Questions involving differentials (again)

In summary, Kleppner is saying that there are two methods for deriving the kinematic equations: one that uses the substitution rule for integration, and another that does not. He also mention that when people integrate the differential, the variable that is being integrated with respect to is already included in the differential.
  • #1
autodidude
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What is the change of variables using differentials trick K&K are referring to here?

http://books.google.com.au/books?id...f variables differentials intractable&f=false

(about halfway down the page)

Are there any formalities behind this?

---

Also, when people derive the kinematic equations using calculus? I notice they rely on differentials

e.g.
http://physics.info/kinematics-calculus/

The first one, they had a=dv/dt then multiplied both sides by dt and integrated with respect to that variable...perhaps it's cause I'm still not all that comfortable with playing around with differentials like that yet but it doesn't seem 'proper' to do that. Are there alternate methods that DON'T involve treating differentials like that?

Another method that canceled the differentials is shown here at the end:

I'm not sure about that either
 
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  • #3
Oh yes, just found it. It looks like the substitution rule for integration...

For the second part (kinematic equations link), when they integrate the differential, don't integral signs already come with the differential, the variable that you're integrating with respect to?
 
  • #4
autodidude said:
For the second part (kinematic equations link), when they integrate the differential, don't integral signs already come with the differential, the variable that you're integrating with respect to?
Are you referring to the dv = a.dt line? That is just saying that in a small interval of time, dt, the velocity increase, dv, will be a.dt. This is the logical first step whether you're integrating or differentiating. From there, you can either divide both sides by dt, then take the limit as dt tends to zero, to get the derivative; or perform a sum of dt's over a range, then take the limit to obtain an integral.
Does that help?
 
  • #5
^ Yes! Thanks a lot!
 

FAQ: Questions involving differentials (again)

What is a differential equation?

A differential equation is a mathematical equation that relates a function to its derivatives. It describes how a variable changes in relation to other variables.

How are differential equations used in science?

Differential equations are used in science to model and understand natural phenomena, such as the motion of objects, the growth of populations, and the behavior of physical systems.

What is the difference between ordinary and partial differential equations?

Ordinary differential equations involve only one independent variable, while partial differential equations involve multiple independent variables. Ordinary differential equations are used to model continuous systems, while partial differential equations are used to model systems that involve space and time.

Can differential equations be solved analytically?

Some simple differential equations can be solved analytically by finding a formula for the function that satisfies the equation. However, most differential equations cannot be solved analytically and require numerical methods for approximation.

What are some common applications of differential equations?

Differential equations are used in many fields, including physics, engineering, biology, economics, and chemistry. They are used to study and predict the behavior of complex systems and to make accurate predictions about the future.

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