Question

if r is a position vector of a point the div r is​

Answer 1

SOLUTION

GIVEN

$\vec{r}$ is a position vector

TO DETERMINE

The divergence of $\vec{r}$

EVALUATION

The divergence of a continuously Differentiable vector point function

$\vec{F}$ is denoted by $div \vec{F}$and defined as

$\displaystyle\nabla . \vec{F} = \bigg(\hat{i} \frac{ \partial \vec{F}}{ \partial x} + \hat{j} \frac{ \partial \vec{F}}{ \partial y} + \hat{k} \frac{ \partial \vec{F}}{ \partial z} \bigg)$

$Let \: \: \vec{r} = x \hat{i} + y \hat{j} + z \hat{k}$

Hence the required divergence

$\displaystyle = \nabla . \vec{r}$

$\displaystyle = \bigg(\hat{i} \frac{ \partial \vec{r}}{ \partial x} + \hat{j} \frac{ \partial \vec{r}}{ \partial y} + \hat{k} \frac{ \partial \vec{r}}{ \partial z} \bigg)$

$\displaystyle = \bigg( \frac{ \partial x}{ \partial x} + \frac{ \partial y}{ \partial y} + \frac{ \partial z}{ \partial z} \bigg)$

$\displaystyle =1 + 1 + 1$

$\displaystyle =3$

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