Functions, Approximations, and Rate of Change

Functions and Approximations

• Approximating complex functions with simpler functions, like polynomials, is a common mathematical technique.
• Sin(x) can be approximated using its Taylor expansion: sin(x) ≈ x - x³/3! + x⁵/5! - ...
• E^x can be similiarly approximated : eˣ ≈ 1 + x + x²/2! + x³/3! + ...
• These approximations are especially useful when x is close to 0.

Function Optimization

• Function optimization involves finding the maximum or minimum values of a function.
• Extrema of f(x) occur where its derivative f'(x) = 0.
• If f'(x) = 0, then f(x) has a critical point.
• If f''(x) > 0, then f(x) has a local minimum.
• If f''(x) < 0, then f(x) has a local maximum.
• Optimization is uesd to minimize costs and maximize profits.

Rate of Change Computation

• The rate of change of a function f(x) is given by its derivative f'(x).
• It represents the ratio change in f(x) with respect to change in x.
• If s(t) is an object's position as a function of time t, then its velocity is calculated as v(t) = s'(t).
• Acceleration a(t) is calculated as the derivative of velocity: a(t) = v'(t).
• Rate of change is used by economists to to analyze market changes and by physicists to describe motion of objects

Static Electricity: Charging by Friction

• Atoms have varying degrees of electron attraction.
• Electrons get transferred when neutral object are rubbed together
• This results in one object becoming positively charged and the other becoming negatively charged.

Example of Charging by Frition

• Rubbing fur with a rubber rod causes the fur to become positively charged and the rubber rod to become negatively charged.

Charging by Conduction

• Direct contact between a charged object and a neutral object which results in the transfer of electrons.
• The neutral object acquires the same type of charge as the charged object.

Example of Charging by Conduction

• Contact between a negatively charged rod and a neutral metal sphere causes electrons transfer from the rod to the sphere, resulting in the sphere also becoming negatively charged.

Charging by Induction

• A charged object is brought near, but does not touch, a neutral object.
• The presence of the charged object causes charge separation in the neutral object.
• Grounding the neutral object allows electron flow in or out.
• The neutral object ends up with a charge that is opposite to the charged object.

Example of Charging by Induction

• A negatively charged rubber rod brought near a neutral metal spear will repel the sphere's electrons away from the rod, creating an area of charge separation.
• If sphere is then grounded, electrons flow out of the sphere until it has positive charge.

Polarization Overview

• The rearrangement of charges within the molecules of a neutral object due to the proximity of a charged object.
• Polarization can occur in both conductors and insulators.

Example of Polarization

• A negatively charged rubber rod is brought near a neutral wall, the molecules in the wall become polarized, with the positive charges oriented towards the rod and the negative charges oriented away from the rod.

Electric Force Basics

• The force between charged objects is a vector quantity that includes both magnitude and direction: like charges repel and opposite charges attract.

Coulomb's Law

• The magnitude of the electric force between two point charges can be calculated by:
• F = k|q1q2|/r^2
  • F is the magnitude of the force
  • k is Coulomb's constant (approximately 8.99 x 10^9 N⋅m²/C²)
  • q1 and q2 are the magnitudes of the charges
  • r is the distance between the charges

Example of finding the Magnitude using Coulomb's Law

• Use the variables q1 = +2 μC, q2 = -4 μC, distance = 3m, and Coulomb's constant to find that the elecgric force magnitude is 0.008 N.

Electric Field Basics

• The electric field is the force per unit charge exerted on a positive test charge at a point in space.
• The electric field is a vector: magnitude and direction.
• Direction of the electric field is the direction of the force on the positive test charge.

Electric Field due to a Point Charge Basics

• The electric field is determined by:
  • E = k|q|/r^2
    • E is the magnitude of the electric field
    • k is Coulomb's constant (approximately 8.99 x 10^9 N⋅m²/C²)
    • q is the magnitude of the charge
    • r is the distance from the charge

Electric Field Lines

• Electric field lines are visualisations of the electric field.
• Start on positive charges and end on negative charges.
• Closeness indicates relative strength.
• Electric field never cross each other
• A re perpendicular to the surface of a charged object.

Electric Field Due to a Point Charge Example

• The electric field at 2m from a +5μC point charge is 11237.5 N/C using the above formula.

Factor Analysis Basics

• A statistical method to reduce data complexity and identify latent variables from observed variables.
• Addresses correlations among a large number of variables.
• Defines sets of variables that are highly correlated, known as factors.

Uses of Factor Analysis

• Data reduction and identification of latent variables

Key Concepts of Factor Analysis

• Factors are underlying latent variables that explain the correlations among the observed variables.
• Factor loadings are Correlation between the original variables and the factors.
• An eigenvalue represents the variance in the original variables explained by each factor. Communality is the proportion of each variable's variance that can be explained by the factors. Rotation simplifies the interpretable factor structure.

Types of Factor Analysis

• Exploratory Factor Analysis (EFA) is used to explore the data and identify the number of factors required to represent the data.
• Confirmatory Factor Analysis (CFA) is used to test specific hypotheses about the structure of the factors.

Steps in Factor Analysis

• Data Collection - ensure data is suitable for factor analysis
• Create a Correlation Matrix for the variables.
• Factor Extraction - determine the number of factors to extract using methods like the Kaiser criterion or scree plot.
• Factor Rotation - rotate the factors improving interpretability using methods like varimax (orthogonal) and promax (oblique).
• Interpretation - based on the variables that load highly on them.
• Validate the factor structure

The Factor Model

• X = ΛF+ϵ
  • X is the matrix of observed variables.
  • Λ is the matrix of factor loadings.
  • F is the matrix of common factors.
  • ϵ is the matrix of unique factors or errors.

Factor Analysis Example

• A researcher wants to understand the factors influencing customer satisfaction in a retail store, and analyzes variables such as store ambiance, product quality, price and service, to find two underliying factors; "Overall Experience" and "Value for Money".

Factor Analysis Assumptions

• Linearity - assumes linear relationships between variables
• Multivariate normality - assumes variables are normally distributed
• Absence of multicollinearity - assumes variables are are not too highly correlated

2. Derivation of Velocity Field

• Derives the velocity field of a fluid around a rotating cylinder.

2.1. Definitions of Variables

• The Cartesian coordinate system (x, y, z) and cylindrical coordinate system (r, θ, z) are used.
• R is the radius of they cylinder
• Ω is the angular velocity of the cylinder
• ω = Ω^ez is the Angular velocity vector of the cylinder
• μ is the dynamic viscosity of the fluid
• ρ is the Density of the fluid
• p is the pressure of the fluid
• u is the Velocity vector of the fluid with the following assumptions
  • Cylinder is infinitely long.
  • Flow is incompressible and Newtonian.
  • Flow is steady and laminar.
  • No external force is acting on the fluid.

2.2. Velocity Field

• The flow is axisymmetric (independent of θ) and purely azimuthal.
• $\mathbf{u} = u_\theta(r) \hat{e_\theta}$, where $u_\theta(r)$ is azimuthal velocity.

2.3. Governing Equations

• The governing equations for the flow are the continuity equation and the Navier-Stokes equations.
• Continuity equation: $\frac{1}{r} \frac{\partial}{\partial r} (ru_r) + \frac{1}{r} \frac{\partial u_\theta}{\partial \theta} + \frac{\partial u_z}{\partial z} = 0$
• The simplified Navier-Stokes equation:
  • $\frac{\partial p}{\partial r} = \rho \frac{u_\theta^2}{r}$
  • $0 = \mu \left( \frac{\partial}{\partial r} \left( \frac{1}{r} \frac{\partial}{\partial r} (ru_\theta) \right) \right)$
  • $\frac{\partial p}{\partial z} = 0$
• Leads to the azimuthal velocity component which is integrated to result in an equation for the azimuthal velocity component: $u_\theta(r) = A r + \frac{B}{r}$.

2.4. Boundary Conditions

• At the surface of the cylinder ($r = R$), the fluid velocity is equal to the velocity of the cylinder surface: $u_\theta(R) = \Omega R$.
• Far away from the cylinder ($r \rightarrow \infty$), the fluid velocity is zero: $u_\theta(\infty) = 0$.
• Implies A=0, B=ΩR², and $\mathbf{u} = \frac{\Omega R^2}{r} \hat{e_\theta}$.

2.5. Pressure Field

• Pressure field is derived from the Navier-Stokes equations and is given by $p(r) = -\frac{\rho \Omega^2 R^4}{2r^2} + C$.
• The pressure decreases as the distance from the cylinder increases

Book Summary

• Linear Algebra: Course and Exercises (MPSI) is written by Patrice Tauvel, Professor at the University of Poitiers
• Topics include :
  • Vector Spaces
  • Linear Applications
  • Matrices
  • Determinants
  • Reduction of Endomorphisms