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The Helmholtz Theorem

The Helmholtz Theorem.

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The Helmholtz Theorem

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  1. The Helmholtz Theorem Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently smooth, rapidly decaying vector field in three dimensions can be resolved into the sum of an irrotational (curl-free) vector field and a solenoidal (divergence-free) vector field; this is known as the Helmholtz decomposition. Existence of a potential If a vector field has such a property that for every closed path L: then there exists a potentialfsuch that The curl of such field vanishes. Indeed, Such vector field is called irrotational or curl-free

  2. If a vector field no source: then there exists vector potential A such that Such vector field B is called solenoidal or divergence-free Note that B is not uniquely defined by A. Indeed, gauge function Helmholtz decomposition: the Newtonian potential

  3. Let us assume that the divergence of a vector field E is: wherer(r) vanishes quickly at large r. This fully determines E. where Indeed:

  4. Let us now assume that the divergence-free vector field B is given by: whereJ(r) vanishes quickly at large r. This determines B. where Indeed:

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