Question about conditions for conservative field

In summary, the conversation discusses the two common assumptions made about regions in conservative vector fields: that they are simply connected and open. The individual asking the question is curious about the necessity of these assumptions and whether they are for computational convenience or if there is a logical reason behind them. The conversation also refers to a relevant webpage and a question on a math forum.
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kelvin490
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Question about conditions for conservative field
In common textbooks' discussions about conservative vector field. There is always two assumptions about the region concerned, namely the region is simply connected and open.

Usually in textbooks there is not much explanations on why these assumptions are necessary, no proof is given on why conservative field is not possible if the region is not simply connected or not open.

I wonder whether these two assumptions are just for computational convenience or it is really logically not possible to have a conservative field in region that is not simply connected or is not open?
 
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Related to Question about conditions for conservative field

1. What is a conservative field?

A conservative field is a type of vector field in which the line integral of the field along any closed path is equal to zero. This means that the work done by the field on a particle moving along a closed path is independent of the path taken.

2. What conditions must a vector field satisfy to be considered conservative?

A vector field must satisfy two conditions to be considered conservative: it must be a continuous function and its partial derivatives must be equal. This is known as the gradient condition or the curl-free condition.

3. Can a vector field be conservative if it does not satisfy the conditions?

No, a vector field must satisfy the conditions of being continuous and having equal partial derivatives to be considered conservative. If these conditions are not met, the field is considered non-conservative.

4. What are some real-world examples of conservative fields?

Some common examples of conservative fields in the real world include gravitational fields, electric fields, and magnetic fields. These fields have a constant force and do not depend on the path taken.

5. How is the concept of a conservative field used in physics and engineering?

The concept of a conservative field is used in many applications in physics and engineering, such as calculating work and energy in mechanical systems, analyzing fluid flow, and solving electrostatic and electromagnetic problems. It is also used in the development of algorithms for solving differential equations and in numerical methods for solving problems in science and engineering.

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