Math and stuff

Proposition Let $A$ be a ring, let $S$ and $T$ be two multiplicatively closed subsets of $A$, and let $U$ be the image of $T$ in $S^{-1}A$. Show that the rings $(ST)^{-1}A$ and $U^{-1}(S^{-1}A)$ are isomorphic. Solution $ST$ and $U$ are both multiplicatively closed, so it makes sense to talk...

Proposition Let $f: A \rightarrow B$ be a homomorphism of rings and let $S$ be a multiplicatively closed subset of $A$. Let $T = f(S)$. Show that $S^{-1}B$ and $T^{-1}B$ are isomorphic as $S^{-1}A$-modules. Solution Let $\phi: S^{-1}B \rightarrow T^{-1}B$ be defined such that $\phi(x / s) = x /...

Proposition Give an example that shows that the product of two harmonic functions is not necessarily harmonic. Solution Let $u(x, y) = x$. Then \(u_{xx} = u_{yy} = 0\). Let $v = u \cdot u$. Then \(v_{xx} = 2, v_{yy} = 0\). Thus $v$ is not harmonic.

Proposition Show that all partial derivatives of a harmonic function are harmonic. Solution We are given that \(u_{xx} + u_{yy} = 0\). Thus \((u_{xx} + u_{yy})_x = 0\), so \(u_{xxx} + u_{yyx} = 0\). By changing the order of partial derivatives, \((u_x)_{xx} + (u_x)_{yy} = 0\). Thus $u_x$ is harmonic....

Proposition Suppose $u(x, y)$ and $v(x, y)$ are harmonic in $G$, and $c \in \mathbb{R}$. Prove that $u(x, y) + cv(x, y)$ is also harmonic in $G$. Solution \((u + cv)_{xx} + (u + cv)_{yy} = u_{xx} + cv_{xx} + u_{yy} + cv_{yy} = (u_{xx} + u_{yy}) + c(v_{xx} +...