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Multivariable Calculus

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Divergence After Multiply a Function

MVCALC-41OMG1

Let $\rho(\vec{x}):=\|\vec{x}\|$ be the magnitude function, $\vec{x}\in \mathbb{R}^3$ and $\rho(\vec{x}) \neq 0$.

Which one of the following identities is true for all vector fields $\vec{F}$?

Assume all the vector fields are in three dimensional space $\mathbb{R}^3$.

A

$\text{div }(\rho \vec{F})=\|\vec{x}\| \text{div }\vec{F}+\cfrac{\vec{x}\cdot \vec{F}}{2\|\vec{x}\|}$

B

$\text{div }(\rho \vec{F})=\|\vec{x}\|^2 \text{div }\vec{F}+\vec{x}\cdot \vec{F}$

C

$\text{div }(\rho \vec{F})=\|\vec{x}\|^2 \text{div }\vec{F}-\vec{x}\cdot \vec{F}$

D

$\text{div }(\rho \vec{F})=\|\vec{x}\| \text{div }\vec{F}+\cfrac{\vec{x}\cdot \vec{F}}{\|\vec{x}\|}$

E

$\text{div }(\rho \vec{F})=\|\vec{x}\| \text{div }\vec{F}-\cfrac{\vec{x}\cdot \vec{F}}{\|\vec{x}\|}$