Gradient and Laplacian in integral.

Question

Let $u,v,f$ be functions of $\mathbb{R}^n$ to $\mathbb{R}$, with compact support in a domain $U$, this formula $$\int_{U} f(x) (Du \cdot Dv) dx = \int_{U} f(x)(u D(Dv)) dx = \int_{U} f(x) u(x) \Delta v(x) dx$$

is true?

In simples words, Can I exchange the gradients?

Answer 1

I don't know what you mean in the middle integral but yes this is Green's first Identity with no boundary term: Green's Identities