Fundamental theorem of vector analysis.html

In mathematics, in the area of vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently smooth, rapidly decaying vector field can be resolved into irrotational (curl-free) and solenoidal (divergence-free) component vector fields.

This implies that any vector field $\mathbf{F}$ can be considered to be generated by a pair of potentials: a scalar potential $φ$ and a vector potential $\mathbf{A}$ .

The resulting Helmholtz decomposition of a vector field, which is twice continuously differentiable and with rapid enough decay at infinity, splits the vector field into a sum of gradient and curl as follows:

$\mathbf{F} = - \nabla\,\mathcal{G} (\nabla \cdot \mathbf{F}) + \nabla \times \mathcal{G}(\nabla \times \mathbf{F})$

where $\mathcal{G}$ represents the Newtonian potential operator.

If $\nabla\cdot\mathbf{F}=0$ , we say $\mathbf{F}$ is solenoidal or divergence-free and thus the Helmholtz decomposition of $\mathbf{F}$ collapses to

$\mathbf{F} = \nabla \times \mathcal{G}(\nabla \times \mathbf{F}) = \nabla \times \mathbf{A}$

In this case, $\mathbf{A}$ is known as the vector potential for $\mathbf{F}$ .

Likewise, if $\nabla\times\mathbf{F}=\mathbf{0}$ then $\mathbf{F}$ is said to be curl-free or irrotational and thus the Helmholtz decomposition of $\mathbf{F}$ collapses then to

$\mathbf{F} = - \nabla\,\mathcal{G} (\nabla \cdot \mathbf{F}) = - \nabla \phi.$

In this case, $φ$ is known as the scalar potential for $\mathbf{F}$ .

In general the negative gradient of the scalar potential is equated with the irrotational component, and the curl of the vector potential is equated with the solenoidal component:

$\mathbf{F} = -\nabla \phi + \nabla \times \mathbf{A}$ .

1 Applicability to differential forms
2 Weaker formulation
3 Longitudinal and transverse fields
4 References
- 4.1 General references
- 4.2 References for the weak formulation
5 External links

Applicability to differential forms

The Hodge decomposition generalizes the Helmholtz decomposition from vector fields to differential forms.

Weaker formulation

The Helmholtz decomposition can also be generalized by reducing the regularity assumptions (the need for the existence of strong derivatives). Suppose $Ω$ is a bounded, simply-connected, Lipschitz domain. Every vector field $\mathbf{u}\in(L^2(\Omega))^3$ has an orthogonal decomposition:

$\mathbf{u}=\nabla\phi+\mathrm{curl}\,\mathbf{A}$

where $\phi\in H^1(\Omega)$ and $\mathbf{A}\in H(\mathrm{curl},\Omega)$ . For a slightly smoother vector field $\mathbf{u}\in H(\mathrm{curl},\Omega)$ , a similar decomposition holds:

$\mathbf{u}=\nabla\phi+\mathbf{v}$

where $\phi\in H^1(\Omega)$ and $\mathbf{v}\in(H^1(\Omega))^d$ .

Longitudinal and transverse fields

A terminology often used in physics is the curl-free component of a vector field is called the longitudinal component and the divergence-free component is called the transverse component.¹ This terminology comes from the following construction: Compute the three-dimensional Fourier transform of the vector field F, which we call $\tilde{\mathbf{F}}$ . Then decompose this field, at each point k, into two components, one of which points longitudinally, i.e. parallel to k, the other of which points in the transverse direction, i.e. perpendicular to k. So far, we have

$\tilde{\mathbf{F}}(\mathbf{k}) = \tilde{\mathbf{F}}_l(\mathbf{k}) + \tilde{\mathbf{F}}_t(\mathbf{k})$

$\mathbf{k} \cdot \tilde{\mathbf{F}}_t(\mathbf{k}) = \mathbf{k}\times \tilde{\mathbf{F}}_l(\mathbf{k}) = 0$

Now we apply an inverse Fourier transform to each of these components. Using properties of Fourier transforms, we derive:

$\mathbf{F} = \mathbf{F}_t+\mathbf{F}_l$

$\nabla \cdot \mathbf{F}_t = \nabla\times \mathbf{F}_l = 0$

so this is indeed the Helmholtz decomposition.²

References

General references

George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists, 4th edition, Academic Press: San Diego (1995) pp. 92-93
George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists International Edition, 6th edition, Academic Press: San Diego (2005) pp. 95-101

References for the weak formulation

C. Amrouche, C. Bernardi, M. Dauge, and V. Girault. "Vector potentials in three dimensional non-smooth domains." Mathematical Methods in the Applied Sciences, 21, 823–864, 1998.
R. Dautray and J.-L. Lions. Spectral Theory and Applications, volume 3 of Mathematical Analysis and Numerical Methods for Science and Technology. Springer-Verlag, 1990.
V. Girault and P.A. Raviart. Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms. Springer Series in Computational Mathematics. Springer-Verlag, 1986.

External links

Helmholtz theorem on MathWorld

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