- #1
quasar987 said:Did you write this Pete?
There's something I don't get at all:
Equation (10c): " [itex]\nabla[/itex] and [itex]\nabla '[/itex] are related by [itex]\nabla = -\nabla '[/itex] "
How can they be related since the primed coordinates and the non-primed are not related?
"Regarding the first integral on the right; as the radius of the surface increases as r then the area of the surface increases as r². However the integrand decreases as r³. Therefore as we let the radius go to infinity we see that the first integral vanishes."
The author seems to be making the assumption that F decreases as r², something that was not mentionned in the original statement of the theorem.
It's a detail important to mention imo.
The Helmholtz Theorem is a fundamental principle in vector calculus that states that any vector field can be decomposed into two components: a solenoidal component and an irrotational component.
The Helmholtz Theorem was first proposed by German physicist and physician Hermann von Helmholtz in the 19th century.
A solenoidal component is a vector field that has zero divergence, meaning that the net flow of the vector field through any closed surface is equal to zero. This component is often referred to as the "curl-free" component.
An irrotational component is a vector field that has zero curl, meaning that the vector field does not rotate or swirl around any point. This component is often referred to as the "divergence-free" component.
The Helmholtz Theorem has many applications in physics, engineering, and other scientific fields. It is used to model fluid flow, electromagnetism, and other phenomena that can be described by vector fields. It is also used in the development of numerical methods for solving differential equations.