Partial derivative query - guidance needed

[petertheta]

I have a question but have not seen an example or find anything in my textbooks so would love some advice on how to understand the problem.

Its a theory question on partial derivatives of the second order...

T=T(x,y,z,t) with x=x(t), y=y(t), z=z(t).

Find the second derivative of T wrt t

So, first find the first derivative to give:

$$\frac{dT}{dt} = \frac{\partial T}{\partial x}\frac{dx}{dt} +\frac{\partial T}{\partial y}\frac{dy}{dt}+\frac{\partial T}{\partial z}\frac{dz}{dt}+\frac{\partial T}{\partial t}$$

So I know the second derivative will be:
$$\frac{d}{dt}\left(\frac{dT}{dt}\right)$$

So now my question. How do i take derivatives of products of mixed derivatives and write it in a correct manner, albeit a theoretical one.

Your help means a lot.

PT

Homework Statement

Homework Equations

The Attempt at a Solution

 

[Ray Vickson]

I have a question but have not seen an example or find anything in my textbooks so would love some advice on how to understand the problem.

Its a theory question on partial derivatives of the second order...

T=T(x,y,z,t) with x=x(t), y=y(t), z=z(t).

Find the second derivative of T wrt t

So, first find the first derivative to give:

$$\frac{dT}{dt} = \frac{\partial T}{\partial x}\frac{dx}{dt} +\frac{\partial T}{\partial y}\frac{dy}{dt}+\frac{\partial T}{\partial z}\frac{dz}{dt}+\frac{\partial T}{\partial t}$$

So I know the second derivative will be:
$$\frac{d}{dt}\left(\frac{dT}{dt}\right)$$

So now my question. How do i take derivatives of products of mixed derivatives and write it in a correct manner, albeit a theoretical one.

Your help means a lot.

PT

Your dT/dt is a sum of four terms. Do you agree that you can take another d/dt of each term separately, then add the results? So, look at the first term
$$\frac{\partial T}{\partial x} \frac{dx}{dt}.$$ This has the form G(x,y,z)*v(t), where
$$G = \partial T/\partial x, \: v = dx/dt \text{ and } x = x(t), \:y = y(t), \:z = z(t).$$ Just apply your (d/dt) rule to G*v. Similarly for the other terms.

RGV

 

[petertheta]

Thanks RGV. So would the first "set" be:

$$\frac{\partial T}{\partial x}\frac{dx}{dt}+\frac{dx}{dt}\frac{dx}{dt}$$
or better still...
$$\frac{\partial T}{\partial x}\frac{dx}{dt}+\frac{d^2x}{dt^2}$$

 

[vela]

Nope. Try again. I suggest you write the derivative first in terms of G(x,y,z,t) and v(t) and then use the fact that G=dT/dx and v=dx/dt.

 

[petertheta]

OK. I'm struggling but I think this might be it...

$$\frac{\partial G}{\partial x}\frac{dx}{dt}+G\frac{dx}{dt}\frac{dx}{dt}$$

 

[vela]

Nope. It'll probably help to write out the variable dependencies explicitly. Applying the product rules gives
$$\frac{d}{dt}[G(x,y,z,t)v(t)] = \left(\frac{d}{dt} G(x,y,z,t)\right) v(t) + G(x,y,z,t)\frac{dv}{dt}.$$ Now look at your expression for dT/dt in your original post. You'll get the same sort of thing for dG/dt. I'll let you take it from here.

 

[petertheta]

I get a different result everytime i try :(

OK. Third time lucky! For the first "element"



$$\frac{\partial G}{\partial x}\frac{dx}{dt}v(t)+G\frac{dv}{dt}$$

Then substitute G and v for what is know...

$$\frac{\partial G}{\partial x}\frac{dx}{dt}\frac{dx}{dt}+\frac{\partial T}{\partial x}\frac{d}{dt}v$$
So
$$\frac{\partial^2 T}{\partial x^2}2\frac{dx}{dt}+\frac{\partial T}{\partial x}\frac{d^2x}{dt^2}+...$$

 

[vela]

You seem to keep missing the point: G is a function of y, z, and t as well as x.

Also, are you really claiming that $\frac{dx}{dt}\times\frac{dx}{dt} = 2\frac{dx}{dt}$?

 

[petertheta]

Im only writing for the very first term from the first derivative to see if I'm on the right track how does it look??...

Sorry that was a typo for 2 times dx/dt

 

[petertheta]

Ahh, I think I see your point. For the first term in the first derivative, when taking the second derivative I must do it for x,y,z and t. The the same for the second term... etc, etc... Is this correct??

 

[vela]

Yup.

 

[petertheta]

thats going to be a big answer! OK. So here we go just for the first term of the first derivative...

$$\frac{\partial^2T}{\partial x\partial x}\frac{dx}{dt}\frac{dx}{dt}+\frac{\partial T}{\partial x}\frac{d^2x}{dt^2} + \frac{\partial^2T}{\partial x\partial y}\frac{dy}{dt}\frac{dx}{dt}+\frac{\partial T}{\partial x}\frac{d^2x}{dt^2} + \frac{\partial^2T}{\partial x\partial z}\frac{dz}{dt}\frac{dx}{dt}+\frac{\partial T}{\partial x}\frac{d^2x}{dt^2} + \frac{\partial^2T}{\partial x\partial t}\frac{dt}{dt}\frac{dx}{dt}+\frac{\partial T}{\partial x}\frac{d^2x}{dt^2}...$$

 

[vela]

Almost. It should be
$$\left(
\frac{\partial^2T}{\partial x^2}\frac{dx}{dt} +
\frac{\partial^2T}{\partial y\partial x}\frac{dy}{dt} +
\frac{\partial^2T}{\partial z\partial x}\frac{dz}{dt} +
\frac{\partial^2T}{\partial t\partial x}\right)\frac{dx}{dt} +
\frac{\partial T}{\partial x}\frac{d^2x}{dt^2}$$ The last term only appears once.