Conservative vector field

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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.[1] 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.


Informal treatment[edit]

In a two and three dimensional space, there is an ambiguity in taking an integral between two points as there are infinitely many paths that you could choose to get 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 potential energy due to gravity. 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. If you push the box in a straight line across the room, you will do noticeably less work against friction than if you pushed the box in a curved path covering a greater distance.

Depiction of two possible paths to integrate. In green in the simplest possible path, blue shows a more convoluted curve

Intuitive explanation[edit]

M. C. Escher's lithograph Ascending and Descending







Path independence[edit]

Main article: Gradient theorem



This holds as a consequence of the chain rule and the fundamental theorem of calculus.

An equivalent formulation of this is to say that



Irrotational vector fields[edit]







Therefore every conservative vector field is also an irrotational vector field.







Irrotational flows[edit]

Main article: Vorticity







Conservative forces[edit]

Examples of potential and gradient fields in physics






is the gravitational potential energy.




See also[edit]

Citations and sources[edit]
