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Conservative vector field
In vector calculus a conservative vector field is a vector field that is the gradient of some function, known in this context as a scalar potential. Conservative vector fields have the property that the line integral is path independent, i.e. the choice of integration path between any point and another does not change the result. Path independence of a line integral is equivalent to the vector field being conservative. A conservative vector field is also irrotational; in three dimensions this means that it has vanishing curl. An irrotational vector field is necessarily conservative provided that a certain condition on the geometry of the domain holds, i.e. the domain is simply connected.
Conservative vector fields appear naturally in mechanics: they are vector fields representing forces of physical systems in which energy is conserved.〔George B. Arfken and Hans J. Weber, ''Mathematical Methods for Physicists'', 6th edition, Elsevier Academic Press (2005)〕 For a conservative system, the work done in moving along a path in configuration space depends only on the endpoints of the path, so it is possible to define a potential energy independently of the path taken.
==Informal treatment==
In a two and three dimensional space, there is an ambiguity in taking an integral between two points as there are infinitely many paths between the two points - apart from the straight line formed between the two points one could choose a curved path of greater length as shown in the figure. Therefore in general the value of the integral depends on the path taken. However, in the special case of a conservative vector field, the value of the integral is independent of the path taken which can be thought of as a large-scale cancellation of all elements ''dR'' which don't have a component along the straight line between the two points. To visualise this, imagine two people climbing a cliff; one decides to scale the cliff by going vertically up it, and the second decides to walk along a winding path that is longer in length than the height of the cliff, but at only a small angle to the horizontal. Although the two hikers have taken different routes to get up to the top of the cliff, at the top they will have both gained the same amount of gravitational potential energy. This is because a gravitational field is conservative. As an example of a non-conservative field, imagine pushing a box from one end of a room to another. Pushing the box in a straight line across the room requires noticeably less work against friction than along a curved path covering a greater distance.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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