Math and stuff

Proposition compute $\int_{\gamma} \frac{dz}{z}$ where $\gamma$ is the unit circle, oriented counterclockwise. More generally, show that for any $w \in \mathbb{C}$ and $r > 0$, \[\begin{align*} \int_{C[w, r]} \frac{dz}{z - w} = 2\pi i. \end{align*}\] Solution It suffices to prove the second statement. Let $w \in \mathbb{C}$ and $r >...

Proposition If $\omega$ is the volume element of $V$ determined by $T$ and $\mu$, and $f: \mathbb{R}^n \rightarrow V$ is an isomorphism such that $f^{\ast}T = \ev{,}$ and such that $[f(e_1), \cdots, f(e_n)] = \mu$, show that $f^{\ast}\omega = \det$. Solution As mentioned on P.83 [Spivak], $\det$ is the unique...

Proposition If $c:[0, 1] \rightarrow (\mathbb{R}^n)^n$ is continuous and each $(c^1(t), \cdots, c^n(t))$ is a basis for $\mathbb{R}^n$, show that $[c^1(0), \cdots, c^n(0)] = [c^1(1), \cdots, c^n(1)]$. Solution Since $\{ e_1, \cdots, e_n \}$ is a basis of $\mathbb{R}^n$, there must exist $a_{11}, \cdots, a_{1n} \in \mathbb{R}$ such that $c_1(0)...

Proposition $a \subset (a:b)$ $(a:b)b \subset a$ $((a:b):c) = (a:bc) = ((a:c):b)$ $(\cap_i a_i:b) = \cap_i (a_i:b)$ $(a:\sum_i b_i) = \cap_i (a:b_i)$. Solution Lemma Let $a, b, c$ be ideas. If $\forall x \in a, xb \subset c$, then $ab \subset c$. (Proof) Let $\sum a_ib_i \in ab$ be given....

Proposition Compute the lengths of the paths from this post: The circle $C[1 + i, 1]$, oriented counter-clockwise. The line segment from $-1 - i$ from $2i$. The top half of the circle $C[0, 34]$, oriented clockwise. The rectangle with vertices $\pm 1 \pm 2i$, oriented counter-clockwise. Solution 1 $\gamma(t)...