Exercise [10.07]

Sameed Zahoor · 14 Jul 2008, 10:37

$\xi=A\frac{\partial}{\partial{X}}+B\frac{\partial}{\partial{Y}}$

$\Rightarrow a\frac{\partial}{\partial{x}}+b\frac{\partial}{\partial{y}}=A(\frac{\partial{x}}{\partial{X}}\frac{\partial}{\partial{x}}+\frac{\partial{y}}{\partial{X}}\frac{\partial}{\partial{y}})+B(\frac{\partial{x}}{\partial{Y}}\frac{\partial}{\partial{x}}+\frac{\partial{y}}{\partial{y}}\frac{\partial}{\partial{y}})$

$\Rightarrow a\frac{\partial}{\partial{x}}+b\frac{\partial}{\partial{y}}=(A\frac{\partial{x}}{\partial{X}}+B\frac{\partial{x}}{\partial{Y}})\frac{\partial}{\partial{x}}+(A\frac{\partial{y}}{\partial{X}}+B\frac{\partial{y}}{\partial{Y}})\frac{\partial}{\partial{y}}$

Comparing b/s we get,

$a=A\frac{\partial{x}}{\partial{X}}+B\frac{\partial{x}}{\partial{Y}}$

$b=A\frac{\partial{y}}{\partial{X}}+B\frac{\partial{y}}{\partial{Y}}$

By analogy,

$A=a\frac{\partial{X}}{\partial{x}}+b\frac{\partial{X}}{\partial{y}}$

$B=a\frac{\partial{Y}}{\partial{x}}+b\frac{\partial{Y}}{\partial{y}}$