Rigorous dirrerential definition and physics

[rkaminski]

My question is rather simple but it puzzles me for a long time actually. If we have a look at differential as physicists usually do we came up with a simple definition of "infinitesimal variable change". And this idea then preserves elsewhere like in the definition of entropy:

$\mathrm{d} S = \frac{\mathrm{d} Q}{T}$

or in calculus (e.g. for differential questions etc.). On the other hand mathematicians define the differential as a function:

$\mathrm{d} f (x_0) (h) = f'(x_0)h$

for a given $h$ and so on. The question is how to find a bridge between those two formulations.

Radek

 

[jvicens]

,

Thank you for your question! The concept of differentials can be confusing, as it is often defined differently in different fields of study. However, the key idea behind differentials is the same in both physics and mathematics.

In physics, differentials are used to describe small changes in a physical quantity, such as temperature or position. This is often done using the concept of infinitesimals, which are very small quantities that can be treated as zero for practical purposes. This allows us to approximate a change in a physical quantity as a ratio of infinitesimals, as shown in your example of entropy.

In mathematics, differentials are used to describe the instantaneous rate of change of a function at a specific point. This is done using the derivative, which is defined as the limit of the ratio of infinitesimals as they approach zero. The differential of a function at a point is then defined as the linear function that approximates the change in the function at that point.

So how do we bridge the gap between these two formulations? The key is to understand that both definitions are describing the same concept – a small change in a quantity. In physics, we use infinitesimals to approximate this change, while in mathematics, we use the derivative. Both approaches have their strengths and weaknesses, but they ultimately lead to the same result.

In fact, the two definitions can be related using the chain rule in calculus. The chain rule allows us to express the differential of a composite function in terms of the differentials of its individual components. This allows us to bridge the gap between the two formulations and see that they are ultimately describing the same concept.

I hope this helps to clarify the concept of differentials for you. Keep asking questions and exploring the connections between different fields of study – that's what science is all about!