Poincare's lemma on a star-shaped domain: every closed C1 field is exact

By [L1], the integrand defining $\varphi (x)$ is defined for every $t\in [0,1]$; it is continuous, so the integral exists. Fix $x\in U$ and a coordinate $j$. Openness and [L1] provide a small closed coordinate interval about $x$ whose radial segments from $a$ remain in $U$.

On that interval, [L3] differentiates the defining integral with respect to $x_j$ and gives ${\mathrm{\partial }}_j\varphi (x)={\int }_0^1(F_j(z_t)+t{\sum }_i{\mathrm{\partial }}_j{F}_i(z_t)(x_i-a_i))dt,$ where $z_t=a+t(x-a)$. The integrand and its parameter derivative are continuous because $F$ is $C^1$.

By closedness in [L2], ${\mathrm{\partial }}_j{F}_i={\mathrm{\partial }}_i{F}_j$. Thus the integrand in step 2.1 is $F_j(z_t)+t{\sum }_i{\mathrm{\partial }}_i{F}_j(z_t)(x_i-a_i)=\frac{d}{dt}(t{F}_j(z_t))\mathrm{.}$

Apply [L4] to step 3.1. The endpoint at $t=0$ is $0\cdot F_j(a)=0$, and the endpoint at $t=1$ is $F_j(x)$, so ${\mathrm{\partial }}_j\varphi (x)=F_j(x)$.

Since this holds for every $x$ and $j$, the partial derivatives of $\varphi$ are the $C^1$ functions $F_j$. In particular they are continuous, so [L5] gives $\mathrm{\nabla }\varphi =F$, and their first partials are continuous; hence $\varphi$ is $C^2$.

By the definition in [L2], step 5.1 makes $F$ exact with the displayed potential.