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Invector calculus, aconservative vector field is avector field that is thegradient of somefunction.^[1] A conservative vector field has the property that itsline 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 alsoirrotational; in three dimensions, this means that it has vanishingcurl. An irrotational vector field is necessarily conservative provided that the domain issimply connected.

Conservative vector fields appear naturally inmechanics: They are vector fields representingforces ofphysical systems in whichenergy isconserved.^[2] For a conservative system, thework done in moving along a path in a configuration space depends on only the endpoints of the path, so it is possible to definepotential 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${\displaystyle dR}$ 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.

M. C. Escher's lithograph printAscending 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.

Avector 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${\displaystyle \phi }$^[3] on${\displaystyle U}$ such that

$${\displaystyle \mathbf{v}=\mathrm{\nabla }\phi .}$$

Here,${\displaystyle \mathrm{\nabla }\phi }$ denotes thegradient of${\displaystyle \phi }$. Since${\displaystyle \phi }$ is continuously differentiable,${\displaystyle \mathbf{v}}$ is continuous. When the equation above holds,${\displaystyle \phi }$ is called ascalar potential for${\displaystyle \mathbf{v}}$.

Thefundamental theorem of vector calculus states that, under some regularity conditions, any vector field can be expressed as the sum of a conservative vector field and asolenoidal field.

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]$${\displaystyle {\int }_{P_1}\mathbf{v}\cdot d\mathbf{r}={\int }_{P_2}\mathbf{v}\cdot d\mathbf{r}}$$

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

$${\displaystyle {\int }_{P_c}\mathbf{v}\cdot d\mathbf{r}=0}$$for anypiecewise 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$${\displaystyle {\int }_{P_c}\mathbf{v}\cdot d\mathbf{r}={\int }_{P_1}\mathbf{v}\cdot d\mathbf{r}+{\int }_{P_2}\mathbf{v}\cdot d\mathbf{r}={\int }_{P_1}\mathbf{v}\cdot d\mathbf{r}-{\int }_{-P_2}\mathbf{v}\cdot d\mathbf{r}=0}$$where${\displaystyle -P_2}$ is the reverse of${\displaystyle P_2}$ and the last equality holds due to the path independence${\displaystyle {\int }_{P_1}\mathbf{v}\cdot d\mathbf{r}={\int }_{-P_2}\mathbf{v}\cdot d\mathbf{r}.}$

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${\displaystyle \mathbf{v}=\mathrm{\nabla }\phi }$ for some${\displaystyle C^1}$ (continuously differentiable) scalar field${\displaystyle \phi }$^[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 adifferentiable function) in${\displaystyle U}$ with an initial point${\displaystyle A}$ and a terminal point${\displaystyle B}$. Then thegradient theorem (also calledfundamental theorem of calculus for line integrals) states that$${\displaystyle {\int }_P\mathbf{v}\cdot d\mathbf{r}=\phi (B)-\phi (A).}$$

This holds as a consequence of thedefinition of a line integral, thechain rule, and thesecond fundamental theorem of calculus.${\displaystyle \mathbf{v}\cdot d\mathbf{r}=\mathrm{\nabla }\phi \cdot d\mathbf{r}}$ in the line integral is anexact differential for anorthogonal coordinate system (e.g.,Cartesian,cylindrical, orspherical 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.^[5]

So far it has been proven that a conservative vector field${\displaystyle \mathbf{v}}$ is line integral path-independent. Conversely,if a continuous vector field${\displaystyle \mathbf{v}}$ is (line integral) path-independent, then it is a conservative vector field, so the followingbiconditional statement holds:^[4]

For a continuousvector field${\displaystyle \mathbf{v}:U\to {\mathbb{R}}^n}$, where${\displaystyle U}$ is an open subset of${\displaystyle {\mathbb{R}}^n}$, it is conservative if and only if its line integral along a path in${\displaystyle U}$ is path-independent, meaning that the line integral depends on only both path endpoints regardless of which path between them is chosen.

The proof of this converse statement is the following.

${\displaystyle \mathbf{v}}$ is a continuous vector field which line integral is path-independent. Then, let's make a function${\displaystyle \phi }$ defined as$${\displaystyle \phi (x,y)={\int }_{a,b}^{x,y}\mathbf{v}\cdot d\mathbf{r}}$$over an arbitrary path between a chosen starting point${\displaystyle (a,b)}$ and an arbitrary point${\displaystyle (x,y)}$. Since it is path-independent, it depends on only${\displaystyle (a,b)}$ and${\displaystyle (x,y)}$ 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-dimensionalCartesian coordinate system is used. The second segment of this path is parallel to the${\displaystyle x}$ axis so there is no change along the${\displaystyle y}$ axis. The line integral along this path is$${\displaystyle {\int }_{a,b}^{x,y}\mathbf{v}\cdot d\mathbf{r}={\int }_{a,b}^{x_1,y}\mathbf{v}\cdot d\mathbf{r}+{\int }_{x_1,y}^{x,y}\mathbf{v}\cdot d\mathbf{r}.}$$By the path independence, itspartial derivative with respect to${\displaystyle x}$ (for${\displaystyle \phi }$ to have partial derivatives,${\displaystyle \mathbf{v}}$ needs to be continuous.) is$${\displaystyle \frac{\mathrm{\partial }\phi }{\mathrm{\partial }x}=\frac{\mathrm{\partial }}{\mathrm{\partial }x}{\int }_{a,b}^{x,y}\mathbf{v}\cdot d\mathbf{r}=\frac{\mathrm{\partial }}{\mathrm{\partial }x}{\int }_{a,b}^{x_1,y}\mathbf{v}\cdot d\mathbf{r}+\frac{\mathrm{\partial }}{\mathrm{\partial }x}{\int }_{x_1,y}^{x,y}\mathbf{v}\cdot d\mathbf{r}=0+\frac{\mathrm{\partial }}{\mathrm{\partial }x}{\int }_{x_1,y}^{x,y}\mathbf{v}\cdot d\mathbf{r}}$$since${\displaystyle x_1}$ and${\displaystyle x}$ are independent to each other. Let's express${\displaystyle \mathbf{v}}$ as${\displaystyle {\displaystyle \mathbf{v}}=P(x,y)\mathbf{i}+Q(x,y)\mathbf{j}}$ where${\displaystyle \mathbf{i}}$ and${\displaystyle \mathbf{j}}$ are unit vectors along the${\displaystyle x}$ and${\displaystyle y}$ axes respectively, then, since${\displaystyle d\mathbf{r}=dx\mathbf{i}+dy\mathbf{j}}$,$${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }x}\phi (x,y)=\frac{\mathrm{\partial }}{\mathrm{\partial }x}{\int }_{x_1,y}^{x,y}\mathbf{v}\cdot d\mathbf{r}=\frac{\mathrm{\partial }}{\mathrm{\partial }x}{\int }_{x_1,y}^{x,y}P(t,y)dt=P(x,y)}$$where the last equality is from thesecond fundamental theorem of calculus.

A similar approach for the line integral path shown in the right of the right figure results in$\frac{\mathrm{\partial }}{\mathrm{\partial }y}\phi (x,y)=Q(x,y)$ so$${\displaystyle \mathbf{v}=P(x,y)\mathbf{i}+Q(x,y)\mathbf{j}=\frac{\mathrm{\partial }\phi }{\mathrm{\partial }x}\mathbf{i}+\frac{\mathrm{\partial }\phi }{\mathrm{\partial }y}\mathbf{j}=\mathrm{\nabla }\phi }$$is proved for the 2-dimensionalCartesian coordinate system. This proof method can be straightforwardly expanded to a higher dimensional orthogonal coordinate system (e.g., a 3-dimensionalspherical coordinate system) so the converse statement is proved. Another proof is foundhere as the converse of the gradient theorem.

Let${\displaystyle n=3}$ (3-dimensional space), and let${\displaystyle \mathbf{v}:U\to {\mathbb{R}}^3}$ be a${\displaystyle C^1}$ (continuously differentiable) vector field, with an open subset${\displaystyle U}$ of${\displaystyle {\mathbb{R}}^n}$. Then${\displaystyle \mathbf{v}}$ is called irrotational if itscurl is${\displaystyle \mathbf{0}}$ everywhere in${\displaystyle U}$, i.e., if$${\displaystyle \mathrm{\nabla }\times \mathbf{v}\equiv \mathbf{0}.}$$

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 aslongitudinal vector fields.

It is anidentity of vector calculus that for any${\displaystyle C^2}$ (continuously differentiable up to the 2nd derivative) scalar field${\displaystyle \phi }$ on${\displaystyle U}$, we have$${\displaystyle \mathrm{\nabla }\times (\mathrm{\nabla }\phi )\equiv \mathbf{0}.}$$

Therefore,every${\displaystyle C^1}$ conservative vector field in${\displaystyle U}$ is also an irrotational vector field in${\displaystyle U}$. This result can be easily proved by expressing${\displaystyle \mathrm{\nabla }\times (\mathrm{\nabla }\phi )}$ in aCartesian coordinate system withSchwarz's theorem (also called Clairaut's theorem on equality of mixed partials).

Provided that${\displaystyle U}$ is asimply 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${\displaystyle U}$ is a${\displaystyle C^1}$ conservative vector field in${\displaystyle U}$.

The above statement isnot true in general if${\displaystyle U}$ is not simply connected. Let${\displaystyle U}$ be${\displaystyle {\mathbb{R}}^3}$ with removing all coordinates on the${\displaystyle z}$-axis (so not a simplyconnected space), i.e.,${\displaystyle U={\mathbb{R}}^3\setminus \{(0,0,z)\mid z\in \mathbb{R}\}}$. Now, define a vector field${\displaystyle \mathbf{v}}$ on${\displaystyle U}$ by$${\displaystyle \mathbf{v}(x,y,z)\stackrel{\text{def}}{=}\left(-\frac{y}{x^2+y^2},\frac{x}{x^2+y^2},0\right).}$$

Then${\displaystyle \mathbf{v}}$ has zero curl everywhere in${\displaystyle U}$ (${\displaystyle \mathrm{\nabla }\times \mathbf{v}\equiv \mathbf{0}}$ at everywhere in${\displaystyle U}$), i.e.,${\displaystyle \mathbf{v}}$ is irrotational. However, thecirculation of${\displaystyle \mathbf{v}}$ around theunit circle in the${\displaystyle xy}$-plane is${\displaystyle 2\pi }$; inpolar coordinates,${\displaystyle \mathbf{v}={\mathbf{e}}_{\varphi }/r}$, so the integral over the unit circle is$${\displaystyle {\oint }_C\mathbf{v}\cdot {\mathbf{e}}_{\varphi }d\varphi =2\pi .}$$

Therefore,${\displaystyle \mathbf{v}}$ does not have the path-independence property discussed above so is not conservative even if${\displaystyle \mathrm{\nabla }\times \mathbf{v}\equiv \mathbf{0}}$ since${\displaystyle U}$ where${\displaystyle \mathbf{v}}$ is defined is not a simply connected open space.

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

More abstractly, in the presence of aRiemannian metric, vector fields correspond todifferential${\displaystyle 1}$-forms. The conservative vector fields correspond to theexact${\displaystyle 1}$-forms, that is, to the forms which are theexterior derivative${\displaystyle d\varphi }$ of a function (scalar field)${\displaystyle \varphi }$ on${\displaystyle U}$. The irrotational vector fields correspond to theclosed${\displaystyle 1}$-forms, that is, to the${\displaystyle 1}$-forms${\displaystyle \omega }$ such that${\displaystyle d\omega =0}$. As${\displaystyle d^2=0}$, any exact form is closed, so any conservative vector field is irrotational. Conversely, all closed${\displaystyle 1}$-forms are exact if${\displaystyle U}$ issimply connected.

Thevorticity${\displaystyle \mathit{\omega }}$ of a vector field can be defined by:$${\displaystyle \mathit{\omega }\stackrel{\text{def}}{=}\mathrm{\nabla }\times \mathbf{v}.}$$

The vorticity of an irrotational field is zero everywhere.^[6]Kelvin's circulation theorem states that a fluid that is irrotational in aninviscid flow will remain irrotational. This result can be derived from thevorticity transport equation, obtained by taking the curl of theNavier–Stokes equations.

For a two-dimensional field, the vorticity acts as a measure of thelocal rotation of fluid elements. The vorticity doesnot imply anything about the global behavior of a fluid. It is possible for a fluid that travels in a straight line to have vorticity, and it is possible for a fluid that moves in a circle to be irrotational.

If the vector field associated to a force${\displaystyle \mathbf{F}}$ is conservative, then the force is said to be aconservative force.

The most prominent examples of conservative forces are gravitational force (associated with a gravitational field) and electric force (associated with an electrostatic field). According toNewton's law of gravitation, agravitational force${\displaystyle {\mathbf{F}}_G}$ acting on a mass${\displaystyle m}$ due to a mass${\displaystyle M}$ located at a distance${\displaystyle r}$ from${\displaystyle m}$, obeys the equation

$${\displaystyle {\mathbf{F}}_G=-\frac{GmM}{r^2}\hat{\mathbf{r}},}$$

where${\displaystyle G}$ is thegravitational constant and${\displaystyle \hat{\mathbf{r}}}$ is aunit vector pointing from${\displaystyle M}$ toward${\displaystyle m}$. The force of gravity is conservative because${\displaystyle {\mathbf{F}}_G=-\mathrm{\nabla }{\mathrm{\Phi }}_G}$, where

$${\displaystyle {\mathrm{\Phi }}_G\stackrel{\text{def}}{=}-\frac{GmM}{r}}$$

is thegravitational potential energy. In other words, the gravitation field${\displaystyle \frac{{\mathbf{F}}_G}{m}}$ associated with the gravitational force${\displaystyle {\mathbf{F}}_G}$ is thegradient of the gravitation potential${\displaystyle \frac{{\mathrm{\Phi }}_G}{m}}$ associated with the gravitational potential energy${\displaystyle {\mathrm{\Phi }}_G}$. It can be shown that any vector field of the form${\displaystyle \mathbf{F}=F(r)\hat{\mathbf{r}}}$ is conservative, provided that${\displaystyle F(r)}$ is integrable.

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

$${\displaystyle W={\oint }_C\mathbf{F}\cdot d\mathbf{r}=0.}$$

The totalmechanical 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 ofkinetic energy, or vice versa.

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

• Acheson, D. J. (1990).Elementary Fluid Dynamics. Oxford University Press.ISBN 0198596790.

• Vector calculus
• Force

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