Question

$\rm If \: u = \sqrt{ {x}^{2} + {y}^{2} + {z}^{2} } ,prove \: that \\ \rm \frac{ { \partial}^{2}u }{ \partial {x}^{2} } + \frac{ { \partial}^{2}u }{ \partial y^{2} } + \frac{ { \partial}^{2}u }{ \partial z^{2} } = \frac{2}{u}$​

Answer 1

$\large\underline{\sf{Solution-}}$

Given function is

$\rm \: u = \sqrt{ {x}^{2} + {y}^{2} + {z}^{2} }$

On squaring both sides, we get

$\rm \: {u}^{2} = {x}^{2} + {y}^{2} + {z}^{2}$

On differentiating partially w. r. t. x, we get

$\rm \: \dfrac{\partial }{\partial x}{u}^{2} = \dfrac{\partial }{\partial x}({x}^{2} + {y}^{2} + {z}^{2} )$

$\rm \: 2u\dfrac{\partial u}{\partial x} = 2x + 0 + 0$

$\rm\implies \:\dfrac{\partial u}{\partial x} = \dfrac{x}{u} - - - (1)$

On differentiating partially w. r. t. x, we get

$\rm \: \dfrac{\partial }{\partial x}\bigg(\dfrac{\partial u}{\partial x}\bigg) = \dfrac{\partial }{\partial x}\bigg( \dfrac{x}{u}\bigg)$

$\rm \: \dfrac{\partial^{2}u}{\partial {x}^{2} } = \dfrac{u\dfrac{\partial }{\partial x}x - x\dfrac{\partial }{\partial x}u}{ {u}^{2} }$

$\rm \: \dfrac{\partial^{2}u}{\partial {x}^{2} } = \dfrac{u \times 1 - x\dfrac{\partial u}{\partial x}}{ {u}^{2} }$

$\rm \: \dfrac{\partial^{2}u}{\partial {x}^{2} } = \dfrac{u - x \times \dfrac{x}{u} }{ {u}^{2} }$

$\rm \: \dfrac{\partial^{2}u}{\partial {x}^{2} } = \dfrac{ \dfrac{ {u}^{2} - {x}^{2} }{u} }{ {u}^{2} }$

$\rm\implies \:\rm \: \dfrac{\partial^{2}u}{\partial {x}^{2}} =\dfrac{ {u}^{2} - {x}^{2} }{ {u}^{3} } - - - (1)$

Similarly,

$\rm\implies \:\rm \: \dfrac{\partial^{2}u}{\partial {y}^{2}} =\dfrac{ {u}^{2} - {y}^{2} }{ {u}^{3} } - - - (2)$

Similarly,

$\rm\implies \:\rm \: \dfrac{\partial^{2}u}{\partial {z}^{2}} =\dfrac{ {u}^{2} - {z}^{2} }{ {u}^{3} } - - - (3)$

On adding equation (1), (2) and (3), we get

$\rm \: \dfrac{\partial^{2}u}{\partial {x}^{2}} + \dfrac{\partial^{2}u}{\partial {y}^{2}} + \dfrac{\partial^{2}u}{\partial {z}^{2}}$

$\rm \: = \: \dfrac{ {u}^{2} - {x}^{2} }{ {u}^{3} } + \dfrac{ {u}^{2} - {y}^{2} }{ {u}^{3} } + \dfrac{ {u}^{2} - {z}^{2} }{ {u}^{3} }$

$\rm \: = \: \dfrac{ {u}^{2} - {x}^{2} + {u}^{2} - {y}^{2} + {u}^{2} - {z}^{2} }{ {u}^{3} }$

$\rm \: = \: \dfrac{3{u}^{2} - ({x}^{2} + {y}^{2}+ {z}^{2} )}{ {u}^{3} }$

$\rm \: = \: \dfrac{3{u}^{2} - {u}^{2} }{ {u}^{3} }$

$\rm \: = \: \dfrac{2{u}^{2}}{ {u}^{3} }$

$\rm \: = \: \dfrac{2}{u}$

Hence,

$\rm\implies \:\boxed{ \rm{ \:\frac{ { \partial}^{2}u }{ \partial {x}^{2} } + \frac{ { \partial}^{2}u }{ \partial y^{2} } + \frac{ { \partial}^{2}u }{ \partial z^{2} } = \frac{2}{u} \: }}$