First Uniqueness Theorem

$\exists \mathcal{S}(V)\to \exists !V:\Delta V=0$

• Says that if $V$ is specified on a boundary $\mathcal{S}$ that surrounds a volume $\mathcal{V}$, then there exists exactly one solution to Laplace’s equation
• In physics terms, this means a specified voltage at the boundaries of a volume imply exactly one voltage scalar function for the entire volume

$\exists \mathcal{S}(V)\to \exists !V:\Delta V=-\frac{\rho }{{\epsilon }_0}$

• Says that if $V$ is specified on a boundary $\mathcal{S}$ that surrounds a volume $\mathcal{V}$, then there exists exactly one solution to Poisson’s equation
• In physics terms, this means a specified voltage at the boundaries and specified charge density of a volume imply exactly one voltage scalar function for the entire volume