Helmholtz's Theorem
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Theorem
Let $R$ be a region of ordinary space.
Let $\mathbf V$ be a vector field over $R$.
Let $\mathbf V$ be both non-conservative and non-solenoidal.
Then $\mathbf V$ can be decomposed into the sum of $2$ vector fields:
- one being conservative, with scalar potential $S$, but not solenoidal
- one being solenoidal, with vector potential $\mathbf A$, but not conservative.
Thus $\mathbf V$ satifies the partial differential equations:
| \(\text {(1)}: \quad\) | \(\ds \operatorname {div} \mathbf V = \operatorname {div} \grad S\) | \(=\) | \(\ds \nabla^2 S \ne 0\) | |||||||||||
| \(\text {(2)}: \quad\) | \(\ds \curl \mathbf V = \curl \curl \mathbf A\) | \(=\) | \(\ds \nabla^2 \mathbf A \ne \bszero\) |
where:
- $\operatorname {div}$ denotes the divergence operator
- $\grad$ denotes the gradient operator
- $\curl$ denotes the curl operator
- $\nabla^2$ denotes the Laplacian.
Proof
Let us write:
- $\mathbf V = \grad S + \curl \mathbf A$
where:
- $S$ is a scalar field
- $\mathbf A$ is a vector field chosen to be solenoidal
 | This article, or a section of it, needs explaining. In particular: It is not clear why this is always possible You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code. |
Then:
| \(\ds \operatorname {div} \mathbf V\) | \(=\) | \(\ds \operatorname {div} \grad S + \operatorname {div} \curl \mathbf A\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \operatorname {div} \grad S\) | Divergence of Curl is Zero | |||||||||||
| \(\ds \) | \(=\) | \(\ds \nabla^2 S\) | Laplacian on Scalar Field is Divergence of Gradient | |||||||||||
| \(\ds \) | \(\ne\) | \(\ds 0\) | as $\mathbf V$ is not solenoidal |
and:
| \(\ds \curl \mathbf V\) | \(=\) | \(\ds \curl \grad S + \curl \curl \mathbf A\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \curl \curl \mathbf A\) | Curl of Gradient is Zero | |||||||||||
| \(\ds \) | \(=\) | \(\ds \grad \operatorname {div} \mathbf A - \nabla^2 \mathbf A\) | Curl of Curl is Gradient of Divergence minus Laplacian | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\nabla^2 \mathbf A\) | as $\mathbf A$ is solenoidal: $\operatorname {div} \mathbf A = 0$ | |||||||||||
| \(\ds \) | \(\ne\) | \(\ds \bszero\) | as $\mathbf A$ is not conservative |
| This article, or a section of it, needs explaining. In particular: It is not clear how it is certain that $\mathbf A$ is not conservative You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code. |
$\blacksquare$
Source of Name
This entry was named for Hermann Ludwig Ferdinand von Helmholtz.
Sources
- 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text {V}$: Further Applications of the Operator $\nabla$: $7$. The Classification of Vector Fields: $\text {(iv)}$