Material Derivative: Show F(x,y,z,t) Moves with Fluid

[coverband]

A fluid moves so that its velocity is [tex] \vec {u} \equiv (2xt,-yt,-zt) [/tex], written in rectangular Cartesian coordinates. Show that the surface F(x,y,z,t) = [tex] x^2exp (-2t^2)+(y^2+2z^2)exp(t^2)=constant [/tex] moves with the fluid (so that it always contains the same fluid particles; that is, DF/Dt=0)

Homework Equations

I got this from http://en.wikibooks.org/wiki/Marine..._Wave_Theory#Kinematic_Free_Surface_Condition

[tex] \frac{DF(x,y,z,t)}{Dt}=w. [/tex] If the surface is defined by z=A(x,y,t), then [tex] w = \frac {\partial A}{\partial t} + u \frac {\partial A}{\partial x}+v \frac{\partial A}{\partial y} [/tex] but I'm confused. If A refers to the surface, my surface has z's in it. Also what does the F stand for in the wiki equations

The Attempt at a Solution

 

[jdwood983]

I believe [itex]F[/itex] is also the surface. If you look at one of the prior sections, it says,

...the total derivative of the water surface will be zero, since we move with the surface. Thus on any surface,
[tex]\frac{DF(x,y,z,t)}{Dt}=0[/tex]

I would also assume that if your surface has a component of [itex]z[/itex] in it, you cannot use that form of the derivative and must use the total derivative.

 

[coverband]

Thanks very much