Implicit Function Theorem Proof

Proof of the Implicit Function Theorem

• $F: \mathbb{R}^{n+k} \rightarrow \mathbb{R}^{k}$ is a $C^{r}$ map, where $r \geq 1$.
• $F(a,b) = 0$ for some $(a,b) \in \mathbb{R}^{n+k}$, where $a \in \mathbb{R}^{n}$ and $b \in \mathbb{R}^{k}$.
• $\frac{\partial F}{\partial y}(a,b)$ is an invertible $k \times k$ matrix.
• There exists a neighborhood $U$ of $a$ in $\mathbb{R}^{n}$ and a $C^{r}$ map $g: U \rightarrow \mathbb{R}^{k}$ such that $g(a) = b$ and $F(x, g(x)) = 0$ for all $x \in U$.
• $g(x)$ is the unique solution of $F(x,y) = 0$ for $y$ near $b$.
• A map $H: \mathbb{R}^{n+k} \rightarrow \mathbb{R}^{n+k}$ is defined by $H(x,y) = (x, F(x,y))$.
• $DH(a,b)$ is invertible ($H(a,b) = (a,0)$).
• The Inverse Function Theorem ensures the existence of neighborhoods and a $C^{r}$ diffeomorphism $H^{-1}$.
• $H^{-1}(x,z) = (\varphi(x,z), \psi(x,z))$, and $F(x, \psi(x,z)) = z$.
• $g: U \rightarrow \mathbb{R}^{k}$ is defined by $g(x) = \psi(x,0)$.
• For uniqueness, $y = g(x)$ if $F(x,y) = 0$ for some $x \in U$ and $y$ near $b$.

Sard's Theorem

• $f: U \rightarrow \mathbb{R}^{k}$ is a $C^{r}$ map, where $U \subset \mathbb{R}^{n}$ is open and $r \geq \max{n-k+1, 1}$.
• Let $C = {x \in U \mid rank(Df(x)) < k}$.
• $f(C)$ has measure zero in $\mathbb{R}^{k}$.

Immersions and Submersions

• $f: X \rightarrow Y$ is a $C^{\infty}$ map of manifolds.
• An immersion means $df_{x}: T_{x}X \rightarrow T_{f(x)}Y$ is injective for all $x \in X$, implying $\dim X \leq \dim Y$.
• A submersion means $df_{x}: T_{x}X \rightarrow T_{f(x)}Y$ is surjective for all $x \in X$, implying $\dim X \geq \dim Y$.
• For an immersion $f$, there exist charts $(U, \varphi)$ about $x$ and $(V, \psi)$ about $f(x)$ such that $\psi \circ f \circ \varphi^{-1}: \varphi(U) \rightarrow \psi(V)$ is the standard inclusion map, i.e. $(x_{1}, \dots, x_{n}) \mapsto (x_{1}, \dots, x_{n}, 0, \dots, 0)$.
• For a submersion $f$, there exist charts $(U, \varphi)$ about $x$ and $(V, \psi)$ about $f(x)$ such that $\psi \circ f \circ \varphi^{-1}: \varphi(U) \rightarrow \psi(V)$ is the standard projection map, i.e. $(x_{1}, \dots, x_{n}) \mapsto (x_{1}, \dots, x_{k})$.

Time Response

• Represents how a dynamic system's state changes over time when subjected to an input.
• Encompasses transient and steady-state responses.

Time Domain

• Involves analyzing mathematical functions, physical signals, or time series (economic or environmental) with respect to time.

Step Function

• Defined as $r(t) = A u(t)$.
• $A$ is the amplitude, and $u(t)$ is the unit step function.

Ramp Function

• Defined as $r(t) = A t \cdot u(t)$.
• $A$ is the slope of the ramp.

Parabolic Function

• Defined as $r(t) = A t^2/2 \cdot u(t)$.
• $A$ is the amplitude of the parabolic.

Impulse Function

• Defined as $r(t) = A\delta(t)$.
• $A$ is the area, and $\delta(t)$ is the unit impulse function.

Transfer Function

• Defined as $G(s) = \frac{K}{\tau s + 1}$.
• $K$ is the steady-state gain, and $\tau$ is the time constant.

Response to a Step Input

• Characterized by $y(t) = K(1 - e^{-t/\tau})$.

Time Constant ($\tau$)

• It's the time it takes for the response to reach 63.2% of its final value.

Settling Time ($t_s$)

• It is the time required for the step response to reach and stay (settle) within a certain percentage of its final value
• $t_{s, 2%} = 4\tau$
• $t_{s, 5%} = 3\tau$

Transfer Function

• Described by $G(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}$.
• $\omega_n$ is natural frequency, and $\zeta$ is damping ratio.

Damping Ratio

• $\zeta > 1$ is overdamped
• $\zeta = 1$ is critically damped
• $0 < \zeta < 1$ is underdamped
• $\zeta = 0$ is undamped

Step Input (Underdamped)

• Step response: $y(t) = 1 - \frac{e^{-\zeta\omega_n t}}{\sqrt{1 - \zeta^2}}sin(\omega_d t + \theta)$
• $\omega_d = \omega_n \sqrt{1 - \zeta^2}$ is the damped frequency and $\theta = tan^{-1}(\frac{\sqrt{1 - \zeta^2}}{\zeta})$

Rise Time ($t_r$)

• It is the time required to go from 10% to 90% of the steady-state value
• $t_r \approx \frac{1.8}{\omega_n}$

Peak Time ($t_p$)

• It is the time required to reach the peak/overshoot
• $t_p = \frac{\pi}{\omega_d}$

Max Overshoot ($M_p$)

• It is the maximum peak value of the response curve measured from unity
• $M_p = e^{-\frac{\pi\zeta}{\sqrt{1 - \zeta^2}}}$

Settling Time ($t_s$)

• Time required for the step response to reach and stay (settle) within a certain percentage of its final value
• $t_{s, 2%} = \frac{4}{\zeta\omega_n}$
• $t_{s, 5%} = \frac{3}{\zeta\omega_n}$

Ácidos

• Substances that release hydronium ions ($H_3O^+$) when ionized in an aqueous solution.
• Ex: $HCl + H_2O \longrightarrow H_3O^+ + Cl^-$

Classificação

• Number of ionizable hydrogens: monoacids (like HCl), diacids (like $H_2SO_4$), triacids (like $H_3PO_4$), tetracids (like $H_4[Fe(CN)_6]$).
• Presence of oxygen: hydracids (no O, like HCl) and oxyacids (have O, like $H_2SO_4$).
• Strength:
  • Hydracids: Strong (HCl, HBr, HI), moderate (HF), or weak (others).
  • Oxyacids: Determined by the difference between the number of oxygen and hydrogen atoms (≥ 2: strong, = 1: moderate, = 0: weak).
    • $HClO_4$ is a strong oxyacid: (4 - 1 = 3)
    • $HClO_3$ is a strong oxyacid: (3 - 1 = 2)
    • $HClO_2$ is a moderate oxyacid: (2 - 1 = 1)
    • $HClO$ is a weak oxyacid: (1 - 1 = 0)
• Volatility: Either fixed ($H_2SO_4, H_3PO_4$) or volatile (others).

Bases (or Álcalis)

• Substances that release hydroxide ions ($OH^-$) when dissociated in an aqueous solution.
• Ex: $NaOH \longrightarrow Na^+ + OH^-$

Classificação

• Number of Hydroxides: monobases (NaOH), dibases ($Ca(OH)_2$), tribases ($Al(OH)_3$).
• Solubility: Soluble (alkali metal bases and $NH_4OH$), slightly soluble (alkaline earth metal bases), insoluble (others).
• Strength: Strong (alkali and alkaline earth metal bases), weak (others).

Óxidos

• Binary compounds where oxygen is the most electronegative element.
• Ex: $Na_2O, CO_2$
• $OF_2$ isn't an oxide because fluorine is more electronegative than oxygen.

Classificação

• Ionic Oxides: Formed of metals + O with non-variable oxidation states.
  • Ex.: $Na_2O, CaO$
• Molecular Oxides: Formed by a non-metal + O, or variable oxidation states metal + O.
  • Ex.: $CO_2, SO_3, Mn_2O_7$
• Amphoteric Oxides: React with acids and bases.
  • Ex.: $ZnO, Al_2O_3$
• Neutral or Inert Oxides: Do not react with acids, bases, or water.
  • Ex.: $CO, NO, N_2O$
• Peroxides: Contain the $O_2^{2-}$ group.
  • Ex.: $H_2O_2, Na_2O_2$
• Superoxides: Contains the $O_2^{-}$ group.
  • Ex.: $KO_2$

Heat Treatment Definition

• Process of heating, holding, and cooling a material (usually metal) to obtain desired properties.

Heat Treatment Purposes

• Improve strength and hardness.
• Improve ductility and toughness.
• Relieve internal stresses.
• Refine grain size.
• Improve machinability.
• Improve corrosion resistance.
• Improve electrical and magnetic properties.

Annealing

• Involves heating a material to a specific temperature, holding it for a set time, and then cooling it slowly.

Purposes of Annealing

• Soften metal
• Relieve internal stresses
• Refine grain structure
• Improve ductility and toughness
• Improve machinability

Types of Annealing

• Full Annealing includes heating to austenite range, soaking, and slow furnace cooling.
• Process Annealing includes heating below critical temperature, holding, and air cooling
• Stress Relief Annealing includes heating to low temperature, holding, and slow cooling.
• Spheroidizing includes prolonged heating at a temperature slightly below the critical temperature.
• Normalizing includes heating to austenite range, soaking, and air cooling to improve ductility and toughness.

Hardening

• Processes are enacted to to increase the hardness of a metal.

Types of Hardening

• Case Hardening makes changes to the surface of metal while leaving the core softer
• It includes: Carburizing and Nitriding and Cyaniding
• Quench Hardening comes from heating to austenite range and, quenching in water, oil, or air.
• Precipitation/Age Hardening comes from uniformly dispersed metals in a matrix to increase strength and hardness

Tempering

• Increases the toughness of a metal, usually after it has been hardened through heating it to a temperature below its critical poin, holding it at that temperature, and cooling it.

Cryogenic Treatment

• Improves mechanical properties by cooling materials to very low temperatures (typically below −150 °C or −238 °F) to increase hardness, wear resistance, reduce stress, and improve stability.

Furnace Heat Treatment

• Uses a furnace to heat materials.
• Provides a controlled environment for heat treatment.

Induction Heat Treatment

• Uses electromagnetic induction to heat materials.
• Allows for selective or localized heating.

Laser Heat Treatment

• Uses a laser beam to heat materials.
• Provides precise and rapid heating.

Salt Bath Heat Treatment

• Uses a molten salt bath to heat materials.
• Provides rapid and uniform heating.

Factors Affecting Heat Treatment

• Temperature: The temperature to which the material is heated.
• Time: The length of time the material is held at a specific temperature.
• Cooling Rate: The rate at which the material is cooled.
• Atmosphere: The composition of the surrounding atmosphere during heat treatment.
• Material Composition: The chemical composition of the material being heat-treated.

Applications of Heat Treatment

• Heat treatment is used in Automotive and Aerospace components production
• It's also used in Cutting tools production, Gears and bearings, Fasteners, Dies and molds

Improved mechanical properties and Increased wear resistance

Disadvantages of Heat Treatment

• Can be energy-intensive
• May cause distortion or cracking
• Requires precise control of process parameters
• May alter the surface finish of the material

Definition of Reaction Rate

• It refers to the speed at which a chemical reaction converts reactants into products. It is expressed as the concentration change of a reactant or product per unit time.

Factors Affecting Reaction Rate

• Higher reactant concentrations lead to increased reaction rates.
• Higher temperatures lead to increased reaction rates.
• Increased surface area leads to increased reaction rates.
• Catalysts increase the reaction rate without being consumed.
• Higher pressure leads to increased reaction rates, especially in gaseous reactions.

Definition of Rate Law

• It's an equation relating reaction rate and reactant concentrations.
• General form can $aA + bB \rightarrow cC + dD$, with $\text{Rate} = k[A]^m[B]^n$
• $k$ is the rate constant and $m$ and $n$ are the reaction orders with respect to $A$ and $B$, respectively.

Reaction Order

• Zero Order: Rate is independent of reactant concentration ($m \text{ or } n = 0$).
• First Order: Rate is directly proportional to reactant concentration ($m \text{ or } n = 1$).
• Second Order: Rate is proportional to the square of reactant concentration ($m \text{ or } n = 2$).

Methods for Determining Reaction Order

• Initial Rates Method involves varying the initial concentrations and measure the initial rates.
• Graphical Method relies on plotting concentration vs. time data to determine the order based on the linearity of the graphs.
• Isolation Method involves concentrations of all reactants except one excess. Then, the order is determined with respect to the reactant present in smaller concentration.

Arrhenius Equation

• The Arrhenius equation is the the temperature dependence of the rate constant $k$:
• $k = Ae^{-\frac{E_a}{RT}}$ can be described:
  • $k$ is the rate constant, $A$ is the pre-exponential factor, $E_a$ is the activation energy, $R$ is the gas constant $(8.314 , \text{J/mol·K})$
  • $T$ is the temperature in Kelvin

Definition of Activation Energy

• Activation energy, $E_a$, is the minimum energy required for a reaction to occur.

Definition of Catalysis

• Catalysis is the process used to increase the rate of a chemical reaction by adding a catalyst, which is not consumed in the reaction.

Types of Catalysis

• Homogeneous Catalysis consists of a catalyst that's in the same phase as reactants.
• Heterogeneous Catalysis consists of a catalyst that's in a different phase from reactants.
• Enzyme Catalysis: Enzymes are biological catalysts that speed up biochemical reactions.

Definition of Reaction Mechanisms

• Rate-Determining Step occurs if there's the slowest step in a reaction mechanism, which determines the overall rate of the reaction

Elementary Reactions

Elementary reactions are single-step reactions that cannot be broken down into simpler steps

Definition of Random Variable

• Function assigning a real number to each outcome in sample space of an experiment.
• Denoted by a capital letter (e.g., $X$); observed value with a small letter (e.g., $x$).

Discrete Random Variable

• Has countable number of values (e.g., defective items).

Continous Random Variable

• Has uncountably infinite number of values (e.g. height, temperature).

Probability Distribution

• List of each value of $X$ and its probability.
• Can be a table, graph, or formula.
• Requirements
• $0 \le P(x) \le 1$ for each $x$
• $\Sigma P(x) = 1$

Cumulative Distribution Function

• $P(X \le x) = \Sigma_{t \le x} P(t)$
• Probability that $X$ has value less than or equal to $x$.
• Sums probabilities up to $x$.

Mean (Expected Value)

• $\mu = E(X) = \Sigma xP(x)$
• Weighted average of possible values.

Variance

• $\sigma^2 = V(X) = \Sigma (x - \mu)^2 P(x) = E(X^2) - \mu^2$
• Measure of spread or dispersion.
  • Where $E(X^2) = \Sigma x^2 P(x)$

Standard Deviation

• $\sigma = \sqrt{V(X)}$
• Square root of variance.

Joint Probability Distributions

• $P(X = x, Y = y)$
• Probability of both $X = x$ and $Y = y$.
• Requirements
  • $0 \le P(x,y) \le 1$ for each $x$ and $y$
  • $\Sigma_x \Sigma_y P(x,y) = 1$

Marginal Probabilility Distributions

• Probability Distribution of single variable alone
• $P_X(x) = \Sigma_y P(x,y)$ is the probability of x alone
• $P_Y(y) = \Sigma_x P(x,y)$ is the probability of y alone

Conditional Probabilility Distributions

• $P(X = x | Y = y) = \frac{P(X = x, Y = y)}{P_Y(y)}$
• $P(Y = y | X = x) = \frac{P(X = x, Y = y)}{P_X(x)}$
• Distribution of x if y is known.

Independent Random Variables

• $P(X = x, Y = y) = P_X(x)P_Y(y)$ for all $x$ and $y$
• $X$ and $Y$ have no effect on each other.

Covariance

• Linear association strength measure
• $Cov(X,Y) = E[(X - \mu_X)(Y - \mu_Y)] = E(XY) - \mu_X \mu_Y$ -- where $\mu_X = E(X)$ and $\mu_Y = E(Y)$ and , $E(XY) = \Sigma_x \Sigma_y xyP(x,y)$

Correlation

• $\rho = Corr(X,Y) = \frac{Cov(X,Y)}{\sigma_X \sigma_Y}$
• Standardized covariance measure.
• $-1 \le \rho \le 1$ is the range
• $Cov(X,Y) = 0$ and $\rho = 0$ if independent

Linear Combinations

• Combination of random variables.
  • $E(aX + bY) = aE(X) + bE(Y)$
  • $V(aX + bY) = a^2V(X) + b^2V(Y) + 2abCov(X,Y)$
  • $V(aX + bY) = a^2V(X) + b^2V(Y)$ for independent random variables

Equations of Stellar Structure:

We must recap the four differential equations governing stellar structure:

• Hydrostatic Equilibrium:

$\frac{dP}{dr} = -\rho g = -\frac{GM_r \rho}{r^2}$ where:

$P(r)$ is the pressure at radius $r$. $ρ(r)$ is the density at radius $r$. $G$ is the gravitational constant. $M_r$ is the mass enclosed within radius $r$.

• Mass Continuity:

$\frac{dM_r}{dr} = 4\pi r^2 \rho$ where:

$M_r$ is the mass enclosed within radius $r$. $ρ(r)$ is the density at radius $r$

• Energy Generation:

$\frac{dL_r}{dr} = 4\pi r^2 \rho \epsilon$ where:

$L_r$ is the luminosity at radius $r$. $ϵ$ is the energy generation rate per unit mass.

• Energy Transport:

• Radiative Transport:

  $\qquad \frac{dT}{dr} = -\frac{3}{4ac} \frac{\kappa \rho}{T^3} \frac{L_r}{4\pi r^2}$
  

• Convective Transport:

  $\qquad \frac{dT}{dr} = \frac{T}{P} \frac{dP}{dr} \frac{\Gamma - 1}{\Gamma}$
  

Where also need:

• An Equation of State (EOS): $P = P(\rho, T, X_i)$

• Opacity: $\kappa = \kappa (\rho, T, X_i)$

• Energy Generation Rate: $\epsilon = \epsilon (\rho, T, X_i)$

Boundary Conditions

It is important to know that to solve these equations, we will need boundary conditions at the center ($r = 0$):

$\qquad M_r = 0, \quad L_r = 0$

At the surface ($r = R_*$):

$\qquad T = T_{eff}, \quad L_r = L_$: However, also take into considerations that we don't know $R_$ initially so we can't directly apply the surface boundary conditions.

Surface Boundary Condition Problem

We approximate the surface boundary conditions using a model atmosphere:

• Photosphere: The thin layer near the surface where photons escape.

• Optical Depth: $\tau \equiv \int \kappa \rho dr$ measures the opacity.

• Approximation: Assume $T(\tau)$ is similar to a gray atmosphere:

$\qquad T^4 (\tau) = T_{eff}^4 \frac{3}{4} (\tau + \frac{2}{3})$

• Surface Condition: At the surface ($\tau \approx 0$):

$\qquad T(\tau = 0) = T_{eff} = (\frac{L_}{4\pi R_^2 \sigma})^{1/4} \qquad T(\tau = 0) = T_{eff} = (\frac{L_}{4\pi R_^2 \sigma})^{1/4} $

Solving the Equations

The Process comes down to:

1. Guess $L_$, $R_$, and $X_i$ (composition).
2. Use the surface boundary conditions to get $T$ and $\rho$ at the surface.
3. Integrate inwards using the four differential equations until you reach the center ($r = 0$).
4. Check if the central boundary conditions ($M_r = 0$, $L_r = 0$) are satisfied.
5. Adjust $L_$ and $R_$ and repeat until all boundary conditions are met.

Example: The Sun

• Observational Constraints are:*

• Mass: $M_\odot = 1.989 \times 10^{30} \text{ kg}$

• Radius: $R_\odot = 6.955 \times 10^8 \text{ m}$

• Luminosity: $L_\odot = 3.846 \times 10^{26} \text{ W}$

• Age: $t_\odot = 4.57 \times 10^9 \text{ years}$

• Surface Composition: Mostly Hydrogen and Helium, with a metallicity of $Z \approx 0.02$

A successful solar model should reproduce these observed parameters.

Central Conditions:

• $\qquad T_c \approx 1.57 \times 10^7 \text{ K} $

• $\qquad \rho_c \approx 1.527 \times 10^5 \text{ kg/m}^3$

Composition: Initial composition of $X = 0.706, Y = 0.274 and Z = 0.020$

Energy Transport: Outer Layers Convection becomes important in the outer layers. Core is where Energy is generated by nuclear fusion and is transported radiatively.

The Vogt-Russell Theorem

The mass and chemical composition of a star uniquely determine its radius, luminosity, and internal structure. This theorem however:

• Neglects the effects of rotation and magnetic fields, which can influence stellar structure
• Must be aware that stars can lose mass over time, changing their structure.
• Only applies to stars in hydrostatic and thermal equilibrium otherwise it does not apply to stars undergoing rapid changes.

Initialization

• Set the distance to infinity for every node.
• Set the distance to zero for the starting node $s$.
• Insert all nodes into the priority queue by the following format $Q$ with the priority for the node $v= a[v]$.

Main Loop

While $Q$ is not empty.

• Take node $u$ of the lowest distance from $Q$
• Repeat through each neighbour $v$ of $u$:
• Alternative distance $alt = d[u] + Length (u, v)$.
• Whenever $alt < d[v]$:
• $d[v] = alt$. Then, update the priority node.

Explanation

• The function finds all shortest paths from starting $s$ to nodes in a graphs.
• Efficient through a priority queue to find the shortest distance node.
• Distance d[v] is the memory for the shortest path found to $s$ to $v$.

Example

Is of the following components: A, B, C, D, E and their edge weights. Starting at node A

Pricing Approaches

Cost-Plus Pricing

• Adds a standard markup to the product's cost, ignoring demand and competition.

Break-Even Analysis and Target Profit Pricing

• Sets prices to break even on the costs or to achieve a target profit.

$$ \text{Break-even point in units} = \frac{\text{Fixed costs}}{\text{Price - Variable costs}} $$

Value-Based Pricing

• Uses buyers' perceptions of value.
• Customer driven.

Competition-Based Pricing

• Sets prices based on competitors' strategies, costs, prices, and market offerings.

New Product Pricing Strategies

Price Skimming

• Sets a high initial price to maximize revenue from segments willing to pay more.
• Requires high product quality, enough buyers, manageable production costs, and difficulty for competitors to undercut prices.

Market Penetration Pricing

• Sets a low initial price to attract many buyers and gain market share.
• Needs price-sensitive market, decreasing production costs with sales volume, and competitive advantage through low prices.

Product Mix Pricing Strategies

Product Line Pricing

• Sets price steps between products in a line based on cost differences and customer evaluations, while keeping competitors' prices in mind.

Optional-Product Pricing

• Includes the pricing of optional or accessory products sold with a main item.

Captive-Product Pricing

• Prices necessary companion products for a main item (e.g., razor blades for a razor).

By-Product Pricing

• Sets prices on by-products to make the main product more competitive.

Product Bundle Pricing

• Offers several products together at a reduced price.

Price Adjustment Strategies

Discount and Allowance Pricing

• Rewards customers for certain responses, like early payments.

Segmented Pricing

• Sells products/services at different prices not based on cost differences.

Psychological Pricing

• Uses prices to suggest product qualities, affecting buyers' psychology.

Promotional Pricing

• Temporarily prices products below list price to boost sales.

Geographical Pricing

• Adjusts prices for different locations.

Dynamic Pricing

• Changes prices continually to meet customer needs and circumstances.

International Pricing

• Adapts prices for international markets.

Scalars

• Characterized by a positive or negative number.
  • Examples: length, area, volume, mass, time, temperature

Vectors

• A quantity that has both magnitude and direction
  • Examples: position, force, moment

Scalar operations

• Multiplication and Division of a Vector by a Scalar
  • The magnitude of the vector changes.
  • The direction of the vector is the same, unless the scalar is negative, in which case the direction is opposite.

Vector Additions:

• Triangle Rule:
• "Tip-to-tail" addition
• The resultant vector R extends from the tail of A to the tip of B.
• Parallelogram Law:
• The tail of A and B are at the same point
• A parallelogram is formed by drawing lines parallel to A and B
• The resultant R extends from the tails of A and B to the intersection of the lines parallel to A and B.

Vector Subtraction:

• R = A - B = A + (-B)
• The negative sign reverses the direction of the vector
• $R = \sqrt{A^2 + B^2 - 2AB\cos(c)}$ - is used for the magnitude of two forces
• $\frac{B}{\sin(a)} = \frac{A}{\sin(b)} = \frac{R}{\sin(c)}$ - is used for the direction of the resultant

Finding Components of a Force: A force can be resolved into two components. The parallelogram law is used to determine the components.

Cartesian Vector Notation

A cartesian direction is right-handed coordinate system.

• Unit vectors i, j, k are dimensionless vectors with a magnitude of 1, pointing in the positive x, y, and z directions, respectively.
• F = $F_x$**i** + $F_y$**j** + $F_z$**k** is the resulting force vector
• Magnitude of F = $\sqrt{F_x^2 + F_y^2 + F_z^2}$ is the description of the resulting magnitude of the force vector
• Direction of F is defined by the coordinate direction angles $\alpha$, $\beta$, and $\gamma$.
• $F_x = F\cos(\alpha)$
• $F_y = F\cos(\beta)$
• $F_z = F\cos(\gamma)$
• $\cos^2(\alpha) + \cos^2(\beta) + \cos^2(\gamma) = 1$- is the resulting formula for Directional Cosines
• The formula u = $\frac{F_x}{F}$i + $\frac{F_y}{F}$j + $\frac{F_z}{F}$k = $\cos(\alpha)$i + $\cos(\beta)$j + $\cos(\gamma)$k is the formula for Unit Vector

Addition/Subtraction of Cartesian Vectors

• R* = $\sum$F = $\sum F_x$i + $\sum F_y$j + $\sum F_z$k

Position Vectors

• A position vector r is defined as a fixed vector which locates a point in space relative to another point
• r goes from point A to point B: r = $(x_B - x_A)$i + $(y_B - y_A)$j + $(z_B - z_A)$k

FDL to a Line

• F* = Fu = $F\frac{\mathbf{r}}{r} = F\frac{(x_B - x_A)\mathbf{i} + (y_B - y_A)\mathbf{j} + (z_B - z_A)\mathbf{k}}{\sqrt{(x_B - x_A)^2 + (y_B - y_A)^2 + (z_B - z_A)^2}}$

Dot Operations

• A$\cdot$B = $AB\cos(\theta)$
• A$\cdot$B = $(A_x$**i** + $A_y$**j** + $A_z$**k**)$\cdot$($B_x$**i** + $B_y$**j** + $B_z$**k**) = $A_xB_x + A_yB_y + A_zB_z$

Applications of Dot Operation

• $\cos(\theta) = \frac{\mathbf{A}\cdot\mathbf{B}}{AB} = \frac{A_xB_x + A_yB_y + A_zB_z}{AB}$- is the formula to solve for the Angle between two vectors

• $A_{a} = A\cos(\theta) = \mathbf{A}\cdot\mathbf{u}$ And $\mathbf{A}_a = (\mathbf{A}\cdot\mathbf{u})\mathbf{u}$- are the formulas to solve for the Component of a vector in a specified direction.