if-H-X-2-Y-2-Z-2-prove-2-H-X-2-2-H-Y-2-2-H-Z-2-2-H-

Question Number 59682 by aliesam last updated on 13/May/19

i f

H = X^{2} + Y^{2} + Z^{2}

p r o v e

\frac{\partial^{2} H}{\partial X^{2}} + \frac{\partial^{2} H}{\partial Y^{2}} + \frac{\partial^{2} H}{\partial Z^{2}} = \frac{2}{H}

Commented by mr W last updated on 13/May/19

H^{2} = X^{2} + Y^{2} + Z^{2}

2 H \frac{\partial H}{\partial X} = 2 X

H \frac{\partial H}{\partial X} = X \Rightarrow \frac{\partial H}{\partial X} = \frac{X}{H}

\frac{\partial H}{\partial X} \times \frac{\partial H}{\partial X} + H \frac{\partial^{2} H}{\partial X^{2}} = 1

\frac{X^{2}}{H^{2}} + H \frac{\partial^{2} H}{\partial X^{2}} = 1

\frac{X^{2} + Y^{2} + Z^{2}}{H^{2}} + H (\frac{\partial^{2} H}{\partial X^{2}} + \frac{\partial^{2} H}{\partial Y^{2}} + \frac{\partial^{2} H}{\partial Z^{2}}) = 3

1 + H (\frac{\partial^{2} H}{\partial X^{2}} + \frac{\partial^{2} H}{\partial Y^{2}} + \frac{\partial^{2} H}{\partial Z^{2}}) = 3

H (\frac{\partial^{2} H}{\partial X^{2}} + \frac{\partial^{2} H}{\partial Y^{2}} + \frac{\partial^{2} H}{\partial Z^{2}}) = 2

\Rightarrow \frac{\partial^{2} H}{\partial X^{2}} + \frac{\partial^{2} H}{\partial Y^{2}} + \frac{\partial^{2} H}{\partial Z^{2}} = \frac{2}{H}

Commented by aliesam last updated on 13/May/19

t h a n k y o u s i r

Answered by tanmay last updated on 13/May/19

p l s r e c h e c k q u e s t i o n \dots

Commented by aliesam last updated on 13/May/19

H^{2}

Commented by aliesam last updated on 13/May/19

= X^{2} + Y^{2} + Z^{2}

Answered by tanmay last updated on 13/May/19

x^{2} + y^{2} + z^{2} = H^{2}

H = \sqrt{x^{2} + y^{2} + z^{2}}

\frac{\partial H}{\partial x} = \frac{1}{2 \sqrt{x^{2} + y^{2} + z^{2}}} \times 2 x

\frac{\partial^{2} H}{\partial x^{2}} = \frac{\sqrt{x^{2} + y^{2} + z^{2}} \times \frac{\partial x}{\partial x} - x \times \frac{\partial}{\partial x} (\sqrt{x^{2} + y^{2} + z^{2}}}{(x^{2} + y^{2} + z^{2})}

\frac{\partial^{2} H}{\partial x^{2}} = \frac{\sqrt{x^{2} + y^{2} + z^{2}} - \frac{x}{2 \sqrt{x^{2} + y^{2} + z^{2}}} \times 2 x}{(x^{2} + y^{2} + z^{2})}

\frac{\partial^{2} H}{\partial x^{2}} = \frac{y^{2} + z^{2}}{{(x^{2} + y^{2} + z^{2})}^{\frac{3}{2}}}

s o \frac{\partial^{2} H}{\partial x^{2}} + \frac{\partial^{2} H}{\partial y^{2}} + \frac{\partial^{2} H}{\partial z^{2}}

= \frac{(y^{2} + z^{2}) + (x^{2} + z^{2}) + (x^{2} + z^{2})}{{(x^{2} + y^{2} + z^{2})}^{\frac{3}{2}}}

= \frac{2 (x^{2} + y^{2} + z^{2})}{{(x^{2} + y^{2} + z^{2})}^{\frac{3}{2}}}

. \frac{2 H^{2}}{{(H^{2})}^{\frac{3}{2}}}

= \frac{2}{H}

Commented by aliesam last updated on 13/May/19

g e n i u s

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