How do you derive differential area without specifying a variable?

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Discussion Overview

The discussion revolves around the derivation of differential area without specifying a variable, exploring the nature of differentials versus derivatives. Participants express confusion regarding the notation and the implications of taking derivatives without a clear variable reference.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about taking derivatives without specifying a variable, noting that traditional math books always relate derivatives to specific variables.
  • Another participant suggests that the confusion arises from misunderstanding the concept of a differential, which is distinct from a derivative.
  • A participant references the product rule for derivatives and mentions that terms can cancel out when all are taken with respect to the same variable.
  • There is a request for clarification on the cancellation of terms, indicating that some participants find this aspect particularly confusing.
  • One participant emphasizes that the notation used is for a differential, not a derivative, which is reiterated by another participant.
  • Participants discuss the manipulation of equations involving differentials and express uncertainty about the legitimacy of multiplying both sides by a differential element.
  • One participant warns that there are conditions under which such manipulations may not be permissible, highlighting the need for caution.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the understanding of differentials versus derivatives, with some expressing confusion and others attempting to clarify the concepts. Multiple viewpoints on the legitimacy of certain mathematical manipulations remain evident.

Contextual Notes

Participants have not fully resolved the assumptions underlying the use of differentials in this context, nor have they clarified the conditions under which certain manipulations are valid.

mfc5200
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Can someone please explain the steps to derive this? I've seen it everywhere, but the explanation is never given.

My confusion arises from the second line. In EVERY math book I've ever picked up, whenever you take a derivative, it is ALWAYS with respective to something, i.e y w.r.t x, or some variable w.r.t time, etc. But in this example as shown in textbooks, it seems as though they are taking a derivative, but it is not w.r.t anything. For example, instead of saying dy/dx, they would just be saying dy, which doesn't really make any sense to me. I'm confused about this.

I haven't seen a good explanation of this type of derivative yet. I attached an example. Thanks
 

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It's just a differential area. You have not paid close enough attention to your math books. :wink:
 
Last edited:
Look up the product rule for derivatives. The dot implies the time derivative. Since all terms on both sides are wrt dt it cancels out.
 
FredGarvin said:
Since all terms on both sides are wrt dt it cancels out.

Could you maybe say a little bit more about that. I think this is the part that is confusing me.
 
It's a differential, it's not a derivative.
 
mfc5200 said:
… But in this example as shown in textbooks, it seems as though they are taking a derivative, but it is not w.r.t anything. For example, instead of saying dy/dx, they would just be saying dy, which doesn't really make any sense to me. I'm confused about this.
Cyrus said:
It's a differential, it's not a derivative.

Hi mfc5200! :wink:

Does this help … http://en.wikipedia.org/wiki/Differential_(infinitesimal) ? :smile:
 
Suppose

m-dot =pVA
then
d(m-dot)/dt = pV dA/dt + VA dp/dt + pA dV/dt
Now multiply both sides by dt.
 
Bob S said:
Suppose

m-dot =pVA
then
d(m-dot)/dt = pV dA/dt + VA dp/dt + pA dV/dt
Now multiply both sides by dt.

Yea, I was able to get that far. I just wasn't sure if you were "allowed" to do that. I've never seen that done before.
 
mfc5200 said:
Yea, I was able to get that far. I just wasn't sure if you were "allowed" to do that. I've never seen that done before.

You're not always allowed to do this. So, be aware of that.
 

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