- #1

observer1

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- 11

(I am a mechanical engineer, trying to make up for a poor math education)'

I understand that:

And it stops right there. I am teaching myself differential forms. And as I ratchet up my understanding, I encounter these words--closed and exact--but I am not yet comfortable with their use.

As a result, I MEMORIZE the two words and their definitions. I do this to get through some rough spots as I continue to learn. But now I am at a point where I am hungering to know why these words matter.

It would help me, I think, if I knew WHY those words were used. In other words, I just just as easily have written:

And, if you can, answer in terms of pure theoretical math AND, if possible, with a meaningful (perhaps physical for a mechanical engineer) example.

For example, I THINK I UNDERSTAND that for the case of 1D integration of a form along a line that is CLOSED (like a closed loop or closed circle), that the signed definite integral of the form from "a" to "a-gain" is zero. Does that word CLOSED have anything to do with a the word describing the form. And is this related to the work done by a conservative force in a closed loop? I am almost at the point of seeing that a closed form can represent a conservative force, and an exact form represents a potential function. However, I cannot disambiguate the words CLOSED and EXACT since they all seem to mean the same thing in physics... I just need to see these two words separated.

I understand that:

- A CLOSED form is a differential form whose exterior derivative is 0.0.
- An EXACT form is the exterior derivative of another form.

And it stops right there. I am teaching myself differential forms. And as I ratchet up my understanding, I encounter these words--closed and exact--but I am not yet comfortable with their use.

As a result, I MEMORIZE the two words and their definitions. I do this to get through some rough spots as I continue to learn. But now I am at a point where I am hungering to know why these words matter.

It would help me, I think, if I knew WHY those words were used. In other words, I just just as easily have written:

- A TOMATO form is a differential form whose exterior derivative is 0.0.
- A POTATO form is the exterior derivative of another form.

And, if you can, answer in terms of pure theoretical math AND, if possible, with a meaningful (perhaps physical for a mechanical engineer) example.

For example, I THINK I UNDERSTAND that for the case of 1D integration of a form along a line that is CLOSED (like a closed loop or closed circle), that the signed definite integral of the form from "a" to "a-gain" is zero. Does that word CLOSED have anything to do with a the word describing the form. And is this related to the work done by a conservative force in a closed loop? I am almost at the point of seeing that a closed form can represent a conservative force, and an exact form represents a potential function. However, I cannot disambiguate the words CLOSED and EXACT since they all seem to mean the same thing in physics... I just need to see these two words separated.

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