# Math Physics-Equation of Continuity

physics2112

## Homework Statement

PARTA:
Consider a fluid in which $$\rho$$ = $$\rho$$(x,y,z,t); that is the density varies from point to point and with time. The velocity of this fluid at a point is
v= (dx/dt, dy/dt, dz/ dt)
Show that
dp/dt = $$\partial$$t$$\rho$$ + v $$\cdot$$ $$\nabla\rho$$

PARTB:
Combine the above equation with the equation of continuity and prove that
$$\rho\nabla$$$$\cdot$$ v + d$$\rho$$ /dt = 0

I have been attempting this problem for over a week. If anyone can solve this problem or help me out I would really appreciate it!

(Between the $$\nabla$$ and v is a dot but I am not sure if it is showing up!)

xboy
For part a, how'd you write the total time derivative of rho in terms of its partial derivatives?

For part b, what do you think the relevant equations are?

physics2112
I used the symbols to write out rho with a subscript...is that what you meant?
That is the equation we have to show...so I have no clue how they got it mathematically.

For the second your suppose to use the equation of continuity...which I'm not sure which one they mean as I find various examples of it.

xboy
What I meant is this : suppose I have a function f(x,y). Can you write down the total derivative of f w.r.t x i.e df/dx in terms of the partial derivatives $$\partial_{x}f$$ and $$\partial_{y}f$$ ? If you can do that for f and x, go ahead and do it for rho and t.

If you aren't sure of which equation of continuity to use, why not let's go ahead and derive it! The key concept is conservation of mass. Let me know if you'd like a bit of help with that.