Helmholtz Theorem on Decomposition of Vectors

[Gonçalo]

Does anyone know anything about this?

I got a look at wolfram.com and I didnt get much.

I would like to prove that in fact, any divergenceless vector field can be decompose in a toroidal part and a poloidal part. And I think the proof of this is somehow related with this theorem...

 

[Alan M. Wolsky]

query about toroidal and poloidal vector fields

Kindly bear with me. I am new to this forum and unfamiliar with it.

A websearch led me here. I wish to learn the meaning of the phrases "toroidal and poloidal vector fields" At the moment they are just words to me. Please tell me their meaning or tell me how to find out more about same.

I gather that a solenoidal field can be decomposed into a "toroidal part and a poloidal part" but again, I am not familiar with this decomposition. Please tell me how to learn about it.

With hope for your guidance,
Alan M. Wolsky

 

[tacman]

The Helmholtz Theorem on Decomposition of Vectors states that any vector field in three-dimensional space can be decomposed into two parts: a solenoidal (divergence-free) part and an irrotational (curl-free) part. This theorem is named after the German physicist Hermann von Helmholtz, who first proposed it in the mid-19th century.

The solenoidal part of a vector field is often referred to as the "toroidal" part, as it represents the rotational component of the field. This part is characterized by having a zero divergence, meaning that the net flow of the field out of any closed surface is zero. The irrotational part, also known as the "poloidal" part, represents the potential component of the field and is characterized by having a zero curl, meaning that the field lines do not form closed loops.

The Helmholtz Theorem is a fundamental result in vector calculus and has numerous applications in physics and engineering. For example, it is used to study fluid dynamics, electromagnetism, and even quantum mechanics. The proof of this theorem involves the use of the vector calculus operators such as divergence and curl, as well as the fundamental theorem of calculus.

In summary, the Helmholtz Theorem on Decomposition of Vectors is a powerful tool for understanding vector fields and their properties. It allows for a more intuitive and simplified analysis of complex vector fields, making it an essential concept in many areas of science and engineering.