Developing Life Span 2

Complex Numbers

• A complex number is of the form $z = a + bi$, where $a, b \in \mathbb{R}$, and $i$ is the imaginary unit such that $i^2 = -1$.
• $a$ is $Re(z)$, the real part of $z$.
• $b$ is $Im(z)$, the imaginary part of $z$.
• Two complex numbers, $z_1 = a_1 + b_1i$ and $z_2 = a_2 + b_2i$, are equal if and only if $a_1 = a_2$ and $b_1 = b_2$.
• The set of all complex numbers is denoted by $\mathbb{C}$, where $\mathbb{C} = {a + bi | a, b \in \mathbb{R}, i^2 = -1}$.

Representation of Complex Numbers

• Algebraic Representation: $z = a + bi$.
• Geometric Representation (Gaussian plane): A coordinate system with $Re(z)$ on the x-axis and $Im(z)$ on the y-axis, where each complex number $z = a + bi$ corresponds to a point $(a, b)$ in the plane.
• The magnitude of a complex number $z = a + bi$ is the distance of the point $(a, b)$ from the origin: $|z| = \sqrt{a^2 + b^2}$.
• Trigonometric Representation: $z = r(cos(\phi) + i sin(\phi))$, where $r = |z|$ and $\phi$ is the angle between the positive real axis and the vector connecting $0$ with $z$.

Linear Algebra

• A linear equation with $n$ unknowns $x_1, x_2,..., x_n$ is of the form $a_1x_1 + a_2x_2 +... + a_nx_n = b$, where $a_1, a_2,..., a_n$ and $b$ are real or complex constants.
• A system of m linear equations with n unknowns is a set of m linear equations:

System Of Equations

$\begin{cases} a_{11}x_1 + a_{12}x_2 +... + a_{1n}x_n = b_1 \ a_{21}x_1 + a_{22}x_2 +... + a_{2n}x_n = b_2 \... \ a_{m1}x_1 + a_{m2}x_2 +... + a_{mn}x_n = b_m \end{cases}$

• where the $a_{ij}$ and the $b_i$ are real or complex constants.
• The solution n-tuple $(x_1, x_2,..., x_n)$ satisfies all equations in the system.
• Solving a system means finding all its solutions.
• A system is compatible if it has at least one solution and incompatible if it has none.
• A system is determined if it has exactly one solution and indetermined if it has an infinite number of solutions.
• In the plane (n=2), a linear equation represents a line, and a system of 2 linear equations represents the intersection of 2 lines.
  • If the lines intersect, the system is determined.
  • If the lines are parallel and distinct, the system is incompatible.
  • If the lines are coincident, the system is indetermined.
• In space (n=3), a linear equation represents a plane, and a system of 3 linear equations represents the intersection of 3 planes.

Methods for Solving

• Substitution: Express one unknown in terms of others and substitute.
• Linear combination (Gaussian elimination): Combine equations to eliminate unknowns.
• Matrix method: Write the system in matrix form $AX = B$ and use elementary row operations on the augmented matrix $[A|B]$.
• Cramer's rule: Applicable if the system has the same number of equations as unknowns and the determinant of the coefficient matrix is non-zero.

Projectile Motion

• A projectile is an object on which the only force acting is gravity. Neglecting air resistance, motion is constant acceleration, with $a_x = 0$ and $a_y = -g = -9.8 m/s^2$.
• If a projectile is launched with initial velocity $v_0$ at angle $\theta_0$ relative to the horizontal, the initial velocity components are $v_{0x} = v_0 \cos\theta_0$ and $v_{0y} = v_0 \sin\theta_0$.

Horizontal Motion

• The horizontal component of velocity, $v_x$, is constant: $v_x = v_0 \cos\theta_0$.
• The horizontal position at time t is: $x = v_{0x} t = (v_0 \cos\theta_0) t$.

Vertical Motion

• The vertical component of velocity changes linearly with time: $v_y = v_{0y} - gt = v_0 \sin\theta_0 - gt$.
• The vertical position at time t is: $y = v_{0y} t - \frac{1}{2} gt^2 = (v_0 \sin\theta_0) t - \frac{1}{2} gt^2$.

Range

• The range R is the horizontal distance traveled before landing (y = 0).
• The time T when the projectile lands is $T = \frac{2 v_0 \sin\theta_0}{g}$.
• Substituting T into the equation for x, the range is $R = (v_0 \cos\theta_0) T = \frac{2 v_0^2 \sin\theta_0 \cos\theta_0}{g}$.
• $R = \frac{v_0^2 \sin(2\theta_0)}{g}$ (using $2 \sin\theta \cos\theta = \sin(2\theta)$).
• The range R is maximum when $\sin(2\theta_0) = 1$, so $2\theta_0 = 90^\circ$ and $\theta_0 = 45^\circ$.
• The maximum range is $R_{max} = \frac{v_0^2}{g}$.

Chemical Kinetics

• For the reaction $A + B \rightarrow C + D$: $rate = -\frac{d[A]}{dt} = -\frac{d[B]}{dt} = \frac{d[C]}{dt} = \frac{d[D]}{dt}$.
• For the reaction $aA + bB \rightarrow cC + dD$: $rate = -\frac{1}{a}\frac{d[A]}{dt} = -\frac{1}{b}\frac{d[B]}{dt} = \frac{1}{c}\frac{d[C]}{dt} = \frac{1}{d}\frac{d[D]}{dt}$.

Rate Law

• For the reaction $aA + bB \rightarrow cC + dD$: $rate = k[A]^x[B]^y$.
• k is the rate constant.
• x is the order with respect to A.
• y is the order with respect to B.
• x + y is the overall order of the reaction.

Collision Theory

For a reaction to occur:

• Molecules must collide.
• Molecules must have sufficient energy to break bonds.
• Molecules must have the correct orientation.

Arrhenius Equation

$k = Ae^{-E_a/RT}$

• k is the rate constant
• A is the frequency factor
• $E_a$ is the activation energy
• R is the gas constant (8.314 J/mol·K)
• T is the temperature (in Kelvin)
• $ln(k) = \frac{-E_a}{R}(\frac{1}{T}) + ln(A)$.
• $ln(\frac{k_2}{k_1}) = \frac{E_a}{R}(\frac{1}{T_1} - \frac{1}{T_2})$.

Catalyst

• A catalyst speeds up a reaction without being consumed.
• A Homogeneous catalyst is in the same phase as the reactants.
• A Heterogeneous catalyst is in a different phase as the reactants.

The Poisson Process

• A Poisson process with rate $\lambda > 0$ is a continuous-time stochastic process ${N(t), t \geq 0}$.
• $N(0) = 0$.
• Independent increments: For any $0 \leq t_1 < t_2 < \dots < t_n$, the random variables $N(t_2) - N(t_1), N(t_3) - N(t_2), \dots, N(t_n) - N(t_{n-1})$ are independent.
• For any $s, t \geq 0$, the increment $N(t+s) - N(s)$ follows a Poisson distribution with mean $\lambda t$: $P(N(t+s) - N(s) = n) = e^{-\lambda t} \frac{(\lambda t)^n}{n!}, \quad n = 0, 1, 2, \dots$
• $N(t)$ represents the number of events that occur up to time $t$.
• The independent increments property means that the number of events in disjoint time intervals are independent.
• The third property implies that the number of events in an interval of length t follows a Poisson distribution with mean $\lambda t$, regardless of the starting time $s$.

Inter-arrival Time

• Let $T_1$ be the time of the first event, $T_2$ the time of the second event, and so on.
• Define the interarrival times $X_i$ as: $X_1 = T_1, \quad X_i = T_i - T_{i-1}, \quad i = 2, 3, \dots$
• The interarrival times $X_i$ are independent and identically distributed (i.i.d.) exponential random variables with rate $\lambda$.
• The probability density function (pdf) of each $X_i$ is: $f_{X_i}(x) = \lambda e^{-\lambda x}, \quad x \geq 0$