Casual Use of Differentials

[GeoMath]

In another thread (now locked) "Dr.Rocket" and others offer a useful perspective on the use of differentials.

 

For example, it is concluded that dx/dy (in usual mathematics) is not a ratio. I have seen differentials defined to be real numbers - really just del(x), del(y) in which case they can be a ratio, but this is not what I want to ask about.

 

I see many cases in the physical sciences, including the earth sciences and especially in thermodynamics, where differentials are used in what I'll call "casual" (or short cut) derivations. I'll include an example below.

 

My particular interest is not to rain on anyone's parade; the contexts in which I find these derivations persuade me that the authors are not schlocks. I just seek some guidance/assistance in making such "derivations" a little more explicit. In the previous related thread, Dr.Rocket pointed out that many of these shortcuts can be made explicit by referring to the chain rule. I don't see how that might apply here, but maybe there are other implicit justifications that are being invoked.

 

In the following excerpt, we start with two simple equations involving differentials (isolated differentials - I don't understand the meaning we should give that, either!) and proceed to combine them into another equation and then integrate. Can we make this derivation explicit?

[edit: the editor is apparently not WYSIWG; I apologize for the awkward math typography.]

-------------------------------------------------------------------------------------------------------------------------

dA = kA A, dB = kB B, where k's are rate constants for the forward reaction. Assume no back reaction at all (e.g., dry wind blowing across a lake, so there's essentially no possibility that an evaporated H2O will get back into the lake).

but Kb/Ba = a ,

and dB/dA = a (B/A), 1/B (dB) = a 1/A (dA) ...integrate

getting ln (B/Bo) = a ln(A/Ao)

...

 

In another thread (now locked) "Dr.Rocket" and others offer a useful perspective on the use of differentials.

 

For example, it is concluded that dx/dy (in usual mathematics) is not a ratio. I have seen differentials defined to be real numbers - really just del(x), del(y) in which case they can be a ratio, but this is not what I want to ask about.

 

I see many cases in the physical sciences, including the earth sciences and especially in thermodynamics, where differentials are used in what I'll call "casual" (or short cut) derivations. I'll include an example below.

 

My particular interest is not to rain on anyone's parade; the contexts in which I find these derivations persuade me that the authors are not schlocks. I just seek some guidance/assistance in making such "derivations" a little more explicit. In the previous related thread, Dr.Rocket pointed out that many of these shortcuts can be made explicit by referring to the chain rule. I don't see how that might apply here, but maybe there are other implicit justifications that are being invoked.

 

In the following excerpt, we start with two simple equations involving differentials (isolated differentials - I don't understand the meaning we should give that, either!) and proceed to combine them into another equation and then integrate. Can we make this derivation explicit?

[edit: the editor is apparently not WYSIWG; I apologize for the awkward math typography.]

-------------------------------------------------------------------------------------------------------------------------

dA = kA A, dB = kB B, where k's are rate constants for the forward reaction. Assume no back reaction at all (e.g., dry wind blowing across a lake, so there's essentially no possibility that an evaporated H2O will get back into the lake).

but Kb/Ba = a ,

and dB/dA = a (B/A), 1/B (dB) = a 1/A (dA) ...integrate

getting ln (B/Bo) = a ln(A/Ao)

...

 

 

I do note that I can choose to interpret dA to mean dA/dt, etc. in which case the conclusion follows.

Edited October 2, 2012 by GeoMath

[Zorgoth]

In another thread (now locked) "Dr.Rocket" and others offer a useful perspective on the use of differentials.

 

For example, it is concluded that dx/dy (in usual mathematics) is not a ratio. I have seen differentials defined to be real numbers - really just del(x), del(y) in which case they can be a ratio, but this is not what I want to ask about.

 

I see many cases in the physical sciences, including the earth sciences and especially in thermodynamics, where differentials are used in what I'll call "casual" (or short cut) derivations. I'll include an example below.

 

My particular interest is not to rain on anyone's parade; the contexts in which I find these derivations persuade me that the authors are not schlocks. I just seek some guidance/assistance in making such "derivations" a little more explicit. In the previous related thread, Dr.Rocket pointed out that many of these shortcuts can be made explicit by referring to the chain rule. I don't see how that might apply here, but maybe there are other implicit justifications that are being invoked.

 

In the following excerpt, we start with two simple equations involving differentials (isolated differentials - I don't understand the meaning we should give that, either!) and proceed to combine them into another equation and then integrate. Can we make this derivation explicit?

[edit: the editor is apparently not WYSIWG; I apologize for the awkward math typography.]

-------------------------------------------------------------------------------------------------------------------------

dA = k

A A, dB = kB B, where k's are rate constants for the forward reaction. Assume no back reaction at all (e.g., dry wind blowing across a lake, so there's essentially no possibility that an evaporated H2O will get back into the lake).

but Kb/Ba

= a ,

and dB/dA = a (B/A), 1/B (dB) = a 1/A (dA) ...integrate

getting ln (B/Bo) = a ln(A/Ao)

...

 

 

 

 

I do note that I can choose to interpret dA to mean dA/dt, etc. in which case the conclusion follows.

 

 

This is an example of the method of separation of variables for ODEs, which is perfectly standard mathematics and is taught in any college level Differential Equations course.

 

Like the chain rule, it intuitively splits up the dx, dy, or what have you in a derivative like it was a ratio. As for what dx, dy, etc are, the precise definition comes from measure theory and/or differential geometry. For the non-mathematician, take things like separation of variables on faith and think of dx, dy, etc as an infinitesimal over which we sum when we integrate, like the delta(x) in the Riemann integral when the step size is tending to zero.