Explanation

Last verified on 05-06-2026

Explanation

∂²u/∂x² + ∂²u/∂y² = f(x,y) is the 2D Poisson equation.  

If f(x,y) = 0, then it becomes Laplace's equation: ∇²u = 0

Last verified on 05-06-2026

Explanation

Jacobian J = ∂(x,y)/∂(r,θ) = |∂x/∂r ∂x/∂θ|

                               |∂y/∂r ∂y/∂θ|

∂x/∂r = cosθ, ∂x/∂θ = -r sinθ  

∂y/∂r = sinθ, ∂y/∂θ = r cosθ

J = cosθ × r cosθ - (-r sinθ) × sinθ  

  = r cos²θ + r sin²θ  

  = r(cos²θ + sin²θ) = r

Last verified on 05-06-2026

Explanation

Sₙ = n/(n³+0) + n/(n³+1) + ... + n/(n³+n-1) = Σₖ₌₀ⁿ⁻¹ n/(n³ + k)

Each term ≤ n/n³ = 1/n²  

There are n terms → Sₙ ≤ n × 1/n² = 1/n → 0 as n→∞

Lower bound: n/(n³+n-1) × n = n²/(n³+n-1) → 0

By squeeze theorem, limit = 0

Last verified on 05-06-2026

Explanation

Orthogonal matrix definition: Aᵀ = A⁻¹  

Multiply both sides by A → AᵀA = I

Last verified on 05-06-2026