How do you derive differential area without specifying a variable?

In summary, this conversation is about a derivative, which is not a derivative. It's a differential, which is used to calculate the change in a function over time.
  • #1
mfc5200
7
0
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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  • #2
It's just a differential area. You have not paid close enough attention to your math books. :wink:
 
Last edited:
  • #3
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.
 
  • #4
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.
 
  • #5
It's a differential, it's not a derivative.
 
  • #6
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:
 
  • #7
Suppose

m-dot =pVA
then
d(m-dot)/dt = pV dA/dt + VA dp/dt + pA dV/dt
Now multiply both sides by dt.
 
  • #8
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.
 
  • #9
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.
 

1. What is differential area?

Differential area refers to a small, infinitesimal area on a surface or in a plane. It is used in calculus to calculate the change in area as a function of another variable.

2. How is differential area calculated?

To calculate differential area, you can use the formula dA = |∂x/∂u x ∂y/∂v - ∂x/∂v x ∂y/∂u| du dv, where ∂x/∂u and ∂y/∂v are the partial derivatives of the surface, and du and dv are the infinitesimal changes in the parameters u and v.

3. What is the purpose of deriving differential area?

The purpose of deriving differential area is to find the change in area as a function of another variable, which is useful in many mathematical and scientific applications, such as calculating volumes, surface areas, and rates of change.

4. Can differential area be negative?

Yes, differential area can be negative. This is because it is a magnitude and does not have a direction. The sign of the differential area only indicates the orientation of the surface or plane with respect to the coordinate system being used.

5. What are some real-world applications of differential area?

Differential area has many real-world applications, such as calculating the rate of change of a physical quantity, finding the volume or surface area of irregular shapes, and determining the flux of a vector field through a surface. It is also used in fields such as engineering, physics, and economics.

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