Math and stuff

Proposition If $f: \mathbb{R}^n \rightarrow \mathbb{R}$, define a vector field $\grad f$ by \[\begin{align*} (\grad f)(p) = D_1f(p) \cdot (e_1)_p + \cdots + D_nf(p) \cdot (e_n)_p. \end{align*}\] For obvious reasons we also write $\grad f = \nabla f$. If $\nabla f(p) = w_p$, prove that $D_vf(p) = \ev{v, w}$ and...

Proposition If $F$ is a vector field on $\mathbb{R}^3$, define the forms \[\begin{align*} \omega^1_F &= F^1dx + F^2dy + F^3dx \\ \omega^2_F &= F^1dy \wedge dx + F^2dz \wedge dx + F^3 dx \wedge dy \end{align*}\] Prove that \[\begin{align*} df &= \omega^1_{\grad f}, \\ d(\omega^1_p) &= \omega^2_{\curl F}, \\ d(\omega^2_p)...

Proposition If $a_1, a_2$ are ideals of $A$ and if $b_1, b_2$ are ideals of $B$, then $(a_1 + a_2)^e = a_1^e + a_2^e, (b_1 + b_2)^c \supset b_1^c + b_2^c$. $(a_1 \cap a_2)^2 \subset a_1^e \cap a_2^e, (b_1 \cap b_2)^c = b_1^c \cap b_2^c$. $(a_1a_2)^e = a_1^ea_2^e, (b_1b_2)^c \supset...

Proposition Show that $\alpha(x^2 + y^2) + \beta x + \gamma y + \delta = 0$ is the equation for a circle or line if and only if $\beta^2 + \gamma^2 > 4\alpha\delta$. Conclude that $x + iy$ is a solution to the equation above if and only if $u...

Proposition Show that $f(z) = \frac{z - 1}{iz + i}$ maps any circle passing through -1 to a line. Solution A circle centered at $a \in \mathbb{C}$ passing through -1 can be expressed as the set of $z \in \mathbb{C}$ satisfying $\abs{z - a} = \abs{-1 - a} = \abs{1...