find grad log ∣r∣


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Question Number 55816 by Rio Mike last updated on 04/Mar/19

$f i n d g r a d l o g ∣ r ∣$
	Answered by tanmay.chaudhury50@gmail.com last updated on 04/Mar/19
	$grade ln∣r∣ ▽^→ =(i(∂/dx)+j(∂/∂y)+k(∂/∂z))(ln(√(x^2 +y^2 +z^2 )) ) =(1/2)i(∂/∂x)(ln(x^2 +y^2 +z^2 )+(1/2)j(∂/∂y)({ln(x^2 +y^2 +z^2 )}+ (1/2)k(∂/∂z){ln(x^2 +y^2 +z^2 )} =(1/2)[((2x)/(x^2 +y^2 +z^2 ))×i+((2yj)/(x^2 +y^2 +z^2 ))+((2zk)/(x^2 +y^2 +z^z ))] =((ix+jy+kz)/(x^2 +y^2 +z^2 )) =(r^→ /(∣r∣^2 ))$
	$g r a d e l n ∣ r ∣$ $\vec{▽} = (i \frac{\partial}{d x} + j \frac{\partial}{\partial y} + k \frac{\partial}{\partial z}) (l n \sqrt{x^{2} + y^{2} + z^{2}})$ $= \frac{1}{2} i \frac{\partial}{\partial x} (l n (x^{2} + y^{2} + z^{2}) + \frac{1}{2} j \frac{\partial}{\partial y} ({l n (x^{2} + y^{2} + z^{2})} +$ $\frac{1}{2} k \frac{\partial}{\partial z} {l n (x^{2} + y^{2} + z^{2})}$ $= \frac{1}{2} [\frac{2 x}{x^{2} + y^{2} + z^{2}} \times i + \frac{2 y j}{x^{2} + y^{2} + z^{2}} + \frac{2 z k}{x^{2} + y^{2} + z^{z}}]$ $= \frac{i x + j y + k z}{x^{2} + y^{2} + z^{2}}$ $= \frac{\vec{r}}{∣ r ∣^{2}}$
	Commented by otchereabdullai@gmail.com last updated on 04/Mar/19

	$wow! this is fantastic Prof chaudhury$
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