Math and stuff

Proposition If $v \in \mathbb{R}^2$, what is $v \times$? If $v_1, \cdots, v_{n - 1} \in \mathbb{R}^n$ are linearly independent, show that $[v_1, \cdots, v_{n - 1}, v_1 \times \cdots \times v_{n - 1}]$ is the usual orientation of $\mathbb{R}^n$. Solution 1 Let $v = (a, b) \in \mathbb{R}^2$. We...

Proposition For $R > 0$ and $n$ an integer, define the singular 1-cube $c_{R, n}: [0, 1] \rightarrow \mathbb{R}^2 \setminus 0$ by $c_{R, n}(t) = (R\cos 2\pi nt, R\sin 2\pi nt)$. Show that there is a singular 2-cube $c:[0, 1]^2 \rightarrow \mathbb{R}^2 \setminus 0$ such that $c_{R_1, n} - c_{R_2,...

Proposition Let $A$ be a local ring, $M$ and $N$ finitely generated $A$-modules. Prove that if $M \otimes_A N = 0$, then $M = 0$ or $N = 0$. Solution Suppose $M \ne 0$ and $N \ne 0$. We will show that $M \otimes_A N \ne 0$. Let $m$ be...

Proposition Compute $\int_{C[0, 2]}\frac{\exp(z)}{(z - w)^2}$ where $w$ is any fixed complex number $\abs{w} \ne 2$. Solution If $\abs{w} < 2$, then by Cauchy’s Integral formula, the integral equals $2\pi i \exp(w)$. If $\abs{w} > 2$, then by Corollary 4.20[A first course in complex analysis, the integral equals 0.

Proposition Compute the following integrals, where $\gamma$ is the boundary of the square with vertices at $\pm 4 \pm 4i$, positively oriented: $\int_{\gamma}\frac{\exp(z^2)}{z^3} dz$ $\int_{\gamma}\frac{\exp(3z)}{(z - \pi i)^2} dz$ $\int_{\gamma}\frac{\sin(2z)}{(z - \pi)^2} dz$ $\int_{\gamma}\frac{\exp(z)\cos(z)}{(z - \pi)^3} dz$. Solution 1 Let $f(z) = \exp(z^2)$. By Theorem 5.1 [A first course in...