Statics Chapter 2: Vectors

Statics - Chapter 2: Vectors

Vector Operations: Addition of Coplanar Forces

• The diagonal of a parallelogram represents the resultant force of two coplanar forces.
• The law of cosines can be used to determine the resultant force $F_R$: $F_R = \sqrt{F_1^2 + F_2^2 - 2F_1F_2\cos(\theta)}$.
• The law of sines can be used to determine the angle $\alpha$: $\frac{F_1}{\sin(\alpha)} = \frac{F_R}{\sin(\theta)}$.

Cartesian Vector Notation

• A 2D force vector can be expressed as $F = F_x i + F_y j$
• The magnitude of the force vector is $F = \sqrt{F_x^2 + F_y^2}$.
• The direction angle is given by $\theta = \tan^{-1}(\frac{F_y}{F_x})$.

Addition of Several Forces

• The resultant force is $F_R = \sum F_x i + \sum F_y j$
• The magnitude of the resultant force is $F_R = \sqrt{(\sum F_x)^2 + (\sum F_y)^2}$.
• The direction angle of the resultant force is $\theta = \tan^{-1}(\frac{\sum F_y}{\sum F_x})$.

Addition of System of 3D Forces

• A 3D force vector can be expressed as $F = F_x i + F_y j + F_z k$
• The magnitude of the force vector is $F = \sqrt{F_x^2 + F_y^2 + F_z^2}$.

Direction Cosines

• Direction cosines satisfy the equation $\cos^2(\alpha) + \cos^2(\beta) + \cos^2(\gamma) = 1$.
• $F_x = F\cos(\alpha)$
• $F_y = F\cos(\beta)$
• $F_z = F\cos(\gamma)$

Unit Vector

• A unit vector is given by $u = \frac{F}{|F|} = \frac{F_x}{|F|}i + \frac{F_y}{|F|}j + \frac{F_z}{|F|}k$

Position Vectors

• A position vector is given by $r = (x_2 - x_1)i + (y_2 - y_1)j + (z_2 - z_1)k$
• The magnitude of the position vector is $r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$

Force Vector Directed Along a Line

• The force vector can be expressed as a product of its magnitude and a unit vector along the line of action, $F = F\frac{r}{|r|} = F_u = F\frac{(x_2 - x_1)}{|r|}i + F\frac{(y_2 - y_1)}{|r|}j + F\frac{(z_2 - z_1)}{|r|}k$

Dot Product

• The dot product can be calculated as $A \cdot B = |A||B|\cos(\theta)$.
• In terms of components, $A \cdot B = A_x B_x + A_y B_y + A_z B_z$.

Determining the Angle Between Two Vectors

• The angle $\theta$ between two vectors can be found using $\cos(\theta) = \frac{A \cdot B}{|A||B|} = \frac{A_x B_x + A_y B_y + A_z B_z}{|A||B|}$

Finding the Component of a Force Parallel to a Line

• $F' = F\cos(\theta) = F \cdot u$
• $F' = (F \cdot u)u$

Cross Product

• Magnitude of Cross Product: $A \times B = |A||B|\sin(\theta)$
• In terms of components: $$A \times B = \begin{vmatrix} i & j & k \ A_x & A_y & A_z \ B_x & B_y & B_z \end{vmatrix} = (A_yB_z - A_zB_y)i - (A_xB_z - A_zB_x)j + (A_xB_y - A_yB_x)k$$

Transformer

Overview

• The Transformer is a deep learning model proposed in 2017, widely used in machine translation and other NLP tasks.
• It has largely replaced RNNs in NLP due to its superior performance.

Overall Architecture

• The Transformer architecture consists of the input/output, encoder module, decoder module, residual connections, and layer normalization.
• Input is the word embeddings of the input words
• Output is the probability distribution of the output word predictions.
• The encoder and decoder modules are composed of N stacked encoder and decoder layers respectively.

Encoder Module

• The encoder is composed of multiple encoder layers stacked on top of each other.
• Each encoder layer primarily contains a multi-head self-attention mechanism and a feedforward neural network.

Multi-Head Self-Attention

• Multi-head self-attention computes the relationships between words in the input sentence.
• It comprises scaled dot-product attention and multiple heads which enable the model to learn information from different representation subspaces.

Scaled Dot-Product Attention

• Attention is calculated as $Attention(Q, K, V) = softmax(\frac{QK^T}{\sqrt{d_k}})V$
• Q refers to the Query matrix
• K is the Key matrix
• V is the Value matrix
• $d_k$ is the dimensionality of the keys.

Multi-Head

• The calculation is $MultiHead(Q, K, V) = Concat(head_1,..., head_h)W^O$
• Where $head_i = Attention(QW_i^Q, KW_i^K, VW_i^V)$
• $W_i^Q$, $W_i^K$, and $W_i^V$ are weight matrices for queries, keys and values.
• $W^O$ is the projection matrix.

FeedForward Neural Network

• The feed-forward network applies a non-linear transformation to the output of the attention layer
• The calculation is $FFN(x) = ReLU(xW_1 + b_1)W_2 + b_2$

Decoder Module

• The decoder module mirrors the encoder by consisting of N decoder layers.
• Each decoder layer contains masked multi-head self-attention, encoder-decoder attention, and a feedforward network.

Masked Attention

• Masking prevents the decoder from attending to future tokens, which is achieved by setting the corresponding values to $-\infty$ before applying the softmax function.

Encoder-Decoder Attention

• Encoder-decoder attention allows the decoder to focus on relevant parts of the input sequence from the encoder

Positional Encoding

• Since the Transformer lacks recurrence, positional encodings are added to the input embeddings.
• These are functions of sine and cosine:
• $PE_{(pos,2i)} = sin(pos/10000^{2i/d_{model}})$
• $PE_{(pos,2i+1)} = cos(pos/10000^{2i/d_{model}})$
• pos is the position
• i is the dimension
• $d_{model}$ is the dimension of the model.

Chemical Engineering Thermodynamics - Chapter 3: Volumetric Properties of Pure Fluids

Virial Equation of State

• $Z = 1 + \frac{B}{V} + \frac{C}{V^2} + \frac{D}{V^3}+...$
• Z = PV/RT, where Z is the compressibility factor
• Capital letters are virial coefficients
• Alternative form $Z = 1 + B'\rho + C'\rho^2 + D'\rho^3 +...$
• $\rho = 1/V$, where $\rho$ is density
• $B' = B, C' = (C-B^2), D' = (D-3BC+2B^3)$

Ideal Gas Model

• $PV = RT$
• V is volume per mole
• $R = 8.314 J/mol.K$

Van der Waals Equation of State

• $P = \frac{RT}{V-b} - \frac{a}{V^2}$
• Accounts for intermolecular forces
• $a = \frac{27(RT_c)^2}{64P_c}$
• $b = \frac{RT_c}{8P_c}$

Redlich/Kwong Equation of State

• $P = \frac{RT}{V-b} - \frac{a}{T^{1/2}V(V+b)}$
• $a = \frac{0.42748R^2T_c^{2.5}}{P_c}$
• $b = \frac{0.08664RT_c}{P_c}$

Soave/Redlich/Kwong Equation of State

• $P = \frac{RT}{V-b} - \frac{\alpha a}{V(V+b)}$
• $a = \frac{0.42748R^2T_c^2}{P_c}$
• $b = \frac{0.08664RT_c}{P_c}$
• $\alpha = [1 + m(1 - T_r^{0.5})]^2$
• $m = 0.480 + 1.574\omega - 0.176\omega^2$
• $T_r = \frac{T}{T_c}$

Peng Robinson

• $P = \frac{RT}{V-b} - \frac{a\alpha}{V^2 + 2bV - b^2}$
• $a = 0.45724\frac{R^2T_c^2}{P_c}$
• $b = 0.07780\frac{RT_c}{P_c}$
• $\alpha = [1 + m(1 - T_r^{0.5})]^2$
• $m = 0.37464 + 1.54226\omega - 0.26992\omega^2$
• $T_r = \frac{T}{T_c}$

Generalized Correlations for the Virial Coefficients

• $B = \frac{RT_c}{P_c}(B^0 + \omega B^1)$
• $B^0 = 0.083 - \frac{0.422}{T_r^{1.6}}$
• $B^1 = 0.139 - \frac{0.172}{T_r^{4.2}}$

Volume as a Function of T & P

• $dV = (\frac{\partial V}{\partial T})_pdT + (\frac{\partial V}{\partial P})_TdP$

Volume Expansivity

• $\beta = \frac{1}{V}(\frac{\partial V}{\partial T})_p$

Isothermal Compressibility

• $\kappa = -\frac{1}{V}(\frac{\partial V}{\partial P})_T$

Fonction et limites: Exercices

Limites

• Exercice 1: Calculation of various limits, including polynomial, rational, and radical functions.

• Exercice 2: Calculation of limits, focusing on indeterminate forms and trigonometric functions.

• Exercice 3: Evaluation of standard limits, like those with exponential, logarithmic, and inverse trigonometric functions.

Fonctions (Functions)

• Exercice 4: Study functions, including determining domain, derivative, variations, and limits.

• Exercice 5: Given $f(x) = \frac{x^2 - 1}{x^2 + 1}$, determination of the domain, calculation of the derivative, study of variations, determining the limits at the boundaries of its domain and graphing.

• Exercice 6: Given $f(x) = x + \sqrt{x^2 + 1}$, determination of the domain, calculation of the derivative, study of variations, determining the limits at the boundaries of its domain and proving $f$ is bijective from $\mathbb{R}$ to $\mathbb{R}$.

• Exercice 7: Given $f(x) = \frac{\cos(x)}{\sin(x)}$, determination of the domain, calculation of the derivative, study of variations on the interval $]0, \pi[$, determining limits at the boundaries of its domain and graphing.

Funciones vectoriales

Definition

• Vector function: a function whose domain is a subset of real numbers and whose range is a set of vectors.
• 2D Vector: $\vec{r}(t) = \langle f(t), g(t) \rangle = f(t) \hat{i} + g(t) \hat{j}$.
• 3D Vector: $\vec{r}(t) = \langle f(t), g(t), h(t) \rangle = f(t) \hat{i} + g(t) \hat{j} + h(t) \hat{k}$, where f,g, h are real-valued functions.

Limits and Continuity

• Limit of a vector function: $\lim_{t \to a} \vec{r}(t) = \langle \lim_{t \to a} f(t), \lim_{t \to a} g(t), \lim_{t \to a} h(t) \rangle$ (provided the limits exist).
• Vector function $\vec{r}(t)$ is continuous at $t = a$ if $\vec{r}(a)$ is defined, the limit exists, and $\lim_{t \to a} \vec{r}(t) = \vec{r}(a)$. Each component function must be continuous.

Derivatives

• Derivative of a vector function: $\frac{d\vec{r}}{dt} = \vec{r}'(t) = \lim_{\Delta t \to 0} \frac{\vec{r}(t + \Delta t) - \vec{r}(t)}{\Delta t}$.
• If $\vec{r}(t) = \langle f(t), g(t), h(t) \rangle$, then: $\vec{r}'(t) = \langle f'(t), g'(t), h'(t) \rangle$.
• $\vec{r}'(t)$ is a tangent vector provided it is nonzero and the derivative exists. The tangent line at $P$ (defined by $\vec{r}(t_0)$) is parallel to $\vec{r}'(t_0)$.

Differentiation Rules

• If $\vec{u}$ and $\vec{v}$ are differentiable vector functions, f is a differentiable scalar function, and c is a constant, then:
• $\frac{d}{dt} [\vec{c}] = \vec{0}$.
• $\frac{d}{dt} [c \vec{r}(t)] = c \vec{r}'(t)$.
• $\frac{d}{dt} [\vec{u}(t) + \vec{v}(t)] = \vec{u}'(t) + \vec{v}'(t)$.
• $\frac{d}{dt} [f(t) \vec{r}(t)] = f'(t) \vec{r}(t) + f(t) \vec{r}'(t)$.
• $\frac{d}{dt} [\vec{u}(t) \cdot \vec{v}(t)] = \vec{u}'(t) \cdot \vec{v}(t) + \vec{u}(t) \cdot \vec{v}'(t)$.
• $\frac{d}{dt} [\vec{u}(t) \times \vec{v}(t)] = \vec{u}'(t) \times \vec{v}(t) + \vec{u}(t) \times \vec{v}'(t)$.
• $\frac{d}{dt} [\vec{u}(f(t))] = f'(t) \vec{u}'(f(t))$.

Integrals

• Indefinite integral: $\int \vec{r}(t) dt = \langle \int f(t) dt, \int g(t) dt, \int h(t) dt \rangle$.
• Definite integral: $\int_a^b \vec{r}(t) dt = \langle \int_a^b f(t) dt, \int_a^b g(t) dt, \int_a^b h(t) dt \rangle$.

Statistical Inference

Introduction

• Statistical inference: Drawing conclusions about a population from a sample.
• Population: Entire group of interest.
• Sample: Subset of population.
• Parameter: Describes population.
• Statistic: Describes sample.

Point Estimation

• Single value estimate of a population parameter
• Method of Moments: Equating sample to population moments.
• Maximum Likelihood Estimation (MLE): Maximizing likelihood function.

Interval Estimation

• Range of values to estimate parameter.
• Confidence Interval (CI): $$ CI = [\hat{\theta} - z_{\alpha/2}SE(\hat{\theta}), \hat{\theta} + z_{\alpha/2}SE(\hat{\theta})] $$

Hypothesis Testing

• Procedure for decisions about parameters
• Null Hypothesis ($H_0$): Parameter statement assumed true unless proven otherwise
• Alternative Hypothesis ($H_1$ or $H_a$): Contradicts null hypothesis, accepted if null rejected
• Test Statistic: Value from sample data to assess evidence
• P-value: Probability of test statistic extreme or more, assuming $H_0$ is true.
• Significance level ($\alpha$): probability of rejecting null hypothesis when it is true (Type I error).

Regression Analysis

• Statistical technique to model relation

Simple Linear Regression

• Relationship w/ dependent variable using equation: $y = \beta_0 + \beta_1x + \epsilon$

Multiple Linear Regression

• Relationship w/ dependent variable using equation: $y = \beta_0 + \beta_1x_1 + \beta_2x_2 +... + \beta_px_p + \epsilon$

Analysis of Variance (ANOVA)

• Method for comparing the means of groups

One-Way ANOVA

• When independent variable has 1 factor

Two-Way ANOVA

• When independent variable has 2+ factors

Non-parametric Methods

• Does not assume a specific distribution

Chi-Square Test

• Used to the test the independence of categorical variables

Sign Test

• Used to text to median of a population

Bayesian Inference

• Combines prior beliefs with sample data

Bayes' Theorem :

• $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$

Algèbre Linéaire et Géométrie Analytique I

Chapitre 1: Systèmes d'Équations Linéaires

1.1 Introduction

• Équation linéaire: $a_1x_1 + a_2x_2 +... + a_nx_n = b$, where $a_i$ and $b$ are real constants.

• Système d'équations linéaires: A set of linear equations with the same variables. Example:

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

• Solution: A list $s_1, s_2,..., s_n$ that satisfy each equation when $x_i = s_i$.

• Ensemble solution: The set of all possible solutions.

• Systèmes équivalents: Two systems with the same solution set.

• Possible solutions for systems of linear equation

  • Exactly one solution (compatible system)
  • Infinitely many solutions (compatible system)
  • No solution (incompatible system)

1.2 Méthodes de Résolution

• Matrice: A rectangular array of numbers.

• Matrice augmentée: Matrix containing the coefficients and constants of a system.

  -$\begin{bmatrix}
  

a_{11} & a_{12} &... & a_{1n} & b_1 \ a_{21} & a_{22} &... & a_{2n} & b_2 \... &... &... &... &... \ a_{m1} & a_{m2} &... & a_{mn} & b_m \end{bmatrix}$

• Opérations élémentaires sur les lignes:

  • Multiply a row by a nonzero constant.
  • Add a multiple of one row to another.
  • Swap two rows.

• Théorème:* Equivalent matrices = equivalent system.

• Élimination de Gauss: Method to transform matrix into an echelon form.

• Élimination de Gauss-Jordan: Method to transform matrix into a reduced echelon form.