Change of Variables: Transform DeltaT/DeltaT with Chain Rule

In summary: So, using the chain rule, you would use:s*(dh/dt)*F+s*h*(deltaF/deltat)-xi*(ds/dt)*h*(deltaF/deltaxi)
  • #1
Candy309
7
0
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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  • #2
What's delta T / delta t? Is this from discreet math?
 
  • #3
Sorry it's [itex]\deltaT[/itex] /[itex]\deltat[/itex]
 
  • #4
Hi It's the greek letter delta. I want to tranform (delta T)/(delta t)
 
  • #5
I have to solve a heat equation but first I must change the variables.
 
  • #6
I think pwsnafu is asking whether [itex] \frac{ \delta T}{\delta t} [/itex] means the derivative of T with respect to t.
 
  • #7
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?
 
  • #8
Yes it is
 
  • #9
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
 

1. What is the purpose of a change of variables in scientific equations?

The purpose of a change of variables is to simplify complex equations and make them easier to solve. It involves replacing one set of variables with another set that is more convenient to work with, often leading to a more efficient and accurate solution.

2. How does the chain rule apply to a change of variables?

The chain rule is used to calculate the derivative of a function with respect to a new set of variables. It allows us to express the derivative of a composite function in terms of the derivatives of its individual components.

3. Can you provide an example of a change of variables using the chain rule?

One example is the transformation of polar coordinates to Cartesian coordinates. In this case, the chain rule is used to express the partial derivatives of the Cartesian variables in terms of the partial derivatives of the polar variables.

4. What is the significance of the Jacobian matrix in a change of variables?

The Jacobian matrix represents the change in variables from one coordinate system to another. It is used to calculate the determinant of the matrix, which is essential in determining the volume element and integrating over a new set of variables.

5. Are there any limitations to using a change of variables?

In some cases, a change of variables may not be possible or may not result in a simpler equation. It may also introduce new complexities, such as singularities or discontinuities, that can affect the accuracy of the solution. Therefore, it is important to carefully consider the appropriateness of a change of variables in each situation.

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