Ethan's Notes

    First Uniqueness Theorem

    #Physics

    $\displaystyle ~\exists~\mathcal{S}(V)\rightarrow ~\exists~!V:\Delta V=0$

    • Says that if $\displaystyle V$ is specified on a boundary $\displaystyle \mathcal{S}$ that surrounds a volume $\displaystyle \mathcal{V}$, 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

    $\displaystyle ~\exists~\mathcal{S}(V)\rightarrow ~\exists~!V:\Delta V=-\frac{\rho}{{\varepsilon}_{0}}$

    • Says that if $\displaystyle V$ is specified on a boundary $\displaystyle \mathcal{S}$ that surrounds a volume $\displaystyle \mathcal{V}$, 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