partial z} } \cdot \paren {\dfrac {\partial U} {\partial x} \mathbf i + \dfrac {\partial U} {\partial y} \mathbf j + \dfrac {\partial U} {\partial z} \mathbf k}\) | Definition of Del Operator | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\partial^2 U} {\partial x^2} + \dfrac {\partial^2 U} {\partial y^2} + \dfrac {\partial^2 U} {\partial z^2}\) |
$\blacksquare$
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: $22.32$