Growth and stages infant child

Algèbre Linéaire

• The mathematics of data representation and manipulation through vectors and matrices
• Used to solve linear systems and understand vector spaces

Opérations sur les vecteurs et matrices

• Vector addition combines vectors component-wise
• Scalar multiplication scales a vector by a constant
• Vector dot product yields a scalar, summing the product of corresponding components
• Matrix multiplication combines matrices to form result
• Transposition switches rows and columns

Properties du produit scalaire

• Commutative i.e. $\vec{u} \cdot \vec{v} = \vec{v} \cdot \vec{u}$
• Distributive i.e. $\vec{u} \cdot (\vec{v} + \vec{w}) = \vec{u} \cdot \vec{v} + \vec{u} \cdot \vec{w}$
• Associative i.e. $(\alpha\vec{u}) \cdot \vec{v} = \alpha(\vec{u} \cdot \vec{v})$
• Important relation formula $|\vec{u} + \vec{v}|^2 = |\vec{u}|^2 + |\vec{v}|^2 + 2(\vec{u} \cdot \vec{v})$

Orthogonalité

• Indicative property: $\vec{u} \cdot \vec{v} = 0$

Propriétés de la multiplication matricielle

• Associative i.e. $A(BC) = (AB)C$
• Distributive i.e. $A(B + C) = AB + AC$
• Distributive i.e. $(A + B)C = AC + BC$
• Associative i.e. $\alpha(AB) = (\alpha A)B = A(\alpha B)$
• Transposition relation i.e. $(AB)^T = B^T A^T$

Special Matrix Typology

• Square, identity, diagonal, triangular (upper/lower), symmetric, and anti-symmetric matrices
• Matrix symmetry: $A^T = A$
• Matrix inverse is $A^{-1}$, so $AA^{-1} = A^{-1}A = I$

Linear System Representation

• Expressed as $Ax = b$, defining component matrices

Methods de résolution

• Gauss elimination creates echelon form
• Matrix inversion yields solution by $x = A^{-1}b$
• Cramer’s rule employs determinants

Espace vectoriel

• Vector spaces possess vector addition and scalar multiplication operations
• Subspaces must fulfill all vector space axioms

Base et dimension

• Base vectors span the vector space
• Dimension is quantity of base vectors

Transformations linéaires

• Linear transformations preserve vector addition and scalar multiplication
• Can be expressed as a matrix
• For matrix A, a vector is $Av = \lambda v$

Algorithmic Complexity

• Evaluates algorithm time and space resource needs

Big O Notation

• Describes upper bound of growth as input size increases

Common complexities

• • Constant O(1), logarithmic O(log n), linear O(n), linearithmic O(n log n), quadratic O(n^2), exponential O(2^n), and factorial O(n!)

Space Complexity

• Assesses memory usage by algorithm

Physics

• Applying laws of motion, energy, and fields to physical phenomena

Kinematics

• Describes motion using displacement, velocity, and acceleration

Projectile Motion

• Constant horizontal velocity and constant vertical acceleration
• Equation for range: $R = \frac{v_0^2 \sin(2\theta)}{g}$

Centripetal Acceleration

• Uniform circular motion acceleration: $a_c = \frac{v^2}{r}$

Dynamics

• Explores forces affecting motion (friction, springs, gravity)

Thermodynamics

• Thermal energy and its relation to work and heat

Equations

• First law i.e. $\Delta U = Q - W$
• $k_B = 1.38 \times 10^{-23} \text{ J/K}$
• Carnot efficiency i.e. $e_{Carnot} = 1 - \frac{T_C}{T_H}$

Electricity & Magnetism

• Studies electric charges and their related forces and fields

Formulae

• $k = 8.99 \times 10^9 \text{ N m}^2/\text{C}^2$
• $\epsilon_0 = 8.85 \times 10^{-12} \text{ F/m}$
• $\mu_0 = 4\pi \times 10^{-7} \text{ T m/A}$

Constants and Conversions

• $g = 9.8 \text{ m/s}^2$
• $N_A = 6.022 \times 10^{23} \text{ mol}^{-1}$
• $1 \text{ eV} = 1.602 \times 10^{-19} \text{ J}$
• $c = 3.0 \times 10^8 \text{ m/s}$

Partial Differential Equations

• Equations with functions of multiple variables and their partial derivatives

Examples

• Include transport, heat, and wave equations

Heat Equation

• Describes temperature distribution over time

Dérivation

• Assuming constant density, specific heat: $u_t = k u_{xx}$
• $k = \frac{K}{\rho c}$ is the thermal diffusivity.

Wave Equation

• Describes wave propagation

Dérivation

• Assuming small deflections, and $\rho$ and $T$ constant i.e. $u_{tt} = c^2 u_{xx}$
• Where, $c = \sqrt{\frac{T}{\rho}}$ is the wave speed

Laplace's Equation

• Model states of equilibrium i.e. $\Delta u = u_{xx} + u_{yy} = 0$ in 2D
• i.e. $\Delta u = u_{xx} + u_{yy} + u_{zz} = 0$ in 3D

Algèbre Linéaire et Analyse Matricielle - Intro

• Linear equations defined as $a_1x_1 + a_2x_2 +... + a_nx_n = b$

Systèmes d'équations linéaires

• Solve jointly i.e.

2x + 3y = 7
x - y = 1

Matrix Representation

• With coefficient, variable and constant matrices $A = \begin{bmatrix} a_{11} & a_{12} &... & a_{1n} \ a_{21} & a_{22} &... & a_{2n} \... &... &... &... \ a_{m1} & a_{m2} &... & a_{mn} \end{bmatrix}$, $x = \begin{bmatrix} x_1 \ x_2 \... \ x_n \end{bmatrix}$, $b = \begin{bmatrix} b_1 \ b_2 \... \ b_m \end{bmatrix}$

Algèbre Linéaire et Analyse Matricielle

• Vector as ordered list of numbers
• Matrix as rectangular number array

Vector Operations

• Vector and matrix operations include addition, scalar multiplication, transposition, trace, and determinant
• Matrix diagonalization uses transformation matrix

Common equation solvers

• Gauss-Jordan, Cramer, and LU factorization methods

Espace vectoriel

• Vector Space: non-empty set with vector addition and scaling rules

Applications linéaires

• Applications linéaires maintain operations of summing vectors and vector scaling

UNIDAD 4: LÍMITES Y CONTINUIDAD

• Describes limits and functional continuity

Limite de una función en un punto

• Limit: intuitive approximation of output as input nears value

Definition

• $\lim_{x \to x_0} f(x) = L$ where $0 < |x - x_0| < \delta$ such that $|f(x) - L| < \epsilon$
• Limits are the same

Infinitésimos equivalentes in calculation

• Equations: $\sin x \sim x$
• Equations: $\tan x \sim x$
• Equations: $\arcsin x \sim x$
• Equations: $\arctan x \sim x$
• Equations: $1 - \cos x \sim \frac{x^2}{2}$
• Equations: $\ln(1 + x) \sim x$
• Equations: $e^x - 1 \sim x$
• Equations: $(1 + x)^a - 1 \sim ax$

Regla de L'Hôpital rule

• Applied to eliminate limits

Continuidad de una función en un punto

• Where the $\lim_{x \to x_0} f(x) = f(x_0)$

Biostatistics

• Bio statistics is the application of mathematical principles in health care, research, etc
• Uses variation, objectivity and inference

Common biostatistics types

• Discrete, continuous, normal etc etc

Hypothesis Testing

• Null hypothesis: A statement of no effect or no difference
• Alternative hypothesis: A statement that contradicts the null hypothesis