Operadores diferenciales - OCW UPM

Factor de escala en coordenadas esféricas
Los factores de escala son
2 2 2
⎛∂x⎞ ⎛∂y⎞ ⎛∂z⎞ 2 2 2
h h1 = h hr = ⎜ ⎟ + ⎜ ⎟ + ⎜ ⎟ = ( sen senθθ cos ϕϕ ) + ( sen senθθ sen senϕϕ
) + (cos ϕϕ
) = 1
⎝∂r ⎠ ⎝∂r ⎠ ⎝∂r⎠ 2 2 2
⎛ ∂x ⎞ ⎛ ∂y ⎞ ⎛ ∂z
⎞
2 2 2
h2 = hθ = ⎜ ⎟ + ⎜ ⎟ + ⎜ ⎟ = ( r cosθcosϕ) + ( r cos θsenϕ) + ( − rsenθ) = r
⎝∂θ ⎠ ⎝∂θ ⎠ ⎝∂θ ⎠
2 2 2
⎛ ∂x ⎞ ⎛ ∂y ⎞ ⎛ ∂z
⎞
2 2
h3 = hϕ = ⎜ ⎟ + ⎜ ⎟ + ⎜ ⎟ = ( − rsenθ senϕ) + ( rsenθcosϕ) = rsenθ
⎝ ∂ ∂ϕϕ ⎠ ⎝ ∂ ∂ϕϕ ⎠ ⎝ ∂ ∂ϕϕ
⎠