Module II·Article II·~1 min read
Polar and Spherical Coordinates
Lines and Planes
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Curvilinear Coordinate Systems
Polar Coordinates
A point on the plane: (r, φ), r ≥ 0, φ ∈ [0, 2π). Relation to Cartesian coordinates: x = r cosφ, y = r sinφ.
Curves: r = a (circle), φ = α (ray), r = aφ (Archimedean spiral), r = a(1+cosφ) (cardioid).
Area in polar coordinates: S = (1/2)∫_{φ₁}^{φ₂} r²(φ) dφ.
Conic sections in polar coordinates: r = p/(1−e cosφ), where e is the eccentricity. A single formula for the ellipse (e<1), parabola (e=1), hyperbola (e>1). Planetary orbits take exactly this form — Kepler's First Law.
Cylindrical Coordinates
(r, φ, z): x = r cosφ, y = r sinφ, z = z. Jacobian = r.
Equation of a cylinder around the z-axis: r = R.
Spherical Coordinates
(ρ, θ, φ): x = ρ sinθ cosφ, y = ρ sinθ sinφ, z = ρ cosθ. Jacobian = ρ² sinθ.
Equation of a sphere: ρ = R. Application in quantum mechanics: spherical harmonics — wave functions of the hydrogen atom in spherical coordinates.
Elliptic Coordinates
Applied when solving equations in regions bounded by ellipses and hyperbolas. Separation of variables in Laplace's equation.
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