- #1

fog37

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- TL;DR Summary
- Understand the meaning of Helmholtz theorem

Hello,

A generic vector field ##\bf {F} (r)## is fully specified over a finite region of space once we know both its divergence and the curl:

$$\nabla \times \bf{F}= A$$

$$\nabla \cdot \bf{F}= B$$

where ##B## is a scalar field and ##\bf{A}## is a divergence free vector field. The divergence and curl equations are PDEs, i.e. equations applied at all the different spatial points of the region of interest. The region of interest also needs boundary conditions on its boundary. The fields ##B## and ##\bf{A}## are not unique.

Helmholtz theorem states that the same vector field ##\bf{F} (r)## can be written as the gradient of a scalar field ##\Phi## + the curl of a vector field ##\bf{C}## which can be obtained through volume integrals involving the fields ##B## and ##\bf{A}##.

Hope this is correct. This is my understanding of the Helmholtz theorem so far...

Questions:

A generic vector field ##\bf {F} (r)## is fully specified over a finite region of space once we know both its divergence and the curl:

$$\nabla \times \bf{F}= A$$

$$\nabla \cdot \bf{F}= B$$

where ##B## is a scalar field and ##\bf{A}## is a divergence free vector field. The divergence and curl equations are PDEs, i.e. equations applied at all the different spatial points of the region of interest. The region of interest also needs boundary conditions on its boundary. The fields ##B## and ##\bf{A}## are not unique.

Helmholtz theorem states that the same vector field ##\bf{F} (r)## can be written as the gradient of a scalar field ##\Phi## + the curl of a vector field ##\bf{C}## which can be obtained through volume integrals involving the fields ##B## and ##\bf{A}##.

Hope this is correct. This is my understanding of the Helmholtz theorem so far...

Questions:

- Why does Helmholtz theorem only apply to vector fields that are only space dependent and not time dependent?
- What must the boundary conditions be on the region of interest? I have read about the vector field needing to decrease fast to zero at infinity. That means the region of interest is free space and has no boundaries...