Vector Functions of a Real Variable

Funciones vectoriales de variable real

• A function assigns a vector to each real number in a set.
• Vectorial function: $\overrightarrow{r}: I \subseteq \mathbb{R} \longrightarrow \mathbb{R}^n$
• Each real number $t \in I$ is assigned a vector $\overrightarrow{r}(t) = (f_1(t), f_2(t),..., f_n(t))$.
• $f_i: I \subseteq \mathbb{R} \longrightarrow \mathbb{R}$ are real-valued functions.
• Example: $\overrightarrow{r}(t) = (t^2 + 1, \cos(t), e^t)$ is a function with $n = 3$.

Límite

• The limit of a vector function $\overrightarrow{r}(t)$ as $t$ approaches $t_0$ is given by:
• $\lim_{t \to t_0} \overrightarrow{r}(t) = (\lim_{t \to t_0} f_1(t), \lim_{t \to t_0} f_2(t),..., \lim_{t \to t_0} f_n(t))$ if all component limits exist.

Continuidad

• A vector function $\overrightarrow{r}(t)$ is continuous at $t_0$ under three conditions:
  • $\overrightarrow{r}(t_0)$ exists.
  • $\lim_{t \to t_0} \overrightarrow{r}(t)$ exists.
  • $\lim_{t \to t_0} \overrightarrow{r}(t) = \overrightarrow{r}(t_0)$
• $\overrightarrow{r}(t)$ is continuous at $t_0$ if each component function $f_i(t)$ is continuous at $t_0$.

Derivada

• The derivative of a vector function $\overrightarrow{r}(t)$ is:
  • $\overrightarrow{r}'(t) = \lim_{h \to 0} \frac{\overrightarrow{r}(t+h) - \overrightarrow{r}(t)}{h} = (f_1'(t), f_2'(t),..., f_n'(t))$
  • This is only if all component function derivatives exist.
• Notation: $\overrightarrow{r}'(t) = \frac{d\overrightarrow{r}}{dt}$
• Geometrically, if $\overrightarrow{r}(t)$ is a particle's position at time $t$, $\overrightarrow{r}'(t)$ is its velocity vector.
• $\overrightarrow{r}'(t)$ is tangent to the path of $\overrightarrow{r}(t)$ at point $\overrightarrow{r}(t)$.

Reglas de derivación

• Given differentiable vector functions $\overrightarrow{r}(t)$ and $\overrightarrow{s}(t)$, and a differentiable scalar function $f(t)$:
  • $(\overrightarrow{r}(t) + \overrightarrow{s}(t))' = \overrightarrow{r}'(t) + \overrightarrow{s}'(t)$
  • $(c\overrightarrow{r}(t))' = c\overrightarrow{r}'(t)$, where $c$ is a constant.
  • $(f(t)\overrightarrow{r}(t))' = f'(t)\overrightarrow{r}(t) + f(t)\overrightarrow{r}'(t)$
  • $(\overrightarrow{r}(t) \cdot \overrightarrow{s}(t))' = \overrightarrow{r}'(t) \cdot \overrightarrow{s}(t) + \overrightarrow{r}(t) \cdot \overrightarrow{s}'(t)$
  • $(\overrightarrow{r}(t) \times \overrightarrow{s}(t))' = \overrightarrow{r}'(t) \times \overrightarrow{s}(t) + \overrightarrow{r}(t) \times \overrightarrow{s}'(t)$
  • $(\overrightarrow{r}(f(t)))' = \overrightarrow{r}'(f(t)) \cdot f'(t)$ (Chain Rule)

Transductores y sensores

• Transducers convert one form of energy into another.
• Sensors are transducers that measure physical quantities and convert them into electrical signals.

Clasificación de los transductores

• Based on Energy Conversion:
  • Resistive: Converts resistance changes to electrical signals. –Capacitive: Converts capacitance changes to electrical signals.
  • Inductive: Converts inductance changes to electrical signals.
  • Piezoelectric: Converts pressure to electrical signals.
  • Thermoelectric: Converts temperature changes to electrical signals.
• Based on Output Signal:
  • Analog: Continuous output proportional to measured quantity.
  • Digital: Discrete output representing measured quantity.
• Based on Measured Quantity:
• Temperature, Pressure, Flow, Level, Position, Velocity, Acceleration.

Características de los transductores

• Sensitivity is the ratio of output signal change to the physical quantity change.
  • $S = \frac{\Delta V_o}{\Delta p}$
• Resolution is the smallest detectable change.
• Precision is the difference between measured and actual values.
• Accuracy is the ability to produce a measurement close to the true value.
• Linearity measures output signal proportionality to measured quantity.
• Range: the interval of measurable values
• Response Time is the time for the signal to reach a stable value.
• Hysteresis is the difference between increasing and decreasing signal.

Aplicaciones de los transductores

• Applications include industrial automation, medicine, automotive systems, and consumer electronics.
• Automated systems employ transducers to control and monitor industrial processes.
• Measure vital signs, such as temperature, blood pressure and heart rate
• Automotive industry uses them to control and monitor engine performance, tire pressure, and vehicle speed.

Transductores resistivos

• Resistive transducers measure physical quantities based on resistance change.

Potenciómetros

• Potentiometers measure linear or angular position.
• Fixed resistor with sliding contact.
• Output voltage is proportional to contact position.
• Equation: $V_o = V_i \cdot \frac{R_2}{R_1 + R_2}$
• Limited by number of wire turns or size of resistive material grains.
• Used for volume control, position sensing, and voltage adjustment.

Galgas extensométricas

• Strain gauges measure object deformation.
• Thin resistive material sheet attached to the object.
  • Resistance change occurs during deformation
• Equation: $R = \rho \frac{L}{A}$, where R is resistance, $\rho$ is resistivity, L is length, and A is cross-sectional area.
• Gauge Factor (GF): $GF = \frac{\Delta R / R}{\epsilon}$, where $\epsilon$ is strain.
• Strain gauges mounted on a Wheatstone bridge
• Temperature compensation needed for accurate strain measurements.
• Used for deformation measurement, force measurement in scales, and pressure measurement in pressure sensors.

Termistores

• Thermistors are temperature-sensitive resistors.
  • NTC (Negative Temperature Coefficient) thermistors decrease resistance with increasing temperature.
  • PTC (Positive Temperature Coefficient) thermistors increase resistance with temperature.
• Steinhart-Hart Equation models resistance-temperature relationship for NTCs: $\frac{1}{T} = A + B \ln(R) + C (\ln(R))^3$
• Linearization circuits improve measurement accuracy.
• Used in thermostats, temperature control systems, and liquid/gas flow measurement.

RTD

• RTD (Resistance Temperature Detector) measures temperature via metal resistance change.
• Common materials are platinum (Pt), nickel (Ni), and copper (Cu).
• Equation: $R_T = R_0 [1 + \alpha (T - T_0)]$, where $R_T$ is resistance at temperature T.
• Different configurations include two-wire, three-wire, and four-wire RTDs.
• RTDs offer high accuracy and reasonable linearity.
• More expensive compared to thermistors and require current that causes self-heating
• Used in industrial processes, lab equipment, and HVAC systems.

Transductores capacitivos

• Capacitive transducers measure physical quantities based on capacitance change.
• Change in capacitance is used to measure the physical quantity
• Capacitance is proportional to plate area and inversely proportional to plate distance.
• $C = \epsilon \frac{A}{d}$, where C is capacitance, $\epsilon$ is permittivity, A is area, and d is distance.
• Vary capacitance by changing plate area, separation, or dielectric permittivity.
• High sensitivity, low output impedance, and low power consumption
• Sensitivity to temperature, humidity, and vibrations

Transductores inductivos

• Inductive transducers measure physical quantities based on inductance change.
• Types:
  • LVDT: Measures linear displacement using a moving core.
  • Resolvers/Syncros: Measure angular position via a rotor and stator.
  • LVDT Advantages
  • High precision
  • High resolution
  • Low output impedance
  • The ability to measure large quantities
  • LVDT Disadvantages
  • Sensitive to temperature
  • Sensitive to humidity
  • No linearity
• Applications:
  • Position
  • Speed -Acceleration

Transductores piezoeléctricos

• Piezoelectric transducers measure physical quantities using the piezoelectric effect.
• Materials generate electric charge when pressure is applied.
• Materials include natural crystals, polarized ceramics, and polymers
• High sensitivity and frequency response
• Sensitivity to temperature, humidity, and vibrations
• Require a charge amplifier and is susceptible to non-liniarity
• Used for pressure, force, and acceleration measurement.

Transductores termoeléctricos

• Thermoelectric transducers measure temperature using the Seebeck effect.
• A voltage is generated when two different metals at the junctions are at different temperatures.
• Types: J, K, T, E, N, S, R, B
• Offers a wide range and is both simple and robust
• Characterized by low sensitivity and requirement for cold junction compensation
• Used in furnaces, engines, chemical processes, and HVAC systems.

Mobile App User Guide

• Introduces mobile app guide for understanding app features.

Getting Started

• App is downloadable from Apple App Store or Google Play Store.
• Installation steps involve opening the app store, searching for the app, and opening it. Creating an Account: steps include pressing on "Create account", entering relevant details, and verifying the email address.
• Logging In Steps: opening the app, inserting credentials and pressing on "Log In"

Navigating the App

• Offers quick access to main features like Home, Search, Add, Notifications, and Profile.
• Provides access to Settings, Help, Contact, and Log Out.

Features

• Characteristic 1: [Feature 1 description]. Use: [Step 1], [Step 2], [Step 3].
• Characteristic 2: [Feature 2 description]. Use: [Step 1], [Step 2], [Step 3].
• Characteristic 3: [Feature 3 description]. Use: [Step 1], [Step 2], [Step 3].

Settings

• Customizing the User Profile, configure notification preferences, and adjust privacy settings from account panel.

Troubleshooting

• Providing some standard solutions to problems the end-user my face.
• Problems and solutions described.

FAQ

• Frequently answered question.
• Answered questions listed.

Contact

• Provided contact us details for questions or more help .

Chapter 14: Social Psychology

• Social psychology studies how people affect each other's thoughts, feelings, and behaviors.
• The field studies both the self and the social world.

Areas of Social Psychology

• Intrapersonal processes include emotions, attitudes, self-concept, and social cognition.
• Interpersonal processes include helping behavior, aggression, prejudice and discrimination, attraction, and relationships.

Attitudes and Persuasion

What is an Attitude?

• An attitude includes affective, behavioral, and cognitive components.
• Feelings, behavior, belief, and knowledge

How Attitudes Change

• Persuasion changes an attitude toward something based on communications.

Yale Attitude Change Approach

• States that certain sources, such as credibility and attractiveness, are key to persuasiveness.
• Elements of the message, such as subtlety or timing, influence persuasiveness
• Audience attention, intelligence, self-esteem, and age

Elaboration Likelihood Model

• Model of persuasion.
• It has two routes: Central route: logic-driven, using facts. Peripheral route: less direct and relies on emotion or celebrity endorsement to associate positivity.

Tension

What is Tension?

• Is pulling force from opposite ends.
• Opposite of compression.

Tension formula

• Formula: $T = (m \times g) + (m \times a)$
  • Where:
  • $T =$ Tension (N)
  • $m =$ mass (kg)
  • $g =$ acceleration due to gravity (9.8 m/s²)
  • $a =$ acceleration of the object (m/s²)
• Example
  • A 5kg mass is hanging from a ceiling using a string. Calculate the tension in the string.
  • Therefore, the tension in the string is 49N.

Lecture 10

The Pendulum

Simple Pendulum

• Idealizations
  • Mass concentrated at point
  • Massless string
  • No air resistance
  • No friction at the hinge
• $\theta$ The angular displacement
• s is the arc length
• $s = L\theta$
• $\theta = s/L$
• Restoring force:
  • $F_t = -mgsin\theta$
  • $ma = -mgsin\theta$
  • $a = -gsin\theta$
• For small angles:
  • $sin\theta \approx \theta$
  • $a = -g\theta = -g\frac{s}{L}$
  • $a = -\frac{g}{L}s$
  • $\omega^2 = \frac{g}{L}$
  • $\omega = \sqrt{\frac{g}{L}}$
  • $T = \frac{2\pi}{\omega} = 2\pi\sqrt{\frac{L}{g}}$

Physical Pendulum

• Not all mass concentrated at one point.
• Extended object of mass M. D is the distance from the pivot to the center of mass.
  • $\tau = -MgDsin\theta = I\alpha$
  • $\alpha = \frac{\tau}{I} = -\frac{MgD}{I}sin\theta$
  • $\alpha \approx -\frac{MgD}{I}\theta$
  • $\omega = \sqrt{\frac{MgD}{I}}$
  • $T = 2\pi\sqrt{\frac{I}{MgD}}$
• If $I = MD^2$
  • $T = 2\pi\sqrt{\frac{MD^2}{MgD}} = 2\pi\sqrt{\frac{D}{g}}$

Example

• A meter stick is pivoted at one end
  • $I = \frac{1}{3}ML^2$ D = $\frac{L}{2}$
  • $T = 2\pi\sqrt{\frac{\frac{1}{3}ML^2}{Mg\frac{L}{2}}} = 2\pi\sqrt{\frac{2L}{3g}} = 2\pi\sqrt{\frac{2(1m)}{3(9.8m/s^2)}} = 1.64s$

Damped Oscillations

• $F_{damping} = -bv$
  • $F = ma = -kx - bv$
  • $ma + bv + kx = 0$
  • $x(t) = Ae^{-\frac{b}{2m}t}cos(\omega't + \phi)$
• $\omega' = \sqrt{\frac{k}{m} - (\frac{b}{2m})^2}$

Types of Damping

• Underdamped: Light damping, oscillates with decreasing amplitude.
• Critically damped: Returns to equilibrium as quickly as possible without oscillating
• Overdamped: Returns to equilibrium slowly without oscillating

Forced Oscillations and Resonance

• Driving force $F(t) = F_0sin(\omega t)$
  • $A = \frac{F_0/m}{\sqrt{(\omega^2 - \omega_0^2)^2 + (b\omega/m)^2}}$
  • $\omega_0 = \sqrt{\frac{k}{m}}$
• Resonance when $\omega = \omega_0$
• If damping is small, the amplitude is large

The Mathematics of Data Science

Course Overview

• Code: Ma 305
• Instructor: Prof Broderick
• Credits: 3
• Level: Intermediate
• Term: Fall 2023-2024
• Prerequisites: Ma 121 and proficiency in Python

Course Description

• Focuses on mathematical concepts essential for data science.
• Application of Concepts with Python.

Learning Objectives

• Apply calculus techniques and linear algebra to solve data science problems.
• Use Python for data analysis and visualization.

Course Schedule

• Introduction to Data Science Mathematics.
• Linear Algebra: Vectors and Matrices, Linear Transformations.
• Calculus: Differentiation, Optimization: Gradient Descent.
• Probability Theory: Random Variables and Statistical Inference.
• Regularization Ridge, Lasso and Elastic Net.
• Midterm Exam Covers topics from weeks 1-8.
• Clustering and Dimensionality Reduction: PCA.

Recommender Systems and Time Series Analysis

• Final Project Presentations & Conclusion: Students present their final projects and course conclusion.
• Comprehensive exam covering all topics from the course.

Grading Policy

Required Textbook

• Linear Algebra and Learning from Data by Gilbert Strang.

Recommended Readings

• Pattern Recognition and Machine Learning by Christopher Bishop.
• The Elements of Statistical Learning by Trevor Hastie, Robert Tibshirani, and Jerome Friedman.

Additional Information

• Academic Integrity Rules applicable to everyone.
• Contact Disability Services for special requests.
• Contact Information of Instructor provided, including office hours.

Teorema de Bayes

• The theorem describes the probability of an event based on prior knowledge of conditions related to the event.
• Basic Formula: $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$
  • $P(A|B)$: The conditional probability of A, given B is true.
  • $P(B|A)$: The conditional probability of B, given A is true.
  • $P(A)$ and $P(B)$: The probabilities of A and B being true independently.

Deducción del Teorema

• The theorem can be derived from basic definitions of conditional probability
  • $P(A \cap B) = P(A|B)P(B) = P(B|A)P(A)$
  • $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$

Ejemplo

• A fictional test is 95% accurate when a person has disease A and 90% when a person does not have the disease.
• $P(A|B) = \frac{0.95 \cdot 0.01}{0.95 \cdot 0.01 + 0.10 \cdot 0.99} = 0.087$
• The probability the person has disease A is approximately 8.7%.