Math and stuff

Proposition Let $c:[0,1] \rightarrow \mathbb{R}^n$ be a curve such that $\abs{c(t)} = 1$ for all $t$. Show that $c(t)_{c(t)}$ and the tangent vector to $c$ at $t$ are perpendicular. Solution \[\begin{align*} \ev{(c^1(t), \cdots, c^n(t))_{c(t)}, ((c^1)'(t), \cdots, (c^n)'(t))_{c(t)}} &= \ev{(c^1(t), \cdots, c^n(t)), ((c^1)'(t), \cdots, (c^n)'(t))} \\ &= c^1(t)(c^1)'(t) + \cdots +...

Proposition In a ring $A$, let $\Sigma$ be the set of all ideals in which every element is a zero divisor. Show that the set $\Sigma$ has maximal elements and that every maximal element of $\Sigma$ is a prime ideal. Hence the set of zero-divisors in $A$ is a union...

Proposition Draw pictures of $\Spec(\mathbb{Z}), \Spec(\mathbb{R}), \Spec(\mathbb{C}[x]), \Spec(\mathbb{R}[x]), \Spec(\mathbb{Z}[x])$. Solution I am not sure how to draw pictures of these spaces, so I will just describe them in words. $\Spec(\mathbb{Z})$ $\Spec(\mathbb{Z}) = \{ (0) \} \cup \{ (p) \mid p \text{ prime} \}$. Let $E \subset \mathbb{Z}$. If $E$ is...

Proposition Let $c$ be a differentiable curve in $\mathbb{R}^n$, that is, a differentiable function $c:[0, 1] \rightarrow \mathbb{R}^n$. Define the tangent vector $v$ of $c$ at $t$ as \(c_{\ast}((e_1)_t) = ((c^1)'(t), \cdots, (c^n)'(t))_{c(t)}\). If $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$, show that the tangent vector to $f \circ c$ at $t$ is...

Proposition Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and define $c: \mathbb{R} \rightarrow \mathbb{R}^2$ by $c(t) = (t, f(t))$. Show that the end point of the tangent vector of $c$ at $t$ lies on the tangent line to the graph of $f$ at $(t, f(t))$. Solution Let $t_0 \in \mathbb{R}$ be given....