Parametric, Polar & Vector

same calculus, fancier coordinate outfits 🎢

🌀 parametric equations

x = x(t), y = y(t) traces a curve. Eliminate t when easy; otherwise differentiate with:

dy/dx = (dy/dt) / (dx/dt), provided dx/dt ≠ 0

Second derivative (careful chain): d²y/dx² = d/dx (dy/dx) = [d/dt(dy/dx)] / (dx/dt)

r(t) = ⟨x(t), y(t)⟩ describes motion in the plane.

v(t) = r′(t) (velocity) a(t) = v′(t) (acceleration)

Speed = ‖v(t)‖ = √((dx/dt)² + (dy/dt)²). Distance traveled = ∫ ‖v(t)‖ dt.

Point described by (r, θ): x = r cos θ, y = r sin θ. Area of a polar "slice":

A = (1/2) ∫_α^β [r(θ)]² dθ

Convert carefully: slopes use dy/dx from x,y in terms of θ via chain rule on parametrics with t = θ.

dy/dx = (dy/dθ)/(dx/dθ) with x = r(θ)cos θ, y = r(θ)sin θ.

Polar arc length element: √(r² + (dr/dθ)²) dθ

last boss: sequences & series ∑