Proving that J[transverse] + J[longitudinal] = J, where J = current density

[bjnartowt]

Homework Statement

see attached .pdf

Homework Equations

see attached .pdf

The Attempt at a Solution

see attached .pdf

Question: am I wrong for using,
$${{\bf{J}}_\ell } \equiv - \frac{1}{{4\pi }}\nabla \int {\frac{{\nabla ' \bullet {\bf{J}}({\bf{x'}})}}{{\left| {{\bf{x}} - {\bf{x'}}} \right|}}{d^3}x'} = - \frac{1}{{4\pi }}\nabla \left( {\nabla \bullet \int {\frac{{{\bf{J}}({\bf{x'}})}}{{\left| {{\bf{x}} - {\bf{x'}}} \right|}}{d^3}x'} } \right)$$

in Eq. [I.4] of the attached .pdf? It seems the only way to get a Dirac delta to appear later, which I need...

 

[mark726]

it is important to always question and evaluate your methods and assumptions. While it may seem like using this equation is the only way to get a Dirac delta to appear later, it is important to consider alternative methods and approaches as well. It is also important to check your calculations and make sure that your use of this equation is valid and consistent with the principles of the field. Consulting with colleagues or seeking guidance from experts in the field can also help to validate your approach and ensure that you are on the right track. Ultimately, as long as your solution is logically and mathematically sound, it is not necessarily "wrong" to use this equation. However, it is always important to critically evaluate your methods and make sure they align with accepted practices in the field.