I have a question about the application of the sequential continuity of convolution of distributions. We know that,
(The chain rule for distributional derivatives of pullbacks)
Let $F=(F_1,\cdots,F_n):\Omega_1\to\Omega_2$ be a diffeomorphism, then for every $u\in\mathcal{D}'(\Omega_2)$, we have $$\partial_j(F^*u)=\sum_{k=1}^n (\partial_jF_k) \cdot F^*(\partial_ku).$$
We can prove this formula by calculating based on the definition, or by using the density of smooth functions in $\mathcal{D}'$, the chain rule for compositions of smooth functions, and the sequential continuity of pullback and multiplication by smooth factors.
My question is:
Q: How to prove the formula by using the sequential continuity of convolution of distributions?
Consider the convolution $u \ast \varphi_{\epsilon}$ where $\varphi_{\epsilon} \to \delta$ with $\varphi_{\epsilon} \in \mathcal{D}$.
Then $u \ast \varphi_{\epsilon}$ is a $C^{\infty}$ function and you can apply the chain rule for smooth functions. Using $u \ast \varphi_\epsilon \to u \ast \delta = u$ and the arguments you mentioned, one can conclude.