Question

Extra Credit: It is a fact that any three-dimensional vector field F can be of vector fields F G+H where G is curl-free (i.e. V x G 0) and H is divergen V. H 0. G and H c known as the Helmholz decomposition. It is important in electromagnetic theory. At any point in space the longitudinal part of the electric field describes the part of the field due to charged particles while the tranverse part describes the part of the electric field due to electromagnetic radiation in the vicinity of the point. Try to find the Helmholz decomposition for the vector field expressed as a sum and H is divergence-free (i.e. are respectively called the longitudinal and tranverse parts of F. This is

Answer 1

Given that, Three dimensional vector field 'F' F = Gt + 1 where 'G' is curl - free nabla times G = 0 and 'H' is divergence free nabla H = 0 G, H are called the longitudinal and transverse parts of F F = (- y z, xe^y, e^x + z^2) steps for solution: - Find F dr collect the terms in F dr with make exact differential call these terms to be Phi (x, y, z) find G using G = nabla Phi find H using F - G Defiled solution: - here F = (-y z)I + x e^y j + (e^x + z^2)k therefore r = xi + yj + zk longrightarrow dr = dx i + dy j + dz k Now F dr = (- yz) dx + x e^y dy + (e^x + z^2) dz = d (-xyz) + x e^y dy + e^x dz + z^2 dz