Differential Forms Homework: Closed But Not Exact

Thread starter Niles
Start date Nov 23, 2009
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Differential Differential forms Forms

In summary, differential forms are mathematical objects used in differential geometry and calculus to describe geometric shapes and their transformations. A closed differential form satisfies the condition of having an exterior derivative equal to zero, while an exact differential form can be expressed as the exterior derivative of another form. To determine if a form is closed but not exact, one can use the exterior derivative operator or the Poincaré lemma. Applications of closed but not exact forms include topology, geometry, and physics. To improve understanding of these forms, a strong foundation in differential geometry and calculus, practice with problem-solving, and studying real-life applications are recommended.

Nov 23, 2009

Niles

Homework Statement

Hi all

I can find a differential form defined on R²\{0,0}, which is closed but not exact, but is it possible to find a differential form defined on all R², which is closed but not exact?

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Nov 23, 2009

HallsofIvy

Science Advisor

Homework Helper

No, it isn't. That is, essentially, Stoke's theorem.

Related to Differential Forms Homework: Closed But Not Exact

1. What are differential forms?

Differential forms are mathematical objects used in differential geometry and calculus to describe the properties of geometric shapes and their transformations. They are a generalization of vector calculus and allow for a more flexible and intuitive way of computing and understanding various mathematical concepts.

2. What does it mean for a differential form to be closed but not exact?

A closed differential form is one that satisfies a certain condition known as the "exterior derivative" being equal to zero. This means that the form is locally conservative, and its integral around any closed loop will always be zero. However, an exact differential form is one that can be expressed as the exterior derivative of another form. So a form that is closed, but not exact, cannot be written in this way and has some unique properties.

3. How can I determine if a differential form is closed but not exact?

To determine if a differential form is closed, you can use the exterior derivative operator to check if it is equal to zero. If it is equal to zero, then the form is closed. To check if it is exact, you can use a method called "Poincaré lemma" which states that if a form is closed and is defined on a contractible domain, then it must be exact. If this lemma does not hold, then the form is closed but not exact.

4. What are some applications of closed but not exact differential forms?

One of the main applications of closed but not exact differential forms is in the study of topology and geometry. These forms can be used to detect holes, voids, and other topological features in a given space. They are also used in physics, specifically in the study of electromagnetic fields and their properties.

5. How can I improve my understanding of closed but not exact differential forms?

To improve your understanding of closed but not exact differential forms, it is important to have a strong foundation in differential geometry and calculus. You can also practice solving problems and working with different types of forms to gain a better understanding. Additionally, studying real-life applications of these forms can also help in developing a deeper understanding of their properties and uses.