Math and stuff

Proposition If $\omega$ is a 1-form $fdx$ on $[0, 1]$ with $f(0) = f(1)$, show that there is a unique number $\lambda$ such that $\omega - \lambda dx = dg$ for some function $g$ with $g(0) = g(1)$. Solution Uniqueness Suppose $\lambda$ and $g$ exist. Let $c: [0, 1] \rightarrow...

Proposition In the polynomial ring $\mathbb{Z}[t]$, the ideal $m = (2, t)$ is a maximal and the ideal $q = (4, t)$ is $m$-primary, but is not a power of $m$. Solution $\mathbb{Z}[t] / (2, t) = \mathbb{Z}/(2)$, which is a field. Therefore, $m$ is a maximal ideal. $\mathbb{Z}[t] /...

Proposition Suppose $u(x, y)$ is a function $\mathbb{R}^2 \rightarrow \mathbb{R}$ that depends only on $x$. When is $u$ harmonic? Solution Since $u$ does not depend on $y$, $u_{yy} = 0$. Therefore, $u_{xx}$ must be 0 if $u$ is harmonic. This is only satisfied by $u(x, y) = ax + b$...

Proposition Show that the integer $n$ of Problem 4-24[Spivak] is unique. Solution Let $c = c_{1, n} + \partial c^2$. \[\begin{align*} \int_{c} d\theta &= \int_{c_{1, n}} d\theta + \int_{\partial c^2} d\theta \\ &= \int_{\partial c_{1, n}} \theta + \int_{\partial^2 c^2} \theta & \text{(Stokes' theorem)} \\ &= \int_{\partial c_{1, n}} \theta...

Proposition Let $c$ be a singular $k$-cube and $p:[0, 1]^k \rightarrow [0, 1]^k$ a 1-1 function such that $p([0, 1]^k) = [0, 1]^k$ and $\det(p’(x)) \geq 0$ for $x \in [0, 1]^k$. If $\omega$ is a $k$-form, show that \[\begin{align*} \int_{c} \omega = \int_{c \circ p} \omega. \end{align*}\] Solution Let...