Irrotational

In vector calculus, a conservative vector field is a vector field that is the gradient of some function (mathematics), function. A conservative vector field has the property that its line integral is path independent; 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 under the line integral being conservative. A conservative vector field is also Laplace equation for irrotational flow, irrotational; in three dimensions, this means that it has vanishing curl (mathematics), 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 conservation of energy, conserved. For a conservative system, the work (physics), 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 $d$ 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.

Intuitive explanation

M. C. Escher, 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 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 print is impossible.

Definition

A vector field $\mathbf: U \to \R^n$, where $U$ is an open subset of $\R^n$, is said to be conservative if and only if there exists a $C^1$ (Smoothness#Multivariate differentiability classes, continuously differentiable) scalar field $\varphi$For $\mathbf = \nabla \varphi$ to be Conservative vector field#Path independence, path-independent, $\varphi$ is not necessarily continuously differentiable, the condition of being differentiable is enough, since the Gradient theorem, that proves the path independence of $\nabla \varphi$, does not require $\varphi$ to be continuously differentiable.

There must be a reason for the definition of conservative vector fields to require $\varphi$ to be Smoothness#Multivariate differentiability classes, continuously differentiable. on $U$ such that

$\mathbf = \nabla \varphi.$

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

The Helmholtz decomposition, 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 $\mathbf$ is said to be path-independent if it depends on only two integral path endpoints regardless of which path between them is chosen:
$\int_ \mathbf \cdot d \mathbf = \int_ \mathbf \cdot d \mathbf$

for any pair of integral paths $P_1$ and $P_2$ between a given pair of path endpoints in $U$.

The path independence is also equivalently expressed as

$\int_ \mathbf \cdot d \mathbf = 0$
for any piecewise smooth closed path $P_c$ in $U$ where the two endpoints are coincident. Two expressions are equivalent since any closed path $P_c$ can be made by two path; $P_1$ from an endpoint $A$ to another endpoint $B$, and $P_2$ from $B$ to $A$, so
$\int_ \mathbf \cdot d \mathbf = \int_ \mathbf \cdot d \mathbf + \int_ \mathbf \cdot d \mathbf = \int_ \mathbf \cdot d \mathbf - \int_ \mathbf \cdot d \mathbf = 0$
where $-P_2$ is the reverse of $P_2$ and the last equality holds due to the path independence $\int_ \mathbf \cdot d \mathbf = \int_ \mathbf \cdot d \mathbf.$

Conservative vector field

A key property of a conservative vector field $\mathbf$ 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 $\mathbf = \nabla \varphi$ for some $C^1$ (Smoothness#Multivariate differentiability classes, continuously differentiable) scalar field $\varphi$ over $U$ as an open subset of $\R^n$ (so $\mathbf$ is a conservative vector field that is continuous) and $P$ is a differentiable path (i.e., it can be parameterized by a differentiable function) in $U$ with an initial point $A$ and a terminal point $B$. Then the gradient theorem (also called ''fundamental theorem of calculus for line integrals'') states that
$\int_ \mathbf \cdot d = \varphi(B) - \varphi(A).$

This holds as a consequence of the Line integral#Line integral of a vector field, definition of a line integral, the chain rule, and the fundamental theorem of calculus, second fundamental theorem of calculus. $\mathbf \cdot d\mathbf = \nabla \cdot d\mathbf$ in the line integral is an exact differential for an orthogonal coordinate system (e.g., Cartesian coordinate system, Cartesian, cylindrical, or Spherical coordinate system, spherical coordinates). Since the gradient theorem is applicable for a differentiable path, the path independence of a conservative vector field over piecewise-differential curves is also proved by the proof per differentiable curve component.

So far it has been proven that a conservative vector field $\mathbf$ is line integral path-independent. Conversely, ''if a continuous vector field $\mathbf$ is (line integral) path-independent, then it is a conservative vector field'', so the following Logical biconditional, biconditional statement holds:

The proof of this converse statement is the following.

$\mathbf$ is a continuous vector field which line integral is path-independent. Then, let's make a function $\varphi$ defined as
$\varphi(x,y) = \int_^ \mathbf \cdot d$
over an arbitrary path between a chosen starting point $(a,b)$ and an arbitrary point $(x,y)$. Since it is path-independent, it depends on only $(a,b)$ and $(x,y)$ regardless of which path between these points is chosen.

Let's choose the path shown in the left of the right figure where a 2-dimensional Cartesian coordinate system is used. The second segment of this path is parallel to the $x$ axis so there is no change along the $y$ axis. The line integral along this path is
$\int_^ \mathbf \cdot d = \int_^ \mathbf \cdot d + \int_^ \mathbf \cdot d.$
By the path independence, its partial derivative with respect to $x$ (for $\varphi$ to have partial derivatives, $\mathbf$ needs to be continuous.) is
$\frac = \frac \int_^ \mathbf \cdot d = \frac \int_^ \mathbf \cdot d + \frac \int_^ \mathbf \cdot d = 0 + \frac \int_^ \mathbf \cdot d$
since $x_1$ and $x$ are independent to each other. Let's express $\mathbf$ as $= P(x,y) \mathbf + Q(x,y) \mathbf$ where $\mathbf$ and $\mathbf$ are unit vectors along the $x$ and $y$ axes respectively, then, since $d \mathbf = dx \mathbf + dy \mathbf$,
$\frac \varphi (x,y) = \frac \int_^ \mathbf \cdot d\mathbf = \frac \int_^ P(t,y) dt = P(x,y)$
where the last equality is from the Fundamental theorem of calculus, second fundamental theorem of calculus.

A similar approach for the line integral path shown in the right of the right figure results in $\frac \varphi (x,y) = Q(x,y)$ so
$\mathbf = P(x,y) \mathbf+ Q(x,y) \mathbf = \frac \mathbf + \frac \mathbf = \nabla \varphi$
is proved for the 2-dimensional Cartesian coordinate system. This proof method can be straightforwardly expanded to a higher dimensional orthogonal coordinate system (e.g., a 3-dimensional spherical coordinate system) so the converse statement is proved. Another proof is found Gradient theorem, here as the converse of the gradient theorem.

Irrotational vector fields

Let $n = 3$ (3-dimensional space), and let $\mathbf: U \to \R^3$ be a $C^1$ (Smoothness#Multivariate differentiability classes, continuously differentiable) vector field, with an open subset $U$ of $\R^n$. Then $\mathbf$ is called irrotational if and only if its Curl (mathematics), curl is $\mathbf$ everywhere in $U$, i.e., if
$\nabla \times \mathbf \equiv \mathbf.$

For this reason, such vector fields are sometimes referred to as curl-free vector fields or curl-less vector fields. They are also referred to as Helmholtz decomposition#Longitudinal and transverse fields , longitudinal vector fields.

It is an Vector calculus identities#Curl of gradient is zero, identity of vector calculus that for any $C^2$ (Smoothness#Multivariate differentiability classes, continuously differentiable up to the 2nd derivative) scalar field $\varphi$ on $U$, we have
$\nabla \times (\nabla \varphi) \equiv \mathbf.$

Therefore, ''every $C^1$ conservative vector field in $U$ is also an irrotational vector field in $U$''. This result can be easily proved by expressing $\nabla \times (\nabla \varphi)$ in a Cartesian coordinate system with Symmetry of second derivatives#Theorem of Schwarz, Schwarz's theorem (also called Clairaut's theorem on equality of mixed partials).

Provided that $U$ is a simply connected space, simply connected open space (roughly speaking, a single piece open space without a hole within it), the converse of this is also true: ''Every irrotational vector field in a simply connected open space $U$ is a $C^1$ conservative vector field in $U$''.

The above statement is ''not'' true in general if $U$ is not simply connected. Let $U$ be $\R^3$ with removing all coordinates on the $z$-axis (so not a simply connected space), i.e., $U = \R^3 \setminus \$. Now, define a vector field $\mathbf$ on $U$ by
$\mathbf(x,y,z) ~ \stackrel ~ \left( - \frac,\frac,0 \right).$

Then $\mathbf$ has zero curl everywhere in $U$ ($\nabla \times \mathbf \equiv \mathbf$ at everywhere in $U$), i.e., $\mathbf$ is irrotational. However, the Circulation (physics), circulation of $\mathbf$ around the unit circle in the $xy$-plane is $2 \pi$; in polar coordinates, $\mathbf = \mathbf_ / r$, so the integral over the unit circle is
$\oint_ \mathbf \cdot \mathbf_ ~ d = 2 \pi.$

Therefore, $\mathbf$ does not have the path-independence property discussed above so is not conservative even if $\nabla \times \mathbf \equiv \mathbf$ since $U$ where $\mathbf$ is defined is not a simply connected open space.

Say again, in a simply connected open region, an irrotational vector field $\mathbf$ has the path-independence property (so $\mathbf$ as conservative). This can be proved directly by using Stokes' theorem,$\oint _ \mathbf \cdot d \mathbf = \iint _(\nabla \times \mathbf)\cdot d \mathbf = 0$for any smooth oriented surface $A$ which boundary is a simple closed path $P_c$. So, it is concluded that ''In a simply connected open region, any'' $C^1$ ''vector field that has the path-independence property (so it is a conservative vector field.) must also be irrotational and vise versa.''

Abstraction

More abstractly, in the presence of a Riemannian metric, vector fields correspond to differential form, differential . The conservative vector fields correspond to the closed and exact differential forms, exact , that is, to the forms which are the exterior derivative $d\phi$ of a function (scalar field) $\phi$ on $U$. The irrotational vector fields correspond to the closed and exact differential forms, closed , that is, to the $\omega$ such that $d\omega = 0$. As any exact form is closed, so any conservative vector field is irrotational. Conversely, all closed are exact if $U$ is simply connected.

Vorticity

The vorticity $\boldsymbol$ of a vector field can be defined by:
$\boldsymbol ~ \stackrel ~ \nabla \times \mathbf.$

The vorticity of an irrotational field is zero everywhere., pp. 194–196. Kelvin's circulation theorem states that a fluid that is irrotational in an inviscid flow will remain irrotational. This result can be derived from the vorticity transport equation, obtained by taking the curl of the Navier-Stokes Equations.

For a two-dimensional field, the vorticity acts as a measure of the ''local'' rotation of fluid elements. Note that the vorticity does ''not'' imply anything about the global behavior of a fluid. It is possible for a fluid to travel in a straight line to have vorticity, and it is possible for a fluid that moves in a circle to be irrotational.

Conservative forces

If the vector field associated to a force $\mathbf$ is conservative, then the force is said to be a conservative force.

The most prominent examples of conservative forces are a gravitational force and an electric force associated to an electrostatic field. According to Newton's law of universal gravitation, Newton's law of gravitation, a gravitational force $\mathbf_$ acting on a mass $m$ due to a mass $M$ located at a distance $r$ from $m$, obeys the equation

$\mathbf_ = - \frac \hat,$

where $G$ is the gravitational constant and $\hat$ is a ''unit'' vector pointing from $M$ toward $m$. The force of gravity is conservative because $\mathbf_ = - \nabla \Phi_$, where

$\Phi_ ~ \stackrel - \frac$

is the gravitational potential energy. In other words, the gravitation field $\frac$ associated with the gravitational force $\mathbf_$ is the gradient of the gravitation potential $\frac$ associated with the gravitational potential energy $\Phi_$. It can be shown that any vector field of the form $\mathbf=F(r) \hat$ is conservative, provided that $F(r)$ is integrable.

For conservative forces, ''path independence'' can be interpreted to mean that the work done in going from a point $A$ to a point $B$ is independent of the moving path chosen (dependent on only the points $A$ and $B$), and that the work $W$ done in going around a simple closed loop $C$ is $0$:

$W = \oint_ \mathbf \cdot d = 0.$

The total conservation of energy, energy of a particle moving under the influence of conservative forces is conserved, in the sense that a loss of potential energy is converted to the equal quantity of kinetic energy, or vice versa.

See also

* Beltrami vector field
* Conservative force
* Conservative system
* Complex lamellar vector field
* Helmholtz decomposition
* Laplacian vector field
* Longitudinal and transverse vector fields
* Solenoidal vector field

References

Further reading

* {{cite book , first= D. J. , last= Acheson , title= Elementary Fluid Dynamics , publisher= Oxford University Press , date= 1990 , isbn= 0198596790

Vector calculus
Force