Math and stuff

Proposition Find the derivative of the function $T(z) = \frac{az + b}{cz + d}$ where $a, b, c, d \in \mathbb{C}$ with $ad - bc \ne 0$. When is $T’(z) = 0$? Solution \[\begin{align*} T'(z) &= \frac{(cz + d)a - (az + b)c}{(cz + d)^2} \\ &= \frac{ad - bc}{(cz...

Proposition Deduce the following properties of the cross product in $\mathbb{R}^3$. \[\begin{align*} \begin{matrix} e_1 \times e_1 = 0 & e_2 \times e_1 = -e_3 & e_3 \times e_1 = e_2 \\ e_1 \times e_2 = e_3 & e_2 \times e_2 = 0 & e_3 \times e_2 = -e_1 \\ e_1...

Proposition Let $A$ be a Noetherian ring. Prove that the following are equivalent: $A$ is Artinian; $\Spec(A)$ is discrete and finite; $\Spec(A)$ is discrete. Solution $1 \rightarrow 2$ By Proposition 8.1[Atiyah], $\Spec(A)$ only contains maximal ideals. By Proposition 8.3[Atiyah], $\Spec(A)$ only contains finitely many elements. As shown previously, a singleton...

Proposition $r(a) \supset a$. $r(r(a)) = r(a)$. $r(ab) = r(a \cap b) = r(a) \cap r(b)$. $r(a) = (1) \iff a = (1)$. $r(a + b) = r(r(a) + r(b))$. if $p$ is prime, $r(p^n) = p$ for all $n > 0$. Solution 1 If $x \in a$, then $x^1...

Proposition Prove that $f(z) = \overline{z}^2$ does not have an antiderivative in any nonempty region. Solution Suppose it has an antiderivative $F(z)$. Then $F(z)$ is analytic, so $F’(z) = f(z)$ is analytic. However, $f(z) = (x^2 - y^2) + i(-2xy)$ does not satisfy the Cauchy-Riemann equations because $u_x = 2x...