Directional Derivatives and Gradients

Directional Derivatives

• Measures the instantaneous rate of change of f at x₀ in the direction of û.
• Defined as D_û f (x₀) = lim ₕ→₀ (f (x₀ + hû) - f (x₀))/h, where f : ℝⁿ → ℝ.

Gradient

• The gradient of f at x₀ is a vector of its partial derivatives.
• Denoted ∇f (_x_₀) = (∂f/∂x₁ (x₀), ..., ∂f/∂xₙ (x₀)).
• Relates to the directional derivative by D_ûf (x₀) = ∇ f (x₀) ⋅ û if f is differentiable at x₀.

Key Facts about Gradients

• Points in the direction of the steepest ascent of f at x₀.
• Its magnitude, ||∇ f (x₀)||, equals the rate of change in the direction of that steepest ascent.
• The direction for steepest descent is -∇ f (x₀).
• Orthogonal to the level set of f at x₀.

Computing Directional Derivative

• To find the directional derivative of f (x, y) = x² + xy at (1, 2) in the direction of v = (3, 4):
• Compute the gradient: ∇f (x, y) = (2x + y, x).
• Evaluate at (1, 2): ∇f (1, 2) = (4, 1).
• Find the unit vector: û = (3/5, 4/5).
• Compute the directional derivative: D_û f (1, 2) = (4, 1) ⋅ (3/5, 4/5) = 16/5.

Tangent Planes

• For a level surface F(x, y, z) = c, its tangent plane at (x₀, y₀, z₀) is: Fₓ(x₀, y₀, z₀)(x - x₀) + F_y(x₀, y₀, z₀)(y - y₀) + F_z(x₀, y₀, z₀)(z - z₀) = 0.

Computing Tangent Plane

• Given the ellipsoid x² + 4y² + z² = 18, and the point (1, 2, 1):
• Define F(x, y, z) = x² + 4y² + z². The gradient is: ∇F(x, y, z) = (2x, 8y, 2z).
• Evaluate the gradient at (1, 2, 1): ∇F(1, 2, 1) = (2, 16, 2).
• The tangent plane is: 2(x - 1) + 16(y - 2) + 2(z - 1) = 0, which simplified to: x + 8y + z = 18.

Bernoulli's Principle

• States that fluid speed increases when pressure or potential energy decreases.
• Increasing air velocity in one region causes pressure decrease.
• Lift = _P_b - _P_a, lift is due to faster airflow on top of a wing and lower pressure above the wing.
• The Venturi effect is the reduction in fluid pressure that occurs when a fluid flows through a constricted section of pipe.

Game Theory Basics

• Game theory provides a mathematical framework for analyzing strategic interactions among decision-makers (players).
• Players choose different strategies, and their payoffs depend on everyone's actions.
• In the context of game theory, it is generally assumed that decision makers are rational.

Types of Games

• Games can be cooperative or non-cooperative.
• Games can be zero-sum or non-zero sum.
• Games can have complete or incomplete information

Algorithmic Game Theory (AGT) Defined

• Combines game theory and computer science, addressing computational aspects.
• Focuses on areas like algorithm design, complexity analysis, and mechanism design.

AGT Key Areas

• Equilibrium Computation: Finding stable states
• Mechanism Design: Designing games to achieve specific outcomes.
• Social choice: Aggregating individual preferences to make collective decisions.
• Network Games: Analyzing strategic interactions in networked systems.

Prisoner's Dilemma

• Two suspects are offered a deal: confess and the other silent results in freedom for the confessor and 10 years for the silent one otherwise.
• The rational choice is to confess, leading to both confessing, even though staying silent is a better outcome.
• The pursuit of individual rationality leads to a suboptimal collective outcome.

Applications

• Algorithmic game theory has applications in Internet Auctions, Network Routing, Social Networks, E-Commerce, and Resource Allocation.
• Used for analyzing and designing systems in environments in which there are incentives.
• Helps in understanding computation and incentives, enabling the creation of robust systems.

Electromagnetic (EM) Spectrum

• The spectrum includes all possible frequencies of EM radiation, from radio waves to gamma rays.

Key Properties of Electromagnetic Radiation

• Electromagnetic radiation possesses both wave- and particle-like characteristics.
• Defined as E = hf, photon energy (E) relates to frequency (f)
• Here, Planck's constant (h) is approximately 6.626 x 10⁻³⁴ J⋅s.
• Defined as c = λf, the speed of light in a vacuum (c) is approximately 3.00 x 10⁸ m/s.