Math and stuff

Proposition Find each Mobius transformation $f$: $f$ maps $0 \rightarrow 1, 1 \rightarrow \infty, \infty \rightarrow 0$. $f$ maps $1 \rightarrow 1, -1 \rightarrow i, -i \rightarrow -1$. $f$ maps the $x$-axis to $y = x$, the $y$-axis to $y = -x$, and the unit circle to itself. Solution We...

Proposition Prove that $\overline{\sin(z)} = \sin(\overline{z})$ and $\overline{\cos(z)} = \cos(\overline{z})$” Solution \(\begin{align*} \overline{\exp(z)} &= \overline{\exp(x + iy)} \\ &= \overline{\exp(x)\exp(iy)} \\ &= \overline{\exp(x)(\cos(y) + i\sin(y))} \\ &= \overline{\exp(x)\cos(y) + i\exp(x)\sin(y))} \\ &= \exp(x)\cos(y) - i\exp(x)\sin(y) \\ &= \exp(x)(\cos(y) - i\sin(y)) \\ &= \exp(x)\exp(-iy) \\ &= \exp(x-iy) \\ &= \exp(\overline{z}). \end{align*}\)...

Proposition Show that $M_x$ consists of the tangent vectors at $t$ of curves $c$ in $M$ with $c(t) = x$. Solution We will use this discussion on tangent vectors in this solution Without loss of generality, we assume that $t = 0$. Let $M \subset \mathbb{R}^n$ be a $k$-dimensional manifold...

Proposition Suppose $\mathcal{C}$ is a collection of coordinate systems for $M$ such that For each $x \in M$, there is $f \in \mathcal{C}$ which is a coordinate system around $x$; if $f, g \in \mathcal{C}$, then $\det(f^{-1} \circ g)’ > 0$. Show that there is a unique orientation of $M$...

Proposition For each $f \in A$, let $X_f$ denote the complement of $V(f)$ in $X = \Spec(A)$. The sets $X_f$ are open. Show that they form a basis of open sets for the Zariski topology, and that $X_f \cap X_g = X_{fg}$; $X_f = \emptyset \iff f$ is nilpotent; $X_f...