separation constant
Noun
Meaning
separation constant (plural separation constants)
(calculus) a constant that may be introduced upon separation of variables
The partial differential equation
a
sin
2
(
θ
)
Φ
(
ϕ
)
+
Φ
(
ϕ
)
cos
(
θ
)
∂
Θ
(
θ
)
∂
θ
+
∂
2
Φ
∂
ϕ
2
=
0
{\displaystyle a\sin ^{2}(\theta )\Phi (\phi )+\Phi (\phi )\cos(\theta ){\frac {\partial \Theta (\theta )}{\partial \theta }}+{\frac {\partial ^{2}\Phi }{\partial \phi ^{2}}}=0}
can be rewritten
a
sin
2
(
θ
)
+
cos
(
θ
)
∂
Θ
(
θ
)
∂
θ
=
−
1
Φ
(
ϕ
)
∂
2
Φ
∂
ϕ
2
≡
K
{\displaystyle a\sin ^{2}(\theta )+\cos(\theta ){\frac {\partial \Theta (\theta )}{\partial \theta }}=-{\frac {1}{\Phi (\phi )}}{\frac {\partial ^{2}\Phi }{\partial \phi ^{2}}}\equiv K}
, where
K
{\displaystyle K}
is a separation constant that depends on neither
θ
{\displaystyle \theta }
nor
ϕ
{\displaystyle \phi }
. This then yields two ordinary differential equations.