Thermal Energy Equation Term - Chain Rule

kevman90
Messages
3
Reaction score
2

Homework Statement


I am going through a derivation of the thermal energy equation for a fluid and am stumped on one of the steps. Specifically, the text I am using converts the term:

P/ρ*(Dρ/Dt)

to:

ρ*D/Dt(P/ρ) - DP/Dt

where:
ρ = density
P = pressure
D/Dt = material derivative

The text says this is done using the chain rule of differentiation but I can't derive it myself. I'm far removed from calculus so maybe I'm missing something simple but any help would be appreciated.

Homework Equations

The Attempt at a Solution

 
Physics news on Phys.org
One of the two expressions you have needs an extra minus sign. Momentarily, I will show the calculus of the second expression with the chain rule... @kevman90 Do you know how to take the derivative of ## \frac{d(uv)}{dt}##? It is ## u (\frac{dv}{dt}) +v(\frac{du}{dt}) ##. In this case, ## u=P ## and ## v=1/\rho ##. With the chain rule, ## \frac{dv}{dt}=(\frac{dv}{d \rho}) (\frac{d \rho}{dt}) ##. Do you know how to compute ## \frac{d v}{d \rho} ## ? With that, you should be able to process the second expression that you have, but I think you will find that it equals the minus of your first expression.
 
Last edited:
  • Like
Likes berkeman
Charles Link said:
One of the two expressions you have needs an extra minus sign. Momentarily, I will show the calculus of the second expression with the chain rule...
Sorry @Charles Link -- I was in the process of deleting the OP and warning for not showing enough work. But if you want to give a couple hints, that's probably okay.
 
Last edited:
This makes sense - didn't think about using the product rule. I will work through it later but I think I've got it. Also my mistake with the minus sign I forgot to include it out in front of the first term. Thanks!
 
  • Like
Likes berkeman and Charles Link
Thread 'Use greedy vertex coloring algorithm to prove the upper bound of χ'
Hi! I am struggling with the exercise I mentioned under "Homework statement". The exercise is about a specific "greedy vertex coloring algorithm". One definition (which matches what my book uses) can be found here: https://people.cs.uchicago.edu/~laci/HANDOUTS/greedycoloring.pdf Here is also a screenshot of the relevant parts of the linked PDF, i.e. the def. of the algorithm: Sadly I don't have much to show as far as a solution attempt goes, as I am stuck on how to proceed. I thought...

Similar threads

Replies
2
Views
4K
Replies
1
Views
995
Replies
3
Views
2K
Replies
3
Views
1K
Replies
3
Views
3K
Replies
4
Views
2K
Replies
4
Views
2K
Back
Top