This article includes a list of general references, but it lacks sufficient corresponding inline citations. Please help to improve this article by introducing more precise citations. (May 2009) (Learn how and when to remove this message)

In vector calculus, a conservative vector field is a vector field that is the gradient of some function.^[1] A conservative vector field has the property that its line integral is path independent; the choice of 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 under the line integral 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.^[2] For a conservative system, the work done in moving along a path in a configuration space depends on only the endpoints of the path, so it is possible to define potential energy that is independent of the actual 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 ${\displaystyle d{R}}$ that do not 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.

Intuitive explanation

M. C. Escher's lithograph print Ascending and Descending illustrates a non-conservative vector field, impossibly made to appear to be the gradient of the varying height above ground (gravitational potential) as one moves along the staircase. The force field experienced by the one moving on the staircase is non-conservative in that one can return to the starting point while ascending more than one descends or vice versa, resulting in nonzero work done by gravity. On a real staircase, the height above the ground is a scalar potential field: one has to go upward exactly as much as one goes downward in order to return to the same place, in which case the work by gravity totals to zero. This suggests path-independence of work done on the staircase; equivalently, the force field experienced is conservative (see the later section: Path independence and conservative vector field). The situation depicted in the print is impossible.

Definition

A vector field ${\displaystyle \mathbf {v} :U\to \mathbb {R} ^{n}}$, where ${\displaystyle U}$ is an open subset of ${\displaystyle \mathbb {R} ^{n}}$, is said to be conservative if there exists a ${\displaystyle C^{1}}$ (continuously differentiable) scalar field ${\displaystyle \varphi }$^[3] on ${\displaystyle U}$ such that

Here, ${\displaystyle \nabla \varphi }$ denotes the gradient of ${\displaystyle \varphi }$. Since ${\displaystyle \varphi }$ is continuously differentiable, ${\displaystyle \mathbf {v} }$ is continuous. When the equation above holds, ${\displaystyle \varphi }$ is called a scalar potential for ${\displaystyle \mathbf {v} }$.

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.

Path independence and conservative vector field

Path independence

A line integral of a vector field ${\displaystyle \mathbf {v} }$ is said to be path-independent if it depends on only two integral path endpoints regardless of which path between them is chosen:^[4]

for any pair of integral paths ${\displaystyle P_{1}}$ and ${\displaystyle P_{2}}$ between a given pair of path endpoints in ${\displaystyle U}$.

The path independence is also equivalently expressed as

for any piecewise smooth closed path ${\displaystyle P_{c}}$ in ${\displaystyle U}$ where the two endpoints are coincident. Two expressions are equivalent since any closed path ${\displaystyle P_{c}}$ can be made by two path; ${\displaystyle P_{1}}$ from an endpoint ${\displaystyle A}$ to another endpoint ${\displaystyle B}$, and ${\displaystyle P_{2}}$ from ${\displaystyle B}$ to ${\displaystyle A}$, so where ${\displaystyle -P_{2}}$ is the reverse of ${\displaystyle P_{2}}$ and the last equality holds due to the path independence ${\textstyle \displaystyle \int _{P_{1}}\mathbf {v} \cdot d\mathbf {r} =\int _{-P_{2}}\mathbf {v} \cdot d\mathbf {r} .}$

Conservative vector field

A key property of a conservative vector field ${\displaystyle \mathbf {v} }$ is that its integral along a path depends on only the endpoints of that path, not the particular route taken. In other words, if it is a conservative vector field, then its line integral is path-independent. Suppose that ${\displaystyle \mathbf {v} =\nabla \varphi }$ for some ${\displaystyle C^{1}}$ (continuously differentiable) scalar field ${\displaystyle \varphi }$^[3] over ${\displaystyle U}$ as an open subset of ${\displaystyle \mathbb {R} ^{n}}$ (so ${\displaystyle \mathbf {v} }$ is a conservative vector field that is continuous) and ${\displaystyle P}$ is a differentiable path (i.e., it can be parameterized by a differentiable function) in ${\displaystyle }$Zdroj:https://en.wikipedia.org?pojem=Irrotational_vector_field
Text je dostupný za podmienok Creative Commons Attribution/Share-Alike License 3.0 Unported; prípadne za ďalších podmienok. Podrobnejšie informácie nájdete na stránke Podmienky použitia.

---