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Trying2Learn

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- TL;DR Summary
- what is the dx in an integral

Good Morning

I understand that d/dx is an "operator" on a function; and that one should never split, say, df, from dx in df/dx

That said, I have seen it in an integral, specifically for calculating work.

I do understand the idea of "differential forms." However, I am trying to understand if there is a difference between how dx is used in an integral (effectively splitting out the denominator in the "operator" of d/dx).

I do understand that in calculating "work" in physics, we integrate the force over dx. In that case, we realize that force

is a vector and dx is a co-vector.

So, in a nutshell, i can see "through a dark glass" (so to speak) that something is going on here.

However, with that background, I hope I can rephrase my initial question as

I am not sure I am asking this properly. Please forgive me for the poor wording -- that is half the problem I face.

__To cut the chase, what is the dx in an integral?__I understand that d/dx is an "operator" on a function; and that one should never split, say, df, from dx in df/dx

That said, I have seen it in an integral, specifically for calculating work.

I do understand the idea of "differential forms." However, I am trying to understand if there is a difference between how dx is used in an integral (effectively splitting out the denominator in the "operator" of d/dx).

I do understand that in calculating "work" in physics, we integrate the force over dx. In that case, we realize that force

is a vector and dx is a co-vector.

So, in a nutshell, i can see "through a dark glass" (so to speak) that something is going on here.

However, with that background, I hope I can rephrase my initial question as

__"What is the difference between the "dx" in, say the integral leading to work done, vs. the dx in the denominator of the operator (which, should NEVER, in theory, stand alone).__I am not sure I am asking this properly. Please forgive me for the poor wording -- that is half the problem I face.

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