Transforming ∂T/∂t with chain rule using ξ=x/s(t) and T=h(t)F(ξ,t)

[Candy309]

Hi there

I am given xi=x/s(t) and T=h(t)F(xi,t) and I need to tranform deltaT/deltat. How do I do it? Do I use the chain rule? The answer to it is : s*(dh/dt)*F+s*h*(deltaF/deltat)-xi*(ds/dt)*h*(deltaF/deltaxi) but I don't know how to get this answer. Please help me. Thank you

 

[pwsnafu]

What's delta T / delta t? Is this from discreet math?

 

[Candy309]

Sorry it's $\deltaT$ /$\deltat$

 

[Candy309]

Hi It's the greek letter delta. I want to tranform (delta T)/(delta t)

 

[Candy309]

I have to solve a heat equation but first I must change the variables.

 

[Stephen Tashi]

I think pwsnafu is asking whether $\frac{ \delta T}{\delta t}$ means the derivative of T with respect to t.

 

[pwsnafu]

I think pwsnafu is asking whether $\frac{ \delta T}{\delta t}$ means the derivative of T with respect to t.

Yeah. I've never seen lower case delta used in that way before. Do you mean partial derivative?

 

[Candy309]

Yes it is

 

[Candy309]

Hi there

I am given xi=x/s(t) and T=h(t)F(xi,t) and I need to tranform deltaT/deltat. How do I do it? Do I use the chain rule? The answer to it is : s*(dh/dt)*F+s*h*(deltaF/deltat)-xi*(ds/dt)*h*(deltaF/deltaxi) but I don't know how to get this answer. Please help me. Thank you

How do you do the change of variable when xi=x-s(t)/1-s(t) and T=(1-s(t))*F(xi,t). I want to transform partial derivative of T with respect to t.

 

[jackmell]

Hi there

I am given xi=x/s(t) and T=h(t)F(xi,t) and I need to tranform deltaT/deltat. How do I do it? Do I use the chain rule? The answer to it is : s*(dh/dt)*F+s*h*(deltaF/deltat)-xi*(ds/dt)*h*(deltaF/deltaxi) but I don't know how to get this answer. Please help me. Thank you

That's really messy . . . Candy. Looks like you have a chained list of variables:

$$\Xi(x,s)=\frac{x}{s}$$

$$s=s(t)$$

$$T(h,F)=h(t)F(\Xi,t)$$

and you want to compute:

$$\frac{dT}{dt}$$

so by the general chain-rule:

$$\frac{dT}{dt}=h(t)\frac{\partial}{\partial t} F(\Xi,t)+F\frac{dh}{dt}$$

and:

$$\frac{\partial}{\partial t} F(\Xi,t)=\frac{\partial F}{\partial \Xi}\frac{\partial \Xi}{\partial t}+\frac{\partial F}{\partial t}$$

anyway, doing all that and simplifying, I still don't get exactly what you posted as the answer (close though) so maybe I'm missing something. Maybe though you can clean it up for me.

 

[Candy309]

The equation have to transform is d^2T/dx^2=dT/dt