Change of Variables: Transform DeltaT/DeltaT with Chain Rule

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Discussion Overview

The discussion revolves around transforming the expression \(\delta T/\delta t\) using the chain rule in the context of a heat equation. Participants explore the implications of variable changes and the application of derivatives in this transformation.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant presents a transformation involving \(\xi = x/s(t)\) and \(T = h(t)F(\xi, t)\) and seeks guidance on how to derive \(\delta T/\delta t\) using the chain rule.
  • Another participant questions the meaning of \(\delta T/\delta t\) and suggests it may refer to a derivative with respect to \(t\), while others clarify that it is indeed a partial derivative.
  • A participant expresses confusion over the notation and suggests that the expression may be interpreted differently, particularly regarding the use of lower case delta.
  • Further elaboration on the chain rule is provided, including the relationship between \(T\), \(F\), and the derivatives involved, but there is no consensus on the exact form of the transformation or the correctness of the proposed answer.
  • Another participant introduces a different transformation scenario involving \(\xi = x - s(t)/(1 - s(t))\) and seeks similar guidance on transforming the partial derivative of \(T\) with respect to \(t\).
  • One participant mentions that they are unable to arrive at the proposed answer despite following the chain rule, indicating potential discrepancies in the calculations or interpretations.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the transformation process or the correctness of the proposed answer. There are multiple interpretations of the notation and differing opinions on the application of the chain rule.

Contextual Notes

There are unresolved aspects regarding the assumptions made in the transformations, the definitions of the variables involved, and the notation used for derivatives. The discussion reflects varying levels of familiarity with the concepts and methods being applied.

Candy309
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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
 
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What's delta T / delta t? Is this from discreet math?
 
Sorry it's [itex]\deltaT[/itex] /[itex]\deltat[/itex]
 
Hi It's the greek letter delta. I want to tranform (delta T)/(delta t)
 
I have to solve a heat equation but first I must change the variables.
 
I think pwsnafu is asking whether [itex]\frac{ \delta T}{\delta t}[/itex] means the derivative of T with respect to t.
 
Stephen Tashi said:
I think pwsnafu is asking whether [itex]\frac{ \delta T}{\delta t}[/itex] 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?
 
Yes it is
 
Candy309 said:
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.
 
  • #10
Candy309 said:
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:

[tex]\Xi(x,s)=\frac{x}{s}[/tex]

[tex]s=s(t)[/tex]

[tex]T(h,F)=h(t)F(\Xi,t)[/tex]

and you want to compute:

[tex]\frac{dT}{dt}[/tex]

so by the general chain-rule:

[tex]\frac{dT}{dt}=h(t)\frac{\partial}{\partial t} F(\Xi,t)+F\frac{dh}{dt}[/tex]

and:

[tex]\frac{\partial}{\partial t} F(\Xi,t)=\frac{\partial F}{\partial \Xi}\frac{\partial \Xi}{\partial t}+\frac{\partial F}{\partial t}[/tex]

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.
 
  • #11
The equation have to transform is d^2T/dx^2=dT/dt
 

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