Vector-Valued Functions of a Real Variable

Vector-Valued Functions of a Real Variable

• A vector-valued function $\overrightarrow{r}$ links a subset of real numbers to a set of vectors, expressed as $\overrightarrow{r}(t)=(f_{1}(t), f_{2}(t), \ldots, f_{n}(t))$.
• $f_{i}(t)$ are real-valued component functions of the real variable $t$.

Vector-Valued Function Representation

• A vector-valued function of a real variable $\overrightarrow{r}(t)$ can be written as $\overrightarrow{r}(t)=f_{1}(t) \overrightarrow{\mathbf{i}}+f_{2}(t) \overrightarrow{\mathbf{j}}+f_{3}(t) \overrightarrow{\mathbf{k}}$.
• $f_{1}(t)$, $f_{2}(t)$, and $f_{3}(t)$ are the component functions of $\overrightarrow{r}$.
• $\overrightarrow{\mathbf{i}}=(1,0,0)$, $\overrightarrow{\mathbf{j}}=(0,1,0)$, and $\overrightarrow{\mathbf{k}}=(0,0,1)$ are the standard unit vectors.

Example Vector-Valued Function

• The component functions for the vector-valued function $\overrightarrow{r}(t)=\left(t^{2}, \ln (t+1), \sqrt{4-t}\right)$ are $f_{1}(t)=t^{2}$, $f_{2}(t)=\ln (t+1)$, and $f_{3}(t)=\sqrt{4-t}$.

Domain of Vector-Valued Functions

• The domain of a vector-valued function $\overrightarrow{r}$ is the intersection of the domains of its component functions: $\operatorname{Dom}(\overrightarrow{r})=\operatorname{Dom}\left(f_{1}\right) \cap \operatorname{Dom}\left(f_{2}\right) \cap \ldots \cap \operatorname{Dom}\left(f_{n}\right)$.

Domain Example

• Given $\overrightarrow{r}(t)=\left(t^{2}, \ln (t+1), \sqrt{4-t}\right)$, with $\operatorname{Dom}\left(f_{1}\right)=\mathbb{R}$, $\operatorname{Dom}\left(f_{2}\right)=(-1, \infty)$, and $\operatorname{Dom}\left(f_{3}\right)=(-\infty, 4]$, the domain of $\overrightarrow{r}(t)$ is $(-1,4]$.

Static Equilibrium Conditions

• An object is in static equilibrium when it is at rest and neither accelerating linearly nor rotationally.
• The first condition is that the net force on the object must be zero: $\sum \overrightarrow{F} = 0$.
• The second condition is that the net torque on the object must be zero: $\sum \overrightarrow{\tau} = 0$.

Force Table Theory

• When a body is in equilibrium, the sum of all vector forces is zero: $\sum \overrightarrow{F} = \overrightarrow{F_1} + \overrightarrow{F_2} + \overrightarrow{F_3} +... = 0$.
• In two dimensions: $\sum F_x = F_{1x} + F_{2x} + F_{3x} +... = 0$ and $\sum F_y = F_{1y} + F_{2y} + F_{3y} +... = 0$.
• The x and y components are $F_{ix}$ and $F_{iy}$ are the $x$ and $y$ components of the $i^{th}$ force, respectively.

Force Table Principles

• To achieve static equilibrium, the ring must be centered to ensure that the net force is zero; components can be calculated through trigonometry.
• $F_{ix} = F_i \cos{\theta_i}$ and $F_{iy} = F_i \sin{\theta_i}$ are equations to find the components of an object on a force table.
• $F_i$ is magnitude of the $i^{th}$, $\theta_i$ is the angle between force and the $x$-axis.

Rigid Bar Torque Theory

• For a rigid body in equilibrium, the vector sum of all torques must be zero: $\sum \overrightarrow{\tau} = \overrightarrow{\tau_1} + \overrightarrow{\tau_2} + \overrightarrow{\tau_3} +... = 0$.
• $\overrightarrow{\tau_i}$ represents the $i^{th}$ torque.
• Torque is influenced by the forces application point

Torque Calculation

• Torque is given by $\tau = rF\sin{\theta}$, where $\theta$ is the angle between the force vector and the vector from the pivot point to the point where the force is applied.
• Positive torque is counterclockwise; negative is clockwise.

Force Table Procedure

• Set up the force table with the ring centered around the pivot.
• Adjust the magnitudes and directions of the three forces until the ring is centered.
• The magnitudes and directions of the three forces acting on the ring.
• Calculate the $x$ and $y$ components of each force.
• Verify that the net force in $x$ and $y$ directions is close to zero.

Rigid Bar Procedure

• Hang the bar on the pivot.
• Apply weights to the bar until static equilibrium is achieved.
• Measure the forces and distances from the pivot point to each force application point.
• Torque for each force with $\tau = rF\sin{\theta}$.
• Net torque is close to zero.

Data Analysis and Report

• Net force and net torque should be calculated for every measurement
• Percentage difference from zero should be calculated
• Sources of error should be discussed.

Linear Block Codes: Definition

• An (n, k) linear block code C represents a k-dimensional subspace within $\mathbb{F}_2^n$.

Encoding

• In linear block codes, encoding is a linear transformation $E: \mathbb{F}_2^k \rightarrow \mathbb{F}_2^n$.

Generator Matrix G

• A $k \times n$ matrix $G$.
• Determines the encoding process $E$ in $x \mapsto xG$, $x \in \mathbb{F}_2^k$.
• $G$ is a generator matrix for the code $C$.
• The rows of $G$ form a basis for code $C$.

Parity Check Matrix H

• An $(n-k) \times n$ matrix denoted as $H$.
• $x \in C$ if and only if $Hx^T = 0$
• H is a parity check matrix for code $C$.
• The rows form a basis for $C^{\perp}$, the dual code of $C$.

Syndrome Decoding

• The syndrome of $y \in \mathbb{F}_2^n$ is defined as $Hy^T \in \mathbb{F}_2^{n-k}$.
• Coset leaders represent the minimum weight elements within each 'coset' of $C$ in $\mathbb{F}_2^n$.

Decoding Procedure

• Compute the syndrome using $s = Hy^T$.
• Find coset leader e which has s syndrome ,decode y as y - e.

Example Code

• Code $C = {0000, 1010, 0101, 1111}$.
• Parameters include $n = 4$, $k = 2$.

Parity Check Matrix, H Parity Matrix properties

• $G = \begin{bmatrix} 1 & 0 & 1 & 0 \ 0 & 1 & 0 & 1 \end{bmatrix}$, $H = \begin{bmatrix} 1 & 0 & 1 & 0 \ 0 & 1 & 0 & 1 \end{bmatrix}$
• Note that $GH^T = 0$.

Error Correction and Decoding Example

• Received signal $y = 1101$, $Hy^T = \begin{bmatrix} 1 \ 0 \end{bmatrix}$
• Coset leader for syndrome $\begin{bmatrix} 1 \ 0 \end{bmatrix}$ is $e = 1000$.
• Decoded signal becomes $y - e = 1101 - 1000 = 0101$.

Hamming Codes: Definition

• Hamming codes have $r \geq 2$, length $n = 2^r - 1$, parity check matrix $H$ and columns that are all nonzero vectors in $\mathbb{F}_2^r$.
• Dimension is $k = n - r = 2^r - 1 - r$.

Error Correction

• Hamming codes designed to correct one error.

Hamming Codes Example: r = 3

• $n = 2^3 - 1 = 7$, $k = 7 - 3 = 4$.
• $H = \begin{bmatrix} 0 & 0 & 0 & 1 & 1 & 1 & 1 \ 0 & 1 & 1 & 0 & 0 & 1 & 1 \ 1 & 0 & 1 & 0 & 1 & 0 & 1 \end{bmatrix}$. Each column is a unique non-zero vector of length 3.

Hamming Code Error Correction Example

• Received code $y = 0110101, Hy^T=\begin{bmatrix} 1 \ 1 \ 0 \end{bmatrix}$.
• Syndrome equals 6th column of $H$, so the error is in the 6th position.
• Therefore, $e = 0000010$ and $y - e = 0110101 - 0000010 = 0110111$.

Matrix Definition

• A rectangular array of numbers or symbols arranged in rows and columns.

Example

• $A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix}$
• The order is 2 x 2, it has 2 rows and 2 columns.

General Matrix Form

• $A = \begin{bmatrix} a_{11} & a_{12} &... & a_{1n} \ a_{21} & a_{22} &... & a_{2n} \... &... &... &... \ a_{m1} & a_{m2} &... & a_{mn} \end{bmatrix}$
• Where
  • i = 1, 2, 3,..., m
  • j = 1, 2, 3,..., n
  • $a_{ij}$ = An element of the $i^{th}$ row and $j^{th}$ column

Row Matrix

• A matrix where each element exists on the same single row. Example: $A = \begin{bmatrix} 1 & 5 & 9 \end{bmatrix}_{1 \times 3}$

Column Matrix

• A matrix where each element exists on the same, single column. Example: $A = \begin{bmatrix} 2 \ 5 \ 8 \end{bmatrix}_{3 \times 1}$

Zero Matrix

• A matrix where each element will always be 0. Example: $A = \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix}_{2 \times 2}$

Square Matrix

• A matrix where each element will always be an equal number of columns and rows. Example: $A = \begin{bmatrix} 1 & 2 \ 5 & 9 \end{bmatrix}_{2 \times 2}$

Diagonal Matrix

• A square matrix where one row or column has non 0 values. Example: $A = \begin{bmatrix} 3 & 0 \ 0 & 9 \end{bmatrix}_{2 \times 2}$

Scalar Matrix

• A diagonal matrix where all the diagonal elements are equal. Example: $A = \begin{bmatrix} 3 & 0 \ 0 & 3 \end{bmatrix}_{2 \times 2}$

Identity Matrix

• A square matrix where rows and columns start with 1 and all other values are 0. Example: $I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}_{2 \times 2}$

Triangular Matrix

• A square matrix if elements either above or below the principal diagonal are 0.

Upper Triangular Matrix

• A square matrix in which all the elements below the principal diagonal are zero.

Lower Triangular Matrix

• A square matrix in which all the elements above the principal diagonal are zero.

Supply & Demand: Demand

• Demand relates to what consumers wish to acquire and the inverse relationship exists between the price of a good and the quantity demanded (Law of Demand).
  • If a price increase happens (↑ Px) then a decrease in quantity demanded will follow (↓ Qd).
  • If a price decrease happens (↓ Px) then an increase in quantity demanded will follow (↑ Qd).

Demand Curve

• Demonstrates the relationship between price and quantity
• Creates a negative slope

Determinants of Demand

• Multiple external factors will influence demand.
  • Consumer Income denoted with (Y).
  • Consumer Preferences denoted with (G).
  • Prices of related goods that are interchangeable and/or complementary denoted with (PR).
  • Expectations for the future denoted as (E).
  • Number of buyers denoted with (N).
• Represented as $Q_d = f(P_x, Y, G, P_R, E, N)$.

Demand Fluctuations

• Curve will shift with any change in the determinants listed except for price. - Increase in demand shifts curve right. Decrease in demand shifts curve left

Changes in "Quantity" Demanded.

• In contrast to moving the demand curve itself, actual changes in an overall quantity relates only to an underlying change in listed market price.

Supply & Demand: What suppliers ideally "want" to sell.

• There exists a direct and/or positive relationships between market price and quantity.
  • If a price increases on a listed item (↑ Px) than an increase on the items overall quantity happens (↑ Qo)
  • If a price reduction happens on a listed item (↓ Px) then an increase in the quantity also happens (↓ Qo)

Supply Curve

• Illustrates relationships solely between direct market price and overal supply
• Generally illustrated with a positive slope

Supply Determinants

• Outside of market price
  • Prices of inputs denoted by (PI). Technology denoted by (T). Future Expectations by (E). The number of direct sellers market denoted by (N).
• Denoted as: $Q_o = f(P_x, P_I, T, E, N)$

Overall Supply Fluctuations

• Curves shift independent of price changes based on listed determinants
  • An increase in supply leads to a shift to the right.
  • A decrease leads to the opposite

Quantity Offered

• Differs from shifts because it is solely based on listed market-price changes that affect the supply.

Supply & Demand Balance (Equilibrium)

• Equilibrium occurs where buyers meet sellers and supply equals demand via quantity
• Equilibrium price. (Pe): quantity meets demand equitably
• Equilibrium quantity (Qe): is bought and sold with "balanced price equilibrium."

Price Dynamics

• Supply excess means quantity is more than quantity demanded $Q_o > Q_d$
• Demand excess means quantity less than quantity demanded $Q_d > Q_o$

Elasticity

• Elasticity is how supply/demand changes versus determinant changes.

Price Elasticity of Demand (EPD)

• How changes in quantity affect price elasticity $E_{PD} = \frac{% \Delta Q_d}{% \Delta P}$

Basic Elasticity

1. Demand is elastic when $|E_{PD}| > 1$.
2. Demand is inelastic when $|E_{PD}| < 1$.
3. Demand is unitary when $|E_{PD}| = 1$. 1. Demand is perfectly elastic when $|E_{PD}| = \infty$. 1. Demand is perfectly inelastic when $|E_{PD}| = 0$.

Various Factors Influencing Demand

• How substitutable items are as availability increases/reduces or becomes more limited.
• How consumers change when an item is a luxury versus a need.
• Time that passes over set periods/horizons and quantity changes with that.
• Overall proportion is income spent versus spent on something else.

Demand Income Elasticity

• Responsivity from demand due to income shifts.
• Expressed via $E_{YD} = \frac{% \Delta Q_d}{% \Delta Y}$
  • Higher/Normal amounts: $E_{YD} > 0$.
  • Low/Inferiority based: $E_{YD} < 0$.
  • Luxuries see it $E_{YD} > 1$.

Demand First-Necessity Responsivity

• Expressed via $0 < E_{YD} < 1$.

Demand Cross-Elasticity (DCE)

• Measures quantity versus another price change
• Expressed via $E_{CD} = \frac{% \Delta Q_{dA}}{% \Delta P_B}$

Relationships

• Good Substitutions with $E_{CD} > 0$.
• Good Complements with $E_{CD} < 0$.
• Item Independency with $E_{CD} = 0$.

Price Responsivity (Supply)

• Response via quantity.
• Expressed as $E_{PO} = \frac{% \Delta Q_o}{% \Delta P}$
  • Supply elasticity measured via $E_{PO} > 1$. Supply inelasticity measured via $E_{PO} < 1$. Supply Unitary measured via: $E_{PO} = 1$ Perfect inelasticity: via $E_{PO} = 0$

Factors Influencing Price Responsivity (Supply)

• Items/periods on the Time Horizon. Overall available production capacity. The industry vs individual/group changes that happen.
• Mobility that is offered for factors during item production.

Mobility that is stored via inventories

Supply Elasticity, Storage and Ease.

• Facility in these categories changes price.

Applications and/or Usages

Taxes are mandatory payments made to a government.

Various Type Tax Payments

Taxes are mandatory payments made to a government. Direct: from wealth/$ revenue. Indirect taxes from Item payments .

Tax Burden

• Distribution relates tax burdens across "buyers vs seller", "supply vs demand".
• The obligation to make payments is independent to tax burden.
• Demand vs supplied elasticity determine if tax burden shifts from customer to seller. Buyers burden majority for elasticity if demand. Sellers handle more via less elastic supply.

Subsidies

• Subsidies exist via Government buyer and/or seller payments. Sub payments are the inverse of taxes. Reductions via supply increase vs buyer price.

Price Caps

• Caps are legal for item prices specifically. Unmodified with caps vs market equilibrium. Shortages vs equilibrium with "demand excess".

Price Bottoms

Legal for price bottom end on items

• Equilibrium still relevant with price under listed bottom dollar. Excludes (aka) Excess with overall supply that has price over listed value.

First Degree Equations

An equation is an alphanumeric algebra-based equivalency Example shown: $3x - 5 = 4 + x$

Algebraic Elements

Members are on either side of a " = " sign