双球座標系
双球座標系のヤコビ行列は
∂ ( x , y , z ) ∂ ( σ , τ , φ ) = a ( cosh  τ − cos  σ ) 2 [ ( cos  σ cosh  τ − 1 ) cos  φ sin  σ sinh  τ cos  φ − ( cosh  τ − cos  σ ) sin  σ cos  φ ( cos  σ cosh  τ − 1 ) sin  φ sin  σ sinh  τ sin  φ ( cosh  τ − cos  σ ) sin  σ sin  φ sin  σ sinh  τ − cos  σ cosh  τ + 1 0 ] {\displaystyle {\frac {\partial (x,y,z)}{\partial (\sigma ,\tau ,\phi )}}={\frac {a}{(\cosh \tau -\cos \sigma )^{2}}}{\begin{bmatrix}(\cos \sigma \cosh \tau -1)\cos \phi &\sin \sigma \sinh \tau \cos \phi &-(\cosh \tau -\cos \sigma )\sin \sigma \cos \phi \\(\cos \sigma \cosh \tau -1)\sin \phi &\sin \sigma \sinh \tau \sin \phi &(\cosh \tau -\cos \sigma )\sin \sigma \sin \phi \\\sin \sigma \sinh \tau &-\cos \sigma \cosh \tau +1&0\\\end{bmatrix}}}