Ψ λ ∞ ∂

Mike Tate Mathematics

Mathematical Universe

e^(iπ) + 1 = 0 • φ = (1+√5)/2 • ∇⋅E = ρ/ε₀ • F = G(m₁m₂)/r² • ζ(s) = Σ 1/n^s • ∂²ψ/∂t² = c²∇²ψ • E = mc² • a² + b² = c² • ∫e^x dx = e^x + C • ∇×B = μ₀J + μ₀ε₀∂E/∂t • Γ(z) = ∫₀^∞ t^(z-1)e^(-t)dt • ΔxΔp ≥ ħ/2 • HΨ = iħ ∂Ψ/∂t • PV = nRT • ∇²φ = 0 • S = k log W • λ = h/p • R_{μν} – ½Rg_{μν} = 8πT_{μν} •

MikeTateMath.org

Solutions to open problems

Archimedes Screw

$Archimedes Screw Mathematical Principle$

Ancient Greek technology demonstrating fundamental principles of physics and mathematics

Mathematical Principles Explained:

E = p/s, P = G(m₁m₂)/r² · ζ(s) = Σ 1/n^s

Energy to Power Ratio: E = p/s represents the relationship between energy (E), momentum (p), and displacement (s) in mechanical systems

Universal Gravitation: P = G(m₁m₂)/r² shows Newton’s law where gravitational force depends on mass product and inverse square distance

Zeta Function: ζ(s) = Σ 1/n^s connects number theory with physics through the Riemann zeta function

Golden Ratio Explorer

Discover the mathematical constant φ = (1+√5)/2 that appears throughout nature, art, and architecture. This irrational number approximately equals 1.618 and creates aesthetically perfect proportions.

Click to reveal the magic of φ

Euler’s Identity

e^(iπ) + 1 = 0 connects five fundamental mathematical constants in one elegant equation. Considered the most beautiful formula in mathematics, linking exponential growth, imaginary numbers, and circular motion.

Click to understand mathematical beauty

Fundamental Theorem

∫e^x dx = e^x + C demonstrates the intimate relationship between differentiation and integration. The exponential function is unique in being its own derivative and integral.

Click to explore calculus foundations