Summary: Consider a function \(u(x)\) in the standard localized Sobolev space \(W^{1,p}_{\text{loc}}(\mathbb{R}^n)\) where \(n\geq 2\), \(1\leq p<n\). Suppose that the gradient of \(u(x)\) is globally \(L^p\) integrable; i.e., \(\int_{\mathbb{R}^n}|\nabla u|^p dx\) is finite. We prove a Poincaré inequality for \(u(x)\) over the entire space \(\mathbb{R}^n\). It follows that the function subtracting a certain constant is in the space \(W_0^{1,p} (\mathbb{R}^n)\), which is the completion of \(C_0^\infty (\mathbb{R}^n)\) functions under the norm \(\|\varphi\|=(\int_{\mathbb{R}^n} |\nabla\varphi|^p dx)^{1/p}\) where \(\varphi\in C_0^\infty(\mathbb{R}^n)\). We then prove a similar inequality for functions whose higher order derivatives are \(L^p\) integrable on \(\mathbb{R}^n\). Next we study functions whose gradients are \(L^p\) integrable on an exterior domain of \(\mathbb{R}^n\) and apply the results to another proof of an existence theorem for irrotational and incompressible flows around a body in space.