Understanding Mass-Energy Equivalence to Fdx and dm in E=mc^2

[rrrright]

Hi I was wondering if anyone could help me with this equation.

$$Fdx &= dm c^2$$

First of all, excuse me for my limited knowledge of calculus, but how exactly can you just use the numerator of a derivative? What do Fdx and dm mean if they are not in respect to anything? Do they simply mean a change in x and a change in m?

Secondly, I have seen this equation used to get to E=mc^2 through integration. How exactly do we integrate either side of this equation? What is the step by step process for doing this? Again, please excuse my limited math knowledge.

 

[Matterwave]

Splitting the derivatives is technically not kosher in math; however, physicists do it all the time. I think there are probably weird pathological functions (such as discontinuous everywhere functions or some such) where splitting the derivative will lead to the wrong answer, but physics usually only deals with well behaved functions. Usually something like dx just means a small displacement in x.

You can immediately integrate that expression to obtain W=mc^2. Integrateing Fdx gives you the work and integrating d(mc^2) just gives you mc^2 back (sorta like the fundamental theorem of calculus).

 

[starthaus]

Hi I was wondering if anyone could help me with this equation.

$$Fdx &= dm c^2$$

First of all, excuse me for my limited knowledge of calculus, but how exactly can you just use the numerator of a derivative? What do Fdx and dm mean if they are not in respect to anything? Do they simply mean a change in x and a change in m?

Secondly, I have seen this equation used to get to E=mc^2 through integration. How exactly do we integrate either side of this equation? What is the step by step process for doing this? Again, please excuse my limited math knowledge.

Hi, welcome on board!

For the answers, look at the first file in my blog. It explains the differentiation as well as the physics you are asking about.