Finding the derivative vs differentiating vs finding the differential of a fn/eqtn

[torquemada]

Even though I've taken 3 semesters of calc, some of the terms are used interchangeably and are a bit obfuscated (at least in my mind lol). I understand how to take the derivative of a function with respect to 1 or more variables, but I'm reading Mary L Boas Math methods in physical science and I'm in the discussion of differentials. She talks about doing this operation called finding the differential of an equation (or function too?).

for example, y = s - t becomes dy = ds - dt

Is this a legal operation according to mathematicians - or is it a hand-waving operation that is only used in the physical sciences and lacks pure mathematical rigor? Can this operation be done on functions too? or just equations?

Also there are often times when she writes that we differentiate a function without specifying with respect to what variable - just differentiating a function period. For example (Boas 2nd edition page 155), in E&M the equation R = kl/r^2 for wire resistance. She says we can find dR/R by differentiating ln R = ln k + ln l - 2ln r which gives us

dR/R = dl/l - 2dr/r

So is this differentiating with respect to some variable that I am not aware of? Or is this taking a differential of the equation ln R = ln k + ln l - 2ln r ?

If the latter, would that mean that differentiating nonspecific to a certain variable is equivalent to taking the differential of an equation/function?

Or are there 3 distinct operations? Differentiating nonspecific, differentiating with respect to a certain variable, and finding the differential of an equation/function? Thanks in advance for any clarification you can provide

 

[lavinia]

Even though I've taken 3 semesters of calc, some of the terms are used interchangeably and are a bit obfuscated (at least in my mind lol). I understand how to take the derivative of a function with respect to 1 or more variables, but I'm reading Mary L Boas Math methods in physical science and I'm in the discussion of differentials. She talks about doing this operation called finding the differential of an equation (or function too?).

for example, y = s - t becomes dy = ds - dt

Is this a legal operation according to mathematicians - or is it a hand-waving operation that is only used in the physical sciences and lacks pure mathematical rigor? Can this operation be done on functions too? or just equations?

Also there are often times when she writes that we differentiate a function without specifying with respect to what variable - just differentiating a function period. For example (Boas 2nd edition page 155), in E&M the equation R = kl/r^2 for wire resistance. She says we can find dR/R by differentiating ln R = ln k + ln l - 2ln r which gives us

dR/R = dl/l - 2dr/r

So is this differentiating with respect to some variable that I am not aware of? Or is this taking a differential of the equation ln R = ln k + ln l - 2ln r ?

If the latter, would that mean that differentiating nonspecific to a certain variable is equivalent to taking the differential of an equation/function?

Or are there 3 distinct operations? Differentiating nonspecific, differentiating with respect to a certain variable, and finding the differential of an equation/function? Thanks in advance for any clarification you can provide

One can think of dR ,the differential of a function - possibly vector valued , as a small increment in R. If R is related to other functions in an equation then small changes in R imply small changes in the other functions. That is all that this really means. Small however is something of a vague idea and really what it means is small enough so that the equation among differentials is a very good approximation.

This idea can be stated in terms of linear functions on the tangent space of the manifold.
Here the idea is mathematically different but the idea is exactly the same.

 

[torquemada]

This idea can be stated in terms of linear functions on the tangent space of the manifold.
Here the idea is mathematically different but the idea is exactly the same.

I appreciate your efforts but that sounds like a formalism that is beyond what I need to understand this and it's unintelligible at my current level. Plus you didn't actually answer any of my questions lol.

Can anyone directly answer my questions in more layman's terms? thank you

 

[lavinia]

I appreciate your efforts but that sounds like a formalism that is beyond what I need to understand this and it's unintelligible at my current level. Plus you didn't actually answer any of my questions lol.

Can anyone directly answer my questions in more layman's terms? thank you

Actually I did answer your question.