- #1
math771
- 204
- 0
Hi there! It looks like you are trying to prove that the second derivatives of the magnetic potential function ##\mathbf{A}## belong to the class ##C(\mathbb{R}^3)##. This is a great question and involves some advanced mathematical techniques.
One approach you can take is to use the dominated convergence theorem, as you mentioned. However, in order to do this, you will need to find a function ##\varphi## that satisfies the conditions of the theorem. One possible choice for ##\varphi## is the function ##\frac{1}{\|\mathbf{y}\|^2}##. This function is positive and integrable on ##\mathbb{R}^3##, and it dominates the integrand ##\frac{\mathbf{J}(\mathbf{y}+\mathbf{x})}{\|\mathbf{y}\|}##, which is also positive.
Using this function as your ##\varphi##, you can then apply the dominated convergence theorem and derive under the integral sign to show that the second derivatives of ##\mathbf{A}## exist and are continuous on ##\mathbb{R}^3##. This will then prove that ##\mathbf{A}\in C^2(\mathbb{R}^3)##, and therefore the second derivatives belong to the class ##C(\mathbb{R}^3)##.
I hope this helps! Let me know if you have any further questions.