show-that-2-log-r-1-r-

Question Number 47189 by 23kpratik last updated on 06/Nov/18

s h o w t h a t ▽^{2} (l o g r) = 1 / r

Commented by Joel578 last updated on 06/Nov/18

w h a t i s ▽ s y m b o l m e a n ?

Commented by ajfour last updated on 06/Nov/18

▽ i s g r a d i e n t

▽^{2} i s l a p l a c i a n .

Answered by tanmay.chaudhury50@gmail.com last updated on 06/Nov/18

\vec{r} = i x + j y + k z

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

l n r = \frac{1}{2} l n (x^{2} + y^{2} + z^{2})

▽^{2} = \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial z^{2}} + \frac{\partial^{2}}{\partial z^{2}}

\frac{\partial^{2}}{\partial x^{2}} {\frac{1}{2} l n (x^{2} + y^{2} + z^{2})

\frac{\partial}{\partial x} {\frac{\partial}{\partial x} \frac{1}{2} l n (x^{2} + y^{2} + z^{2})}

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

= \frac{(x^{2} + y^{2} + z^{2}) \times 1 - x (2 x)}{{(x^{2} + y^{2} + z^{2})}^{2}}

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

t h u s o n a d d i t i o n w e g e t

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

= \frac{x^{2} + y^{2} + z^{2}}{{(x^{2} + y^{2} + z^{2})}^{2}} = \frac{1}{x^{2} + y^{2} + z^{2}} = \frac{1}{r^{2}}