Why are vector fields called conservative when the curl = 0?

Question

In English please.

Answer 1

A conservative vector field can be thought of as a conservative force field. An example of a conservative force field would gravity exterted by a planet (or any object). If you move any object from point A to point B in this conservative force field, the energy required to do it is independent of the path. In fact, moving any object from any point A back to point A neither requires energy nor yields surplus energy. The mathematical requirement for this vector property is that the curl is 0 everywhere, which happens to be a property of virtually all force fields in physics, be it gravity, electromagnetic, Van der Waals, etc, since otherwise energy wouldn't be conserved. Hence, we speak of CONSERVATIVE vector fields when the curl = 0.

To help visualize what a non-conservative force field could be like, check Escher's drawing of an never-ending staircase. If this was physically possible, it would be an example of a non-conservative force field. This might help you understand the requirement that curl = 0 for a conservative vector field, because for this physically unrealizable staircase, the curl is not 0.

Answer 2

The term comes from Physics. The word conservative applies to energy and the amount of work is done. So we (both in Physics and Math) say that a field is conservative if traveling along any closed path doesn't waste any energy.

The best example of a conservative field is the gravitational field. So, if I throw a ball up and it comes back down to my hand from where it left, the ball has not lost or gained anything. Because it returned to its point of origin in a conservative field. In fact, it doesn't even matter which path the ball took as long it came back. I can thrown it straight up and down. I can throw a curve and it will still be true.

An example of a non-conservative field is a system with friction. So if I start with a certain amount of energy, travel a path and then come back to that same point, I would have have lost some speed for example because of friction. In fact, this is path dependent too. The longer my path, the more I lost.

If you haven't had any physics, I can't help you then.

Answer 3

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