Undergrad Retaining Linear Terms of v and δ in Expansion Calculations

[Apashanka]

While doing some calculations I came across some terms which are $\frac{(v•\vec \nabla)v}{a(t)}$ and $\vec \nabla•[\rho(1+\delta)v]$ where all quantities have spatial dependence other than "a" which has only time dependence ,
the first term here is canceled and the the second term is rewritten as $\frac{\rho(\nabla•v)}{a}$ and the argument is given "retaining only linear terms of $v /\delta$"
Can anyone please help me in how to check this out??

 

[mathman]

Presumably $|\frac{\nu}{\delta}|$ is small.

 

[Apashanka]

Presumably $|\frac{\nu}{\delta}|$ is small.

Sorry I didn't got ,it's linear terms of "$v$" or "$\delta$"

 

[pasmith]

Sorry I didn't got ,it's linear terms of "$v$" or "$\delta$"

Why would "v/\delta" mean anything other than v divided by \delta? If you mean terms linear in two independent variablews, say "linear in v and \delta".

If |\epsilon| \ll 1 is small, then the most significant terms are those independent of \epsilon and those which are linear in \epsilon. Higher order terms can be neglected.

 

[mathman]

Sorry I didn't got ,it's linear terms of "$v$" or "$\delta$"

Linear in the ratio $\frac{\nu}{\delta}$.

 

[Mark44]

and the argument is given "retaining only linear terms of $v /\delta$"

Sorry I didn't got ,it's linear terms of "v" or "δ"

Linear in the ratio $\frac{\nu}{\delta}$.

I believe that linear in the ratio is the only reasonable meaning. I sincerely doubt that "/" should be interpreted as "or".