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If
r=| vec(r)|, where
vec(r)=x hat(ı)+y hat(ȷ)+z hat(k), then prove that
vec(grad).
*** END OF PAPER ***

If $r=|\vec{r}|$ , where $\vec{r}=x \hat{\imath}+y \hat{\jmath}+z \hat{k}$ , then prove that \vec{\(\newline\)abla} . $\newline$ * END OF PAPER *

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Simplify radical expressions with variables II

Full solution

Q. If

r=|\vec{r}|

, where

\vec{r}=x \hat{\imath}+y \hat{\jmath}+z \hat{k}

, then prove that \vec{\(\newline\)abla} .

\newline

*** END OF PAPER ***

Identify vector and magnitude: Identify the vector $\vec{r}$ and its magnitude $r$ . $\newline$ $\vec{r} = x \hat{i} + y \hat{j} + z \hat{k}$ $\newline$ $r = \sqrt{x^2 + y^2 + z^2}$
Calculate gradient of $r$ : Calculate the gradient of $r$ .
\(\newlineabla(r) = $\newline$ abla(\sqrt{x^2 + y^2 + z^2})\)
Using the chain rule, \(\newlineabla(r) = \frac{1}{2}(2x, 2y, 2z) / \sqrt{x^2 + y^2 + z^2}\)
= $\left(\frac{x}{r}, \frac{y}{r}, \frac{z}{r}\right)$
Recognize unit vector: Recognize that $(\frac{x}{r}, \frac{y}{r}, \frac{z}{r})$ is the unit vector of $\vec{r}$ . Since $r = \sqrt{x^2 + y^2 + z^2}$ , dividing each component by $r$ normalizes the vector. Thus, $(\frac{x}{r}, \frac{y}{r}, \frac{z}{r}) = \frac{\vec{r}}{r}$

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