Dimensions, Vectors, Scalars

Dimensions of Physical Quantities

• Area is measured in $L^2$.
• Volume is measured in $L^3$.
• Speed is measured in $L/T$.
• Acceleration is measured in $L/T^2$.
• Dimensions represent the fundamental types of physical quantities.
• Units are the standards used to assign numerical values to dimensions.

Vector vs. Scalar

• A scalar is fully defined by its numerical value and units.
• A vector requires both magnitude and direction for complete specification.
• Examples of scalars include temperature, mass, and time.
• Examples of vectors include velocity, force, and displacement.

Component of a Vector

• A component of a vector cannot be greater than the vector's magnitude.
• The component is the projection of the vector onto an axis, with the maximum value equaling the vector's magnitude.

Unit Vectors

• Unit vectors provide a way to express a vector in terms of its components along coordinate axes.
• Vectors can be represented as $\vec{A} = A_x\hat{i} + A_y\hat{j}$.

Projectile Motion

• When a gun is fired horizontally at a falling monkey, the bullet will hit the monkey.
• Both bullet and monkey experience the same gravitational acceleration.

Newton's First Law

• An example is a soccer ball remaining at rest unless kicked.

Newton's Third Law

• During a mosquito hitting a car's windshield, they experience an equal force.

Static Friction

• Static friction opposes the start of motion.
• Kinetic friction is the force opposing motion once it has started.
• Static friction is generally greater than kinetic friction.

Work and Energy

• No work is done when carrying a heavy bag horizontally because there is no displacement in the direction of the force.

Potential Energy

• A ball thrown upwards has decreasing kinetic energy and increasing potential energy as it ascends.

Conservation of Energy

• Total mechanical energy will not be conserved if non-conservative forces are acting.
• Non-conservative forces like friction, dissipate energy as heat.

Center of Mass

• A baseball bat thrown in the air has a center of mass that follows a parabolic trajectory.

Momentum

• A particle with zero kinetic energy also has zero momentum.

Impulse

• Impulse can be zero if a force acts for a short time or if the net force is zero.

Elastic Collision

• Elastic collisions conserve both momentum and kinetic energy.
• Inelastic collisions conserve momentum, but not kinetic energy.

Planck's Law

• Planck's law describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium.
• The law was discovered by Max Planck in 1900 and is a foundational concept in quantum mechanics.
• Spectral radiance of a black body is proportional to the inverse fifth power of the wavelength.
• It is exponentially decreasing with the inverse of the product of wavelength and temperature.

Planck's Law Formula

• $B(\lambda, T) = \frac{2hc^2}{\lambda^5} \cdot \frac{1}{e^{\frac{hc}{\lambda k_B T}} - 1}$
• $B(\lambda, T)$ is the spectral radiance.
• $\lambda$ is the wavelength of the radiation.
• T is the absolute temperature of the black body.
• h is Planck's constant ($6.62607015 \times 10^{-34} J \cdot s$).
• c is the speed of light in a vacuum ($2.99792458 \times 10^8 m/s$).
• $k_B$ is Boltzmann's constant ($1.380649 \times 10^{-23} J/K$).

Planck's Law - Key Points

• As temperature increases, the total radiation increases, and the spectrum shifts to shorter wavelengths.
• Resolves the ultraviolet catastrophe, unexplained by classical physics, accurately describes black body radiation spectrum.

Planck's Law - Applications

• In astrophysics, used for determining the temperature of stars.
• In thermography, it helps in measuring temperature using infrared cameras.
• In lighting, it assists in designing efficient light sources.

Black Body Spectrum

• The intensity and peak wavelength emitted change with temperature.
• Higher temperatures produce higher peaks and shorter peak wavelengths.

Bernoulli's Principle

• States that an increase in fluid speed occurs simultaneously with a decrease in pressure or potential energy for inviscid flow.
• It can be derived from the principle of conservation of energy.
• It can be derived from Newton's second law.

Incompressible Flow Equation

• A common form of Bernoulli's equation, valid at any arbitrary point along a streamline, is: $v^{2}/2 + gz + p/ρ = constant $.
• v = fluid flow speed at a point on streamline.
• g = acceleration due to gravity.
• z = elevation of the point above a reference plane, with the positive z-direction pointing upward.
• p = pressure at the chosen point.
• $\rho$ = the density of the fluid at all points in the fluid.

Simplified Bernoulli's Equation

• Used when the change in the gz term along the streamline is so small it can be ignored.
• Aircraft in flight is a practical example.
• $p + 1/2 * ρv^{2} = constant $, where p is the static pressure of the fluid and $1/2 * ρv^{2}$ is dynamic pressure.

Compressible Flow Equation

• Bernoulli developed his principle from his observations on liquids, and his equation is only applicable to incompressible fluids.
• For compressible fluids, a similar equation can be derived for certain flow conditions.

Compressible Flow in Thermodynamics

• The most general form of Bernoulli's equation is: $\frac{v^{2}}{2} + Ψ + w = constant$
• $\frac{v^{2}}{2}$ is the kinetic energy per unit mass of the flow.
• Ψ is the potential energy per unit mass at the point considered.
• w is the enthalpy per unit mass at the point considered.
• Potential energy (Ψ) must me conservative.

Compressible Flow in Fluid Dynamics

• For an isentropic process, and assuming that changes in potential energy can be ignored a very useful form of the Bernoulli equation is: $p + ρv^{2}/2 = p_{0}$
• $p_{0}$ is "total pressure" and is constant along a streamline.
• p is the static pressure.
• $ρ$ is the density of the fluid.
• v is the flow speed

The Wave Equation

• Deals with waves in 1D, is related to a flexible string of length $L$.
• Displacement $u(x,t)$ is important.
• There is assumed to be small displacement in the $y$ direction.
• The tension $T$ is constant.
• The mass density $\rho$ is constant.

Derivation of the Wave Equation

• The forces acting on a small element of the string can be described by: $$\sum F_y = T \sin(\theta + d\theta) - T\sin(\theta) = (dm) \frac{\partial^2 u}{\partial t^2}$$
• For small angles $\sin(\theta) \approx \tan(\theta) = \frac{\partial u}{\partial x}$. Also, $dm = \rho dx$.
• $$T\left(\frac{\partial u}{\partial x}(x+dx, t) - \frac{\partial u}{\partial x}(x, t)\right) = \rho dx \frac{\partial^2 u}{\partial t^2}$$
• Gives rise to the final equation after simplification and calculus derivations: $$\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$$, where $c = \sqrt{T/\rho}$ is the speed of wave propagation.

Initial Conditions for the Wave Equation

• We need two initial conditions:
• $u(x, 0) = f(x)$
• $\frac{\partial u}{\partial t}(x, 0) = g(x)$

Boundary Conditions for the Wave Equation

• Three common types of boundary conditions:
• Dirichlet boundary condition
  • $u(0, t) = 0$
  • $u(L, t) = 0$
• Neumann boundary condition
  • $\frac{\partial u}{\partial x}(0, t) = 0$
  • $\frac{\partial u}{\partial x}(L, t) = 0$
• Robin boundary condition
  • $\frac{\partial u}{\partial x}(0, t) + au(0, t) = 0$
  • $\frac{\partial u}{\partial x}(L, t) + bu(L, t) = 0$

Solving Wave Equation by Seperation of Variables

• Let $u(x, t) = X(x)T(t)$
• Sub into the wave equation:
  • $X(x)T''(t) = c^2 X''(x)T(t)$
  • $\frac{T''(t)}{c^2T(t)} = \frac{X''(x)}{X(x)} = -\lambda$
• We get two ordinary differential equations:
  • $X''(x) + \lambda X(x) = 0$
  • $T''(t) + c^2 \lambda T(t) = 0$
• Solve the spatial equation first using the boundary conditions.

Example Wave Equation

• $$\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}, \quad 0 < x < L, \quad t>0$$
• $$u(0, t) = u(L, t) = 0$$
• $$u(x, 0) = x(L-x)$$
• $$\frac{\partial u}{\partial t}(x, 0) = 0$$

Algoritmos de Busca

Busca não Informada (Uninformed Search)

• Busca em Largura (BFS) - Breadth-First Search
  • Explores level by level.
  • Complete.
  • Sub-optimal (for uniform cost).
  • Time complexity: $O(b^d)$
• Busca em Profundidade (DFS) - Depth-First Search
  • Explores branch by branch.
  • Incomplete.
  • Sub-optimal.
  • Time complexity: $O(b^m)$
• Busca de Custo Uniforme - Uniform Cost Search
  • Expands the node with the lowest cost first.
  • Complete.
  • Optimal.
  • Time complexity: $O(b^{\lceil C*/\epsilon \rceil})$
• Busca em Profundidade Limitada - Depth-Limited Search
  • DFS with a depth limit.
  • Incomplete.
  • Sub-optimal.
  • Time complexity: $O(b^l)$
• Busca de Aprofundamento Iterativo (IDS) - Iterative Deepening Search
  • Iterative depth-limited search.
  • Complete.
  • Optimal (for uniform cost).
  • Time complexity: $O(b^d)$

Busca Informada (Informed Search)

• Busca Gulosa (Greedy) - Greedy Search
  • Expands the node "closest" to the goal.
  • Incomplete.
  • Sub-optimal.
  • Time complexity: $O(b^m)$
• Busca A - A Search**
  • Uses the function $f(n) = g(n) + h(n)$
  • Complete.
  • Optimal.
  • Time complexity: $O(b^m)$

Legenda (Legend)

• b: branching factor
• d: solution depth
• m: maximum search tree depth
• l: depth limit
• C*: optimal solution cost
• $\epsilon$: cost of the "cheapest" action

Matematicas

Derivación

Definición

• Derivation is an operation that relates a function $f$ to its rate of change with respect to its variable.

Notación

• $f'(x)$
• $\frac{df}{dx}$
• $\frac{d}{dx}f(x)$

Reglas de Derivación

Regla de la Potencia (Power Rule)

• The derivative of $x^n$, where $n$ is a constant, is:
• $\frac{d}{dx}(x^n) = nx^{n-1}$

Regla de la Suma/Resta (Sum/Subtract Rule)

• The derivative of a sum or subtraction of functions is the sum or subtraction of their derivatives:
• $\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$

Regla del Producto (Product Rule)

• The derivative of the product of two functions is:
• $\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$

Regla del Cociente (Quotient Rule)

• The derivative of the quotient of two functions is:
• $\frac{d}{dx}[\frac{f(x)}{g(x)}] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$

Regla de la Cadena (Chain Rule)

• The derivative of a composite function is:
• $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$

Derivadas de Funciones Comunes (Derivatives of Common Functions)

• Seno (Sine): $\frac{d}{dx}(sin(x)) = cos(x)$
• Coseno (Cosine): $\frac{d}{dx}(cos(x)) = -sin(x)$
• Tangente (Tangent): $\frac{d}{dx}(tan(x)) = sec^2(x)$
• Exponencial (Exponential): $\frac{d}{dx}(e^x) = e^x$
• Logaritmo Natural (Natural Logarithm): $\frac{d}{dx}(ln(x)) = \frac{1}{x}$

Ejemplos (Examples)

• Derivar $f(x) = 3x^2 + 2x - 1$
• $f'(x) = 6x + 2$
• Derivar $f(x) = sin(x) \cdot cos(x)$
• $f'(x) = cos^2(x) - sin^2(x)$
• Derivar $f(x) = (x^2 + 1)^3$
• $f'(x) = 3(x^2 + 1)^2 \cdot 2x = 6x(x^2 + 1)^2$

Aplicaciones (Applications)

• Finding the slope of a curve at a given point.
• Determining the maxima and minima of a function.
• Calculating the velocity and acceleration of a moving object.
• Optimizing designs and processes.