Why consider Differential Element

[koolraj09]

Hi all.
In most of our analysis of continuous systems ex: in fluid mechanics, solid mechanics etc. we consider a differential element(infinitesimally small element) and obtain the differential equations that govern the phenomenon. But why do we consider a differential element? What is it's significance? Or does it have any advantages? Does it simplify our analysis?
Also do we have another method instead of the differential approach to obtain the governing differential equations for a particular phenomenon?
Thanks.

 

[SteveL27]

Hi all.
In most of our analysis of continuous systems ex: in fluid mechanics, solid mechanics etc. we consider a differential element(infinitesimally small element) and obtain the differential equations that govern the phenomenon. But why do we consider a differential element? What is it's significance? Or does it have any advantages? Does it simplify our analysis?
Also do we have another method instead of the differential approach to obtain the governing differential equations for a particular phenomenon?
Thanks.

I can think of two reasons.

1) Using infinitesimals is a way of doing calculus without having to be bothered with the modern niceties of mathematical rigor. You get to the same place, but quicker.

2) The first derivative (which is what you're really looking at when you look at the infinitesimal change in output) gives the best physical intuition of the motion. Say you're driving up the freeway. What piece of information gives the most important information about the motion? Clearly the first thing you want to know is your velocity.

That's how I think about it. You want to look at the infinitesimal behavior because that's what tells how the system is changing at any moment in time. It's important information.

But instead of using the modern mathematical formalisms of epsilons and deltas, they let you think about it in terms of infinitesimals. In calculus we've banned infinitesimals [obligatory disclaimer about nonstandard analysis] but if you want to understand a situation intuitively, thinking about an infinitesimal change is often very helpful.

 

[AlephZero]

Also do we have another method instead of the differential approach to obtain the governing differential equations for a particular phenomenon?
Thanks.

Yes. Oten you can take a finite part of the system (e.g. a finite sized volume of a fluid) and apply laws like conservation of energy and momentum to the whole volume (and the flows into and out of the volume, across the finite sized boundaries). This is sometimes called a "weak formulation" of the problem, because it at first sight it doesn't give you any equations that apply just at one point (called a "strong formulation"). But that isn't really a problem, because the conservation laws must apply to every possible region, so you can usually show mathematically (e.g. using general vector calulus theorems) that the weak and strong formulations are equivalent.

Both methods have advantages and disadvantages, but the idea of looking at what happens to a "very small" part of the system and turning that into a differential equation is probably more intuitive, at least for non-mathematicans.