# Understanding Exact vs. Closed Forms for Mechanical Engineers

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• observer1
In summary, as a mechanical engineer trying to make up for a poor math education, I have come across the terms "closed" and "exact" in my self-study of differential forms. Closed forms are those whose exterior derivative is 0.0, while exact forms are the exterior derivative of another form. These terms are important because they help us understand the relationship between forms and their derivatives, and can be applied in various fields such as physics where they can represent conservative forces and potential functions. However, not all closed forms are exact, as demonstrated by counterexamples such as the Dirac magnetic monopole.
observer1
(I am a mechanical engineer, trying to make up for a poor math education)'

I understand that:
1. A CLOSED form is a differential form whose exterior derivative is 0.0.
2. 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:
1. A TOMATO form is a differential form whose exterior derivative is 0.0.
2. A POTATO form is the exterior derivative of another form.
Please forgive my sarcasm, but I am trying to get BEYOND memorizing the words. Why were those two words chosen?

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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fresh_42 said:
Maybe it gets you closer to the exact usage of the terms.

It is so bad that it is funny ...

atyy
observer1 said:
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.

An exact form can be derived from a potential, so it is related to conservative forces, eg. the circumstances in which E=gradΦ, or in which B=curlA.

Every exact form is closed, eg. curl(gradΦ)=0 and div(curlA)=0.

But is every closed form exact? For example, if we see that div(B)=0, can we infer that B=curlA?

We cannot because we can produce counterexamples, eg. the Dirac magnetic monopole. This is mentioned eg. in Abanov's notes on differential forms http://felix.physics.sunysb.edu/~abanov/Teaching/Spring2009/phy680.html or in Deschamps's article http://www-liphy.ujf-grenoble.fr/pagesperso/bahram/biblio/Deschamps1981_dif_forms.pdf.

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## 1. What is the difference between exact and closed form solutions in mechanical engineering?

Exact form solutions in mechanical engineering refer to solutions that can be determined through mathematical equations and formulas without approximation. These solutions provide precise and accurate results. On the other hand, closed form solutions in mechanical engineering are obtained through approximations and simplifications of complex equations. They are often used when exact solutions are not feasible or when quick estimations are needed.

## 2. How do mechanical engineers determine whether to use exact or closed form solutions?

Mechanical engineers consider various factors such as the complexity of the problem, available resources, and desired level of accuracy when deciding whether to use exact or closed form solutions. If the problem is simple and resources are limited, closed form solutions may be sufficient. However, for more complex problems that require high accuracy, exact form solutions are preferred.

## 3. What are the advantages of using exact form solutions in mechanical engineering?

Exact form solutions offer precise and accurate results, making them ideal for critical engineering calculations. They also provide a deeper understanding of the problem and its underlying principles. Furthermore, by using exact solutions, engineers can verify the accuracy of their approximations and determine the margin of error.

## 4. Are there any disadvantages of using exact form solutions in mechanical engineering?

The main disadvantage of using exact form solutions is that they can be time-consuming and resource-intensive. In some cases, the complexity of the equations may make it impossible to find an exact solution, leading engineers to rely on closed form solutions or numerical methods. Additionally, exact solutions may not always be necessary, and using them may result in unnecessary effort and costs.

## 5. Can mechanical engineers use a combination of exact and closed form solutions?

Yes, mechanical engineers often use a combination of exact and closed form solutions to solve complex problems. They may use exact solutions to verify the accuracy of their approximations or to gain a better understanding of the problem, while using closed form solutions for quick estimations. This approach allows engineers to balance accuracy and efficiency in their calculations.

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