helmholtz decomposition

Helmholtz decomposition has many application, especially physics.

 

Let, F be a vector field on a bounded domain V in R^3, which is twice continuously differentiable inside V, and let S be the surface that encloses the domain V.

 

Then, F can be decomposed into a curl-free component and divergence-free component.

 

Maxwell's equation for electric and magnetic fields in the static case corresponds to the the 2 equations given specified curl and divergence.

 

weak formulation of helmholtz decomposition

 

derivation from the fourier transform

Fourier transform of vector field is guaranteed to exist.

Now, we can derive fourier transformed helmholtz decomposition

G(k) = - ikG_{\Phi}(k) + ik \times G_A (k)

then, by applying inverse Fourier transform, we get the one we want.

 

 

 

so far, we are thinking of 3 dimensional vector fields, but we can extend it to higher dimensional one.

 

let us define scalar potential as the integral of the product (divegence of vector fields and kernel) over the space.

 

let us define rotational potential as the similar one. (curl of vector fields, which is similar to antisymmetric matrix, and kernel)

 

In this case, we think of matrix. If you want more generalized one, you can get tensor.