Physics: Understanding Vectors

Physics: Vectors

A vector is a quantity possessing both magnitude and direction.
Vectors are exemplified by displacement, velocity, acceleration, and force.
Vectors are denoted using boldface (e.g., A) or with an arrow above (e.g., $\vec{A}$).
Magnitude refers to the length of a vector and represents a scalar quantity; for vector A, it's written as |A| or A.
A unit vector, symbolized as $\hat{A}$, has a magnitude of 1 and points in the direction of A, defined as $\hat{A} = \frac{\vec{A}}{|\vec{A}|} = \frac{\vec{A}}{A}$.
Vectors can be represented as the sum of their components in a coordinate system such as $\vec{A} = A_x\hat{i} + A_y\hat{j}$ in a 2D Cartesian system.
$A_x$ and $A_y$ represent the x and y components of vector A, respectively, while $\hat{i}$ and $\hat{j}$ are the unit vectors along the x and y axes.
Vector addition can occur graphically, by placing the tail of one vector at the head of another, or analytically, by adding their components such as $\vec{A} + \vec{B} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j}$.
Vector subtraction involves adding the negative of the vector to be subtracted, e.g., $\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$ or $\vec{A} - \vec{B} = (A_x - B_x)\hat{i} + (A_y - B_y)\hat{j}$.
The scalar product (dot product) of two vectors is $ \vec{A} \cdot \vec{B} = AB\cos\theta$ or $\vec{A} \cdot \vec{B} = A_xB_x + A_yB_y + A_zB_z$.
The vector product (cross product) is defined as $|\vec{A} \times \vec{B}| = AB\sin\theta$ or written in terms of components as $\vec{A} \times \vec{B} = (A_yB_z - A_zB_y)\hat{i} + (A_zB_x - A_xB_z)\hat{j} + (A_xB_y - A_yB_x)\hat{k}$.

Calculus

Calculus is the study of continuous change, involving two primary branches: differential calculus and integral calculus.
Differential calculus addresses instantaneous rates of change and curve slopes.
Integral calculus deals with the accumulation of quantities and the areas under or between curves.
It has been under development for centuries, with significant contributions of Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century.
Calculus is vital in science, engineering, and economics for modeling phenomena like motion, heat, light, and sound.

Differential Calculus

Concerned with derivatives, measuring the instantaneous rate of change of a function.
The derivative of $f(x)$ at $x=a$ is $f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$, representing the slope of the tangent line at $(a, f(a))$.
Key rules include the Power Rule ($f(x) = x^n$, then $f'(x) = nx^{n-1}$), Constant Multiple Rule ($f(x) = cf(x)$, then $f'(x) = cf'(x)$), Sum Rule ($h(x) = f(x) + g(x)$, then $h'(x) = f'(x) + g'(x)$), Product Rule ($h(x) = f(x)g(x)$, then $h'(x) = f'(x)g(x) + f(x)g'(x)$), Quotient Rule ($h(x) = \frac{f(x)}{g(x)}$, then $h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$), and Chain Rule ($h(x) = f(g(x))$, then $h'(x) = f'(g(x)) \cdot g'(x)$).
Example: The derivative of $f(x) = 3x^2 + 2x + 1$ is $f'(x) = 6x + 2$.

Integral Calculus

Centered on integrals, used to determine areas, volumes, and accumulation.
The integral of $f(x)$ over $[a, b]$ denoted as $\int_{a}^{b} f(x) , dx$ symbolizes the signed area between the curve $y = f(x)$ and the x-axis from $x = a$ to $x = b$.
The Fundamental Theorem of Calculus links differential and integral calculus.
The First Fundamental Theorem states that if $F(x) = \int_{a}^{x} f(t) , dt$, then $F'(x) = f(x)$.
The Second Fundamental Theorem states that $\int_{a}^{b} f(x) , dx = F(b) - F(a)$, where $F'(x) = f(x)$.
Key rules include Power Rule ($\int x^n , dx = \frac{x^{n+1}}{n+1} + C$), Constant Multiple Rule ($\int cf(x) , dx = c \int f(x) , dx$), and Sum Rule ($\int [f(x) + g(x)] , dx = \int f(x) , dx + \int g(x) , dx$).
The indefinite integral of $f(x) = x^3 + 4x + 2$ is $\frac{x^4}{4} + 2x^2 + 2x + C$, where C is the constant of integration.

Quantum Mechanics

It is a fundamental theory describing nature's physical properties at atomic and subatomic levels.
Key principles include wave-particle duality, superposition, quantization, and the uncertainty principle.
Wave-particle duality states that all matter and energy exhibit both wave-like and particle-like characteristics. -Superposition allows quantum systems to exist in multiple states simultaneously until measured.
Quantization means physical quantities can only take discrete values, unlike classical mechanics.
The Uncertainty Principle, $\Delta x \Delta p \geq \frac{\hbar}{2}$, limits how precisely position and momentum can be known simultaneously.
Its mathematical framework relies on linear algebra, complex numbers, and differential equations.
Quantum system states are described by a wave function, $\Psi$.
The Schrödinger equation, $i\hbar\frac{\partial}{\partial t}\Psi(r, t) = \hat{H}\Psi(r, t)$, governs the time evolution of the wave function.
Applications span quantum computing, laser tech, semiconductor devices, and MRI.
Challenges involve quantum gravity, quantum-mech interpretation, and development of quantum technologies.

Lecture 7: October 12, 2023 - Linear Transformations

Vector spaces: revisited $\mathbb{R}^n, \mathbb{C}^n$, $M_{m \times n} (\mathbb{R})$ and $P(\mathbb{R})$.
A linear transformation is $T: V \rightarrow W$ such that $T(x + y) = T(x) + T(y)$ and $T(cx) = cT(x)$.
Zero Transformation: $T: V \rightarrow W$, $T(x) = 0$ for all $x \in V$.
Identity Transformation: $I: V \rightarrow V$, $I(x) = x$ for all $x \in V$.
Example: $T: \mathbb{R}^2 \rightarrow \mathbb{R}^3$, $T(x, y) = (x + y, x - y, y)$.
Derivative Transformation: $T: P(\mathbb{R}) \rightarrow P(\mathbb{R})$, $T(f(x)) = f'(x)$.
Transpose Transformation: $T: M_{n \times n}(\mathbb{R}) \rightarrow M_{n \times n} (\mathbb{R})$, $T(A) = A^t$.
If $T: V \rightarrow W$ is a linear transformation, then $T(0) = 0$, $T(-x) = -T(x)$ and $T(x - y) = T(x) - T(y)$.
$T(\sum_{i=1}^{n} a_i v_i) = \sum_{i=1}^{n} a_i T(v_i)$ holds for linear transformations.
Given a basis ${v_1, \dots, v_n}$ for $V$ and elements $w_1, \dots, w_n \in W$, a unique linear transformation $T: V \rightarrow W$ can be defined such that $T(v_i) = w_i$.

Bayes' Theorem

The Theorem describes the probability of an event, based on prior knowledge of conditions related to the event.
Bayes' Theorem formula: $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$.
$P(A|B)$ is the probability of A given B.
$P(B|A)$ is the probability of B given A.
$P(A)$ is the prior probability of A.
$P(B)$ is the prior probability of B.
Deduction of Bayes' Theorem from conditional probabilities: $P(A|B) = \frac{P(A \cap B)}{P(B)}$ and $P(B|A) = \frac{P(B \cap A)}{P(A)}$.
The example calculates the probability of a person having a disease given a positive test result with 99% accuracy and a disease prevalence of 1%. Even with a 99% accurate test, the probability of actually having the disease is only 50% due to low prevalence.