Exploring the Mathematical Conversion between ∆ and dX

In summary, the conversation discusses the use of the integral of dQ/T in thermodynamics, which is a path-independent function known as entropy. The mathematical expression for the conversion between delta and dX is given, along with the concept of integrating factor. The conversation ends with a clarification on the difference between definite and indefinite integrals, and the use of ΔS to represent the change in entropy between two points.
  • #1
mraptor
37
0
hi,

Reading a book on thermodynamics and the guy often uses something like this :

∫1/T dQ = ΔS

and then he says "this in differential form" :

dQ/T = dS

I kind of get the idea visually that one slice of "integral" will be dQ and you can think of it this way.
But my question is how do you mathematically express this conversion between delta <==> dX (both ways).
Some Examples would be nice.

thank you
 
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  • #2
hi mraptor! :smile:
mraptor said:
I kind of get the idea visually that one slice of "integral" will be dQ and you can think of it this way.
But my question is how do you mathematically express this conversion between delta <==> dX (both ways).

integrate dQ/T = dS, and obviously you get …

∫1/T dQ = ∫ dS :wink:​

then write ∫ dS = ∆S :smile:
 
  • #3
While I do not know much Physics

The integral of dQ/t in a reversible system is independent of path. This means that the integral defines a differentiable function on the thermodynamic phase space. This function I think is called entropy and its derivative is dQ/T.

The mathematical fact that you want is that when you have an integral that is path independent - AKA conservative - starting from any point one get a well defined function by integrating along any path from the starting point to another point. You should try to prove for yourself that this function can be differentiated.
 
  • #4
In other words, dQ is not an exact 1-form, it needs an integrating factor (1/T) to make it exact. Caratheodory re-wrote known (as of 1910) thermodynamics from the point of view of differential geometry (as known to him).
 
  • #5
tiny-tim said:
hi mraptor! :smile:integrate dQ/T = dS, and obviously you get …

∫1/T dQ = ∫ dS :wink:​

then write ∫ dS = ∆S :smile:

I thought :

∫ dS = S

not ΔS ! that is what I can't get..
and also what about the reverse conversion..
 
  • #6
mraptor said:
I thought :

∫ dS = S

not ΔS ! that is what I can't get..

ΔS is the change in S between the endpoints of the integral.
 
  • #7
pasmith said:
ΔS is the change in S between the endpoints of the integral.

So you are saying that he is implying that he is doing definite-integral, rather than indefinite-integral..
That is why he is getting ΔS, rather than S
 
  • #8
mraptor said:
… he is doing definite-integral, rather than indefinite-integral..
That is why he is getting ΔS, rather than S

yup! :biggrin:
 

1. What is the relationship between ∆ and dX?

The symbol ∆, often read as "delta", represents a change or difference in a variable. The symbol dX, often read as "delta X", represents an infinitesimal change in a variable. In mathematical terms, dX is the derivative of X with respect to another variable, which can be represented as dX/dY. Therefore, ∆ and dX are related through the concept of differentiation.

2. How is ∆ calculated?

The calculation of ∆ depends on the context in which it is used. In general, ∆ can be calculated by subtracting the initial value of a variable from its final value. For example, if the initial value of a variable is 5 and the final value is 10, then ∆ would be 10-5 = 5. In the context of calculus, ∆ can be calculated using the limit definition of a derivative.

3. What does dX/dY represent?

dX/dY represents the derivative of variable X with respect to variable Y. It can be thought of as the rate of change of X with respect to Y. In other words, it shows how much X changes when Y changes by a small amount (represented by dY).

4. How is dX calculated?

The calculation of dX also depends on the context in which it is used. In general, dX is calculated using the limit definition of a derivative, which involves taking the limit of a function as the change in the independent variable approaches 0. In other cases, dX may be given as a constant value or calculated using other mathematical methods.

5. What is the importance of exploring the mathematical conversion between ∆ and dX?

Understanding the relationship between ∆ and dX is crucial in many areas of mathematics and science. It allows us to analyze and model complex systems and phenomena using calculus. This conversion is also important in fields such as physics, engineering, and economics, as it helps us make predictions and solve problems that involve continuously changing variables.

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