Problem about rate of change (multivariable calculus)

[supermiedos]

Homework Statement

Let T = f(x, y, z), where dx/dt = 4, dy/dt = 4 and dz/dt = -3
Calculate dT/dt if dT/dx = 4, dT/dy = 7 and dT/dz = 9

Homework Equations

dT/dt = ∂T/∂x (dx/dt) + ∂T/∂y (dy/dt) + ∂T/∂z (dz/dt)

The Attempt at a Solution

I tried to get an explicit formula for T. I thought that, if dT/dx = 4, then T = 4x + c. Similarly, T = 7y + c and T = 9z + c. If i add the above expressions, I get:

3T = 4x + 7y + 9z + C, so, T would be T = 4x/3 + 7y/3 + 3z + C, and I could use the chain rule and just substitute dx/dt and so on...

Is my reasoning fine?

 

[voko]

Why do you need to do anything about T?

You have the equation for dT/dt, and you know the values of everything on its right-hand side.

 

[HallsofIvy]

First, you can't get an explicit expression for T, there are infinitely many possibililties. But you don't need that, just the "chain rule"":
$$\frac{dT}{dt}= \frac{\partial T}{\partial x}\frac{dx}{dt}+ \frac{\partial T}{\partial y}\frac{dy}{dt}+ \frac{\partial T}{\partial z}\frac{dz}{dt}$$

 

[supermiedos]

But I need ∂T/∂x and I have dT/dx = 4 (same for the others). Don't tell me ∂T/∂x = dT/dx, because I'm going to cry

 

[Karnage1993]

Using d when you have a function with more than one variable is not correct notation. ∂T/∂x is the correct notation.

 

[supermiedos]

Omg you are right. My teacher wrote it like dT/dx, and I never realized he wrote it wrong. Thank you for your help guys