What Do Mathematicians Think of Physicists Cancelling dx?

In summary, physicists often divide equations by dx or \partial x without fully considering the mathematical properties of differentials. This can lead to errors, particularly when dividing by zero. Mathematicians treat dy/dx as a fraction and define differentials to make this property precise, but it is important to ensure that all necessary conditions are met before dividing by dx.
  • #1
ShayanJ
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In some places in physics like proving the wave equation,we see that the physicist divides the equation by [itex] dx [/itex] or [itex] \partial x [/itex] or maybe cancels some of them and then says with a smile:You shouldn't tell a mathematician!
I want to know the ideas of mathematicians about that and want to know why its wrong?
thanks
 
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  • #2
"dy/dx" is NOT a fraction as defined so "dy" and "dx" are not separate values. However, because it is defined as a limit of fractions, we can always "go back before the limit, use the fraction property, then take the limit again". So "dy/dx" can be treated like a fraction and to make that precise, mathematics defines "differentials", dy and dx, which, while not really the derivative have the property that the derivative, dy/dx, is the fraction "dy" divided by "dx".

It is perfectly legitimate, mathematically, to "divide by dx" provided you have checked all of the properties required to be sure the differentials exist. It is that part that physicists typically play "fast and loose" with, assuming that, since everything has a specific physical meaning, they must exist.
 
  • #3
The only thing you need to watch out for is dividing by zero(such is the only minor problem with the chain rule proof using dz/dx = dz/dy*dy/dx
 
  • #4
Skrew said:
The only thing you need to watch out for is dividing by zero(such is the only minor problem with the chain rule proof using dz/dx = dz/dy*dy/dx

I don't understand what could be that zero!
 

1. What is the concept of "cancelling dx" in mathematics and physics?

The concept of "cancelling dx" is a mathematical technique used in calculus to simplify equations by removing the "dx" term from the numerator and denominator. In physics, this technique may also be used to simplify equations involving infinitesimal changes in physical quantities.

2. Is "cancelling dx" a valid mathematical operation?

Yes, "cancelling dx" is a valid mathematical operation in certain contexts, such as in calculus or physics. However, it is important to understand the underlying principles and assumptions involved in order to use this technique correctly.

3. How do mathematicians and physicists view the concept of "cancelling dx" differently?

Mathematicians and physicists may view the concept of "cancelling dx" differently because they approach problems from different perspectives. Mathematicians tend to focus on the theoretical aspects and rigor of the mathematics, while physicists are more concerned with practical applications and using mathematical tools to understand the physical world.

4. Are there any limitations to using "cancelling dx" in calculations?

Yes, there are limitations to using "cancelling dx" in calculations. This technique should only be applied in certain situations where the underlying assumptions are valid. Additionally, it is important to consider the potential for errors when using this technique, as it can lead to incorrect results if not used correctly.

5. How does the concept of "cancelling dx" relate to the broader fields of mathematics and physics?

The concept of "cancelling dx" is just one small aspect of the broader fields of mathematics and physics. It is a useful technique for simplifying equations, but it is not the only tool used in either field. Both mathematics and physics are vast and diverse disciplines that involve many different concepts and techniques beyond "cancelling dx".

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