The distance of the variable point P which has coordinates x, y, z from the fixed points (0, 0, 1) and (0, 0, -1) are denoted by u and v respectively. New variable ξ, η, ϕ are defined by \(ξ = \frac{1}{2}(u+v), η = \frac{1}{2}(u-v)\) and ϕ is the angle between the plane y = 0 and the plane containing the three points, i.e. \(ϕ = \tan^{-1}\left(\frac{y}{x}\right)\) over 1 ≤ ξ < ∞, -1 ≤ η < 1, 0 ≤ ϕ < 2π. The Jacobian of \(\frac{\partial (\xi, \eta, \phi)}{\partial (x, y, z)}\) has the value , then \(\int\int\int_{all\:space}\frac{u-v)^2}{uv}\:exp\left(-\frac{u+v}{2}\right)dxdydz=\)

Concept:

\(\frac{\partial (\xi, \eta, \phi)}{\partial (x, y, z)}\times \frac{\partial (x, y, z)}{\partial (\xi, \eta, \phi)}=1\)

\({1\over ζ^2-\eta^2}\times \frac{\partial (x, y, z)}{\partial (\xi, \eta, \phi)}=1\)

\( \frac{\partial (x, y, z)}{\partial (\xi, \eta, \phi)}=ζ^2-\eta^2\) .............(i)

Calculation:

Given, \(ξ = \frac{1}{2}(u+v)\)

 \(u+v=2ξ\)...........(ii)

\(η = \frac{1}{2}(u-v)\)

\( u-v=2η\) ............(iii)

Solving (ii) and (iii), we get:

\(u=ξ+\eta\) and \(v=ξ-\eta\)

\(J= \frac{\partial (x, y, z)}{\partial (\xi, \eta, \phi)}=ζ^2-\eta^2=uv\) .......(iv)

Putting values of equations (ii), (iii), and (iv), we get:

\(=\int\int\int\frac{(u-v)^2}{uv}\:e^\left(-\frac{u+v}{2}\right)dx\space dy\space dz\)

\(=\int_{\zeta=1}^{\infty}\int_{\eta=-1}^{1}\int_{\phi=0}^{2\pi}4\eta^2 d\zeta \space d\eta \space d\phi\)​​

\(=\frac{16\pi}{3e}\)