If-v-r-m-where-r-x-2-y-2-z-2-show-that-2-v-x-2-2-v-y-2-2-v-z-2-m-m-1-r-m-2-

Question Number 135935 by Engr_Jidda last updated on 17/Mar/21

I f v = r^{m} w h e r e r = \sqrt{x^{2} + y^{2} + z^{2}}, s h o w t h a t

\frac{\partial^{2} v}{\partial x^{2}} + \frac{\partial^{2} v}{\partial y^{2}} + \frac{\partial^{2} v}{\partial z^{2}} = m (m - 1) r^{m - 2}

Answered by Olaf last updated on 17/Mar/21

v = r^{m} (x_{i}^{2}) in Einstein notation

\frac{\partial v}{\partial x_{j}} = {m r}^{m - 1} \frac{2 x_{j}}{2 r} = {m x}_{j} r^{m - 2}

\frac{\partial^{2} v}{\partial x_{j}^{2}} = m (r^{m - 2} + x_{j} (m - 2) r^{m - 3} \frac{2 x_{j}}{2 r})

\frac{\partial^{2} v}{\partial x_{j}^{2}} = {m r}^{m - 4} (r^{2} + (m - 2) x_{j}^{2})

In Einstein notation :

\frac{\partial^{2} v}{\partial x_{j}^{2}} = {m r}^{m - 4} (3 r^{2} + (m - 2) r^{2})

\frac{\partial^{2} v}{\partial x_{j}^{2}} = m (m + 1) r^{m - 2}

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