Bio 5.9 easy

Riemann Mapping Theorem

• Given an open, simply connected proper subset $\Omega$ of $\mathbb{C}$, and a point $\omega \in \Omega$, there exists a unique conformal map $F: \Omega \rightarrow \mathbb{D}$ (the open unit disk) such that $F(\omega) = 0$ and $F'(\omega) > 0$.

Proof: Step 1: Establishing Non-Emptiness of the Family $\mathcal{F}$

• Define $\mathcal{F}$ as the family of injective holomorphic functions $f: \Omega \rightarrow \mathbb{D}$.
• Since $\Omega$ is a proper subset of $\mathbb{C}$, there exists $w \in \mathbb{C} \backslash \Omega$.
• Construct $g(z) = z - w$, which is holomorphic and non-vanishing on $\Omega$.
• Due to $\Omega$'s simple connectivity, there exists a holomorphic function $h$ on $\Omega$ such that $h^2(z) = g(z)$.
• $h$ is injective as if $h(z_1) = h(z_2)$ or $h(z_1) = -h(z_2)$, then $z_1 = z_2$.
• $h(\Omega)$ is open; hence, a radius $r>0$ exists such that $B(h(z), r) \subset h(\Omega)$.
• $B(-h(z), r)$ and $h(\Omega)$ are disjoint, proving $r < |2h(z)|$.
• A function $\phi(z) = \frac{r}{2(h(z)+h(z_0))}$ for some $z_0 \in \Omega$ is constructed, which is holomorphic and injective and maps $\Omega$ into $\mathbb{D}$. Thus, $\mathcal{F}$ is non-empty.

Proof: Step 2: Finding a Maximizer

• Define $M = \sup {|f'(\omega)| : f \in \mathcal{F}}$.
• There exists a sequence ${f_n}$ in $\mathcal{F}$ such that $|f_n'(\omega)| \rightarrow M$.
• Montel's Theorem argument: A subsequence ${f_{n_k}}$ converges uniformly on compact subsets of $\Omega$ to a holomorphic function $F$.
• $F: \Omega \rightarrow \overline{\mathbb{D}}$ and $|F'(\omega)| = M > 0$.

Proof: Step 3: $F \in \mathcal{F}$

• $F$ is injective and $F:\Omega \rightarrow \mathbb{D}$
• Assuming $F$ is not injective leads to the existence of $z_1 \neq z_2$ in $\Omega$ such that $F(z_1) = F(z_2) = w$.
• An automorphism $\tilde{f}{n_k}(z) = \frac{f{n_k}(z) - f_{n_k}(z_1)}{1 - \overline{f_{n_k}(z_1)}f_{n_k}(z)}$ of $\mathbb{D}$ is defined with $\tilde{f}{n_k}(z_1) = 0$ and $\tilde{f}{n_k}(z_2) \neq 0$.
• As $f_{n_k} \rightarrow F$ uniformly, $\tilde{f}_{n_k} \rightarrow \tilde{F}$ where $\tilde{F}(z) = \frac{F(z) - F(z_1)}{1 - \overline{F(z_1)}F(z)}$ and $\tilde{F}(z_1) = \tilde{F}(z_2) = 0$.
• Define $g_{n_k}(z) = \frac{\tilde{f}{n_k}(z)}{z - z_1}$, which is holomorphic on $\Omega$ and $g{n_k}(z) \neq 0$ for $z \neq z_1$.
• Also, $G(z) = \frac{\tilde{F}(z)}{z - z_1}$ is holomorphic on $\Omega$ but $G(z_2) = 0$.
• Since $g_{n_k} \rightarrow G$ uniformly, we get a contradiction to $G(z_2) = 0$.
• A proof by contradiction shows that $F: \Omega \rightarrow \mathbb{D}$ by supposing there exists $w \in \mathbb{D} \backslash F(\Omega)$.
• Defining a holomorphic $h$ where $h^2(z) = \frac{F(z) - w}{1 - \bar{w}F(z)}$.
• Constructing $H(z) = \frac{G(z) - G(\omega)}{1 - \overline{G(\omega)}G(z)}$ where $G(z) = h(z)$.
• $H \in \mathcal{F}$ with $H(\omega) = 0$.
• $\left|H'(\omega)\right| > \left|F'(\omega)\right|$, which contradicts the maximality of $F'(\omega)$.

Proof: Step 4: Uniqueness

• Assume $G: \Omega \rightarrow \mathbb{D}$ is another conformal map with $G(\omega) = 0$ and $G'(\omega) > 0$.
• $G \circ F^{-1}: \mathbb{D} \rightarrow \mathbb{D}$ is an automorphism of $\mathbb{D}$ with $G \circ F^{-1}(0) = 0$.
• $G(z) = e^{i\theta}F(z)$ for some $\theta \in \mathbb{R}$.
• Given $G'(\omega) > 0$ and $F'(\omega) > 0$, $e^{i\theta} = 1$.
• $G(z) = F(z)$ for all $z \in \Omega$ proving uniqueness.

Reaction Rate

• For a reaction $A + B \rightarrow C + D$, it is the measure of the rate of disappearance of reactants or the rate of appearance of products.

Rate Expression

• $r = -\frac{1}{a}\frac{d[A]}{dt} = -\frac{1}{b}\frac{d[B]}{dt} = \frac{1}{c}\frac{d[C]}{dt} = \frac{1}{d}\frac{d[D]}{dt}$, where a, b, c, d are stoichiometric coefficients.

Rate Law

• $r = k[A]^x[B]^y$, where k is the rate constant, x and y are orders with respect to reactants A and B respectively, and x+y is the overall reaction order.

Arrhenius Equation

• $k = Ae^{-\frac{E_a}{RT}}$, where k is the rate constant, A is the frequency factor, $E_a$ is the activation energy, R is the gas constant, and T is the temperature.

Integrated Rate Laws

• 0 order: Rate Law $r = k$, Integrated Rate Law $[A]_t = -kt + [A]_0$, Linear Plot $[A]_t$ vs t, Slope $-k$, Intercept $[A]0$, Half-life $t{1/2} = \frac{[A]_0}{2k}$.
• 1st order: Rate Law $r = k[A]$, Integrated Rate Law $ln[A]_t = -kt + ln[A]_0$, Linear Plot $ln[A]_t$ vs t, Slope $-k$, Intercept $ln[A]0$, Half-life $t{1/2} = \frac{0.693}{k}$.
• 2nd order: Rate Law $r = k[A]^2$, Integrated Rate Law $\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}$, Linear Plot $\frac{1}{[A]_t}$ vs t, Slope $k$, Intercept $\frac{1}{[A]0}$, Half-life $t{1/2} = \frac{1}{k[A]_0}$.

Planck's Constant

• The quantum of action in quantum mechanics, symbol $h$.
• Reduced Planck constant (Dirac's constant): $\hbar = \frac{h}{2\pi} = 1.054571817 \times 10^{-34} \text{ joule seconds (J s)}$.

Physical Dimensions

• Planck's constant has dimensions of energy × time, momentum × distance, or angular momentum.
• SI units: joule seconds (J⋅s) or (N⋅m⋅s) or (kg⋅m²/s).
• Reduced Planck constant has the same physical dimension as angular momentum.
• SI units: (J⋅s).

Radiation Properties

• α: Absorptivity: Fraction of incident radiation absorbed by a body.
• ρ: Reflectivity: Fraction of incident radiation reflected by a body.
• τ: Transmissivity: Fraction of incident radiation transmitted through a body.

Surface Characteristics

• Opaque Surface: $\alpha + \rho = 1$.
• Transparent Surface: $\alpha + \rho + \tau = 1$.
• Black Body: Absorbs all incident radiation, emits maximum radiation, diffuse emitter and absorber.
• Gray Body: Emissive power independent of wavelength, constant emissivity.
• Real Surface: Emissivity depends on temperature and wavelength.
• Specular Surface: Angle of incidence equals angle of reflection.
• Diffuse Surface: Incident radiation is scattered equally in all directions.

Emissive Power

• Black Body ($E_b$): $E_b = \sigma T^4$, where $\sigma = 5.67 \times 10^{-8} W/m^2K^4$ (Stefan Boltzmann Constant).
• Real Body (E): $E = \epsilon \sigma T^4$, where $\epsilon$ is the emissivity, $0 \le \epsilon \le 1$.
• Radiosity (J): Total radiation leaving a surface, $J = \rho G + \epsilon E_b$, where G is irradiation.
• Emissive Power: $E_b = \int_0^\infty E_{b\lambda} d\lambda$ and $E = \int_0^\infty E_\lambda d\lambda$.

Wien's Displacement Law

• Relates the wavelength of maximum emission to temperature: $\lambda_{max} T = 2.9 \times 10^{-3} m.K$.

Intensity of Radiation (I)

• Radiation emitted per unit time, per unit area, per unit solid angle: $I = \frac{E}{\pi}$.

Solid Angle (ω)

• $\omega = \frac{A}{r^2}$, where A is the area intercepted by a cone on a sphere of radius r.

Shape Factor

• Fraction of radiation leaving one surface that strikes another directly.
• $F_{i-j} = \frac{\text{Radiation reaching surface j directly from surface i}}{\text{Total radiation leaving surface i}}$.

Properties of Shape Factor

• Reciprocity Theorem: $A_i F_{i-j} = A_j F_{j-i}$.
• Summation Rule: $\sum_{j=1}^n F_{i-j} = 1$.
• $F_{i-i} = 0$ for flat or convex surfaces.

Radiation Shield

• Used to reduce heat transfer between two surfaces.
• $Q_{\text{without shield}} = \frac{\sigma (T_1^4 - T_2^4)}{\frac{1}{\epsilon_1} + \frac{1}{\epsilon_2} - 1}$.
• $Q_{\text{with shield}} = \frac{\sigma (T_1^4 - T_2^4)}{\left(\frac{1}{\epsilon_1} + \frac{1}{\epsilon_3} - 1\right) + \left(\frac{1}{\epsilon_3} + \frac{1}{\epsilon_2} - 1\right)}$.
• If $\epsilon_1 = \epsilon_2 = \epsilon_3 = \epsilon$, then $Q_{\text{with shield}} = \frac{1}{2} Q_{\text{without shield}}$.