Conservative vector field

In vector calculus, a conservative vector field is a vector field that is the gradient of some function. Conservative vector fields have the property that the line integral is path independent, i.e., the choice of any path between two points does not change the value of the line integral. Path independence of the 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 the domain is simply connected. In vector calculus, a conservative vector field is a vector field that is the gradient of some function. Conservative vector fields have the property that the line integral is path independent, i.e., the choice of any path between two points does not change the value of the line integral. Path independence of the 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 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. 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 that is independent of the actual path taken. 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 d R {displaystyle d{R}} that don't have a component along the straight line between the two points. To visualize 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. M. C. Escher's painting Ascending and Descending illustrates a non-conservative vector field, impossibly made to appear to be the gradient of the varying height above ground as one moves along the staircase. It is rotational in that one can keep getting higher or keep getting lower while going around in circles. It is non-conservative in that one can return to one's starting point while ascending more than one descends or vice versa. On a real staircase, the height above the ground is a scalar potential field: If one returns to the same place, one goes upward exactly as much as one goes downward. Its gradient would be a conservative vector field and is irrotational. The situation depicted in the painting is impossible. A vector field v : U → R n {displaystyle mathbf {v} :U o mathbb {R} ^{n}} , where U {displaystyle U} is an open subset of R n {displaystyle mathbb {R} ^{n}} , is said to be conservative if and only if there exists a C 1 {displaystyle C^{1}} scalar field φ {displaystyle varphi } on U {displaystyle U} such that Here, ∇ φ {displaystyle abla varphi } denotes the gradient of φ {displaystyle varphi } . When the equation above holds, φ {displaystyle varphi } is called a scalar potential for v {displaystyle mathbf {v} } . The fundamental theorem of vector calculus states that any vector field can be expressed as the sum of a conservative vector field and a solenoidal field. A key property of a conservative vector field v {displaystyle mathbf {v} } is that its integral along a path depends only on the endpoints of that path, not the particular route taken. Suppose that P {displaystyle P} is a rectifiable path in U {displaystyle U} with initial point A {displaystyle A} and terminal point B {displaystyle B} . If v = ∇ φ {displaystyle mathbf {v} = abla varphi } for some C 1 {displaystyle C^{1}} scalar field φ {displaystyle varphi } so that v {displaystyle mathbf {v} } is a conservative vector field, then the gradient theorem states that This holds as a consequence of the chain rule and the fundamental theorem of calculus.