# Conservative vector field

A **conservative vector field** is a vector field whose line integral between two points depends only on those points and not on the curve connecting them. Equivalently, its line integral around every closed curve in its domain is zero, and, on a simply connected domain, it can be written as the gradient of a scalar function called a potential.<sup>[1](https://www.maths.usyd.edu.au/u/daners/publ/vector-calculus/part-line-integrals-3-3.html)</sup><sup> • </sup><sup>[2](https://tutorial.math.lamar.edu/Classes/CalcIII/FundThmLineIntegrals.aspx)</sup>

| Key fact | Detail |
|---|---|
| Definition | A continuous vector field on an open set is conservative if \( \int_C \mathbf{f} \cdot d\mathbf{x} \) has the same value for every piecewise smooth curve \( C \) with the same start and end points.<sup>[1](https://www.maths.usyd.edu.au/u/daners/publ/vector-calculus/part-line-integrals-3-3.html)</sup> |
| Loop property | A field is conservative if and only if \( \oint_C \mathbf{f} \cdot d\mathbf{x} = 0 \) for every piecewise smooth closed curve \( C \) in the domain.<sup>[1](https://www.maths.usyd.edu.au/u/daners/publ/vector-calculus/part-line-integrals-3-3.html)</sup> |
| Potential | A conservative field is a gradient field: \( \mathbf{F} = \nabla f \), and \( \int_C \mathbf{F} \cdot d\mathbf{r} = f(\mathbf{b}) - f(\mathbf{a}) \) for any curve from \( \mathbf{a} \) to \( \mathbf{b} \).<sup>[1](https://www.maths.usyd.edu.au/u/daners/publ/vector-calculus/part-line-integrals-3-3.html)</sup><sup> • </sup><sup>[2](https://tutorial.math.lamar.edu/Classes/CalcIII/FundThmLineIntegrals.aspx)</sup> |
| Uniqueness | A potential function is unique only up to an additive constant on a connected domain.<sup>[3](https://mathworld.wolfram.com/PotentialFunction.html)</sup> |
| Curl test | On an open, simply connected domain, \( \mathbf{F} = \langle P, Q, R \rangle \) is conservative if and only if \( P_y = Q_x \), \( P_z = R_x \), and \( Q_z = R_y \) throughout the domain.<sup>[4](https://openstax.org/books/calculus-volume-3/pages/6-3-conservative-vector-fields)</sup> |
| Physics meaning | A conservative force does work independent of path, and the total mechanical energy (kinetic plus potential) is conserved when only such forces act.<sup>[11](https://www.physicsclassroom.com/getattachment/Physics-Video-Tutorial/Work-Energy-and-Power/Forces-and-System-Analysis/Lecture-Notes/LessonNotes.pdf?lang=en-US)</sup> |

## Equivalent characterizations

Three properties of a vector field on a connected open domain turn out to be the same condition, and textbooks use any of them as the definition.<sup>[6](https://sites.millersville.edu/bikenaga/calculus3/conservative-fields/conservative-fields.html)</sup>

- Path independence: for any two points \( P \) and \( Q \), the integral \( \int_C \mathbf{F} \cdot d\mathbf{r} \) yields the same value for every path connecting \( P \) to \( Q \).<sup>[7](https://math.mit.edu/~dunkel/Teach/18.04_2019S/review/orloff-greenstheorem.pdf)</sup>
- Zero circulation: the integral around every closed curve is zero. The equivalence with path independence follows by joining a path to the reverse of another path with the same endpoints, forming a closed curve whose integral is the difference of the two path integrals.<sup>[1](https://www.maths.usyd.edu.au/u/daners/publ/vector-calculus/part-line-integrals-3-3.html)</sup><sup> • </sup><sup>[2](https://tutorial.math.lamar.edu/Classes/CalcIII/FundThmLineIntegrals.aspx)</sup>
- Gradient form: there is a scalar function \( f \) with \( \mathbf{F} = \nabla f \). Such a field is also called a gradient field or potential field.<sup>[1](https://www.maths.usyd.edu.au/u/daners/publ/vector-calculus/part-line-integrals-3-3.html)</sup><sup> • </sup><sup>[8](https://encyclopediaofmath.org/wiki/Potential_field)</sup>

The Encyclopedia of Mathematics states the same ideas in the language of circulation: if the circulation of a field along an arbitrary closed piecewise-smooth curve in a domain is zero, the field is called potential (or conservative) in that domain.<sup>[9](https://encyclopediaofmath.org/wiki/Vector_analysis)</sup>

## The Fundamental Theorem for Line Integrals

The central computational tool is the Fundamental Theorem for Line Integrals, the analog of the Fundamental Theorem of Calculus. If \( C \) is a smooth curve given by \( \mathbf{r}(t) \), \( a \le t \le b \), and \( f \) is a function whose gradient is continuous on \( C \), then

\[ \int_C \nabla f \cdot d\mathbf{r} = f(\mathbf{r}(b)) - f(\mathbf{r}(a)). \]

The proof uses the chain rule to write the integrand as \( \frac{d}{dt} f(\mathbf{r}(t)) \) and then applies the single-variable Fundamental Theorem of Calculus.<sup>[2](https://tutorial.math.lamar.edu/Classes/CalcIII/FundThmLineIntegrals.aspx)</sup> Consequently, evaluating a line integral of a conservative field is a two-step process: find a potential function \( f \), then compute \( f \) at the endpoints and subtract.<sup>[4](https://openstax.org/books/calculus-volume-3/pages/6-3-conservative-vector-fields)</sup> No parameterization of the path is needed, which matters when the path is complicated or only its endpoints are known.<sup>[4](https://openstax.org/books/calculus-volume-3/pages/6-3-conservative-vector-fields)</sup>

As an illustration, for \( \mathbf{F}(x,y,z) = \langle 2xe^y z + e^x z,\ x^2 e^y z,\ x^2 e^y + e^x \rangle \), whose domain \( \mathbb{R}^3 \) is simply connected, the cross-partial test confirms the field is conservative, and a potential is \( f(x,y,z) = x^2 e^y z + e^x z \). The integral along any smooth curve from the origin to \( (1,1,1) \) is then \( f(1,1,1) - f(0,0,0) = 3e \).<sup>[4](https://openstax.org/books/calculus-volume-3/pages/6-3-conservative-vector-fields)</sup>

## Testing for conservativeness

Every conservative field with continuous partial derivatives satisfies the cross-partial property: for \( \mathbf{F} = \langle P, Q, R \rangle \), the equalities \( P_y = Q_x \), \( P_z = R_x \), and \( Q_z = R_y \) hold throughout the domain. In vector language, \( \nabla \times \mathbf{F} = \mathbf{0} \), that is, the field is curl-free or rotation-free.<sup>[4](https://openstax.org/books/calculus-volume-3/pages/6-3-conservative-vector-fields)</sup><sup> • </sup><sup>[5](https://opentextbooks.library.arizona.edu/mathematicalmodeling/chapter/appendix-vector-calculus/)</sup>

The converse holds only under a topological hypothesis. On an open, simply connected domain, meaning a connected domain with no holes, the cross-partial property implies conservativeness, so the property is both necessary and sufficient there.<sup>[4](https://openstax.org/books/calculus-volume-3/pages/6-3-conservative-vector-fields)</sup> MathWorld states the same result: a continuously differentiable vector field with zero curl has a potential function on a simply connected domain.<sup>[3](https://mathworld.wolfram.com/PotentialFunction.html)</sup>

**Why simple connectivity matters.** The punctured plane \( \mathbb{R}^2 \setminus \{(0,0) \} \) is the standard warning example. A field defined on all of \( \mathbb{R}^2 \) except the origin can satisfy the cross-partial property everywhere on its domain and still fail to be conservative, because the domain has a hole and the test does not apply there.<sup>[4](https://openstax.org/books/calculus-volume-3/pages/6-3-conservative-vector-fields)</sup> The MIT supplementary notes give the physical version: in a region of space threaded by a current-carrying wire, the resulting electromagnetic force field satisfies \( \operatorname{curl} \mathbf{F} = \mathbf{0} \) but is not conservative.<sup>[10](https://math.mit.edu/~jorloff/suppnotes/suppnotes02/v14.pdf)</sup> Conversely, restricting a conservative field on \( \mathbb{R}^2 \) to a non-simply-connected subregion keeps it conservative, so holes do not by themselves destroy the property; they only invalidate the curl test as a sufficient condition.<sup>[4](https://openstax.org/books/calculus-volume-3/pages/6-3-conservative-vector-fields)</sup>

## Finding a potential function

When a field is known to be conservative, its potential is recovered by integration. For \( \mathbf{F} = \langle P, Q \rangle \) in two variables, integrate \( P \) with respect to \( x \); because \( y \) is held constant, the constant of integration may be a function \( h(y) \). Matching the remaining component against \( Q \) determines \( h \), up to a constant.<sup>[4](https://openstax.org/books/calculus-volume-3/pages/6-3-conservative-vector-fields)</sup> For example, from \( f_x = 2xy^2 \) one obtains \( f(x,y) = x^2y^2 + h(y) \), and the condition \( f_y = 2x^2y \) forces \( h'(y) = 0 \), so \( f(x,y) = x^2y^2 \) is a potential.<sup>[4](https://openstax.org/books/calculus-volume-3/pages/6-3-conservative-vector-fields)</sup>

The potential itself is never unique: adding a constant changes nothing about the gradient, so a potential is determined only up to an additive constant.<sup>[1](https://www.maths.usyd.edu.au/u/daners/publ/vector-calculus/part-line-integrals-3-3.html)</sup><sup> • </sup><sup>[3](https://mathworld.wolfram.com/PotentialFunction.html)</sup>

## Role in physics

In mechanics and electrostatics the sign convention \( \mathbf{F} = -\nabla V \) is commonly used, so the force points toward decreasing potential \( V \).<sup>[3](https://mathworld.wolfram.com/PotentialFunction.html)</sup> For a force with a potential \( V \), the work done as a point mass moves along a path equals the decrease in \( V \), which equals the gain in kinetic energy \( \frac{1}{2}mv^2 \); rearranging gives

\[ \frac{1}{2}mv^2(t_0) + V(\mathbf{r}(t_0)) = \frac{1}{2}mv^2(t_1) + V(\mathbf{r}(t_1)), \]

so the total energy, kinetic plus potential, is conserved. This is the origin of the word conservative.<sup>[5](https://opentextbooks.library.arizona.edu/mathematicalmodeling/chapter/appendix-vector-calculus/)</sup> Gravity forces and spring forces are the standard examples of conservative forces, and when only such forces do net work on an object, mechanical energy is exchanged between kinetic and potential forms but its total stays constant.<sup>[11](https://www.physicsclassroom.com/getattachment/Physics-Video-Tutorial/Work-Energy-and-Power/Forces-and-System-Analysis/Lecture-Notes/LessonNotes.pdf?lang=en-US)</sup>

The concept extends across physics. The majority of force fields and electric fields can be treated as potential fields; for an ideal fluid, the equilibrium pressure force on a volume element is \( \mathbf{F} = -\operatorname{grad} f \cdot d\omega \) in terms of the pressure \( f \), and heat flow in a heated body is \( \mathbf{F} = -k \cdot \operatorname{grad} f \) in terms of the temperature.<sup>[8](https://encyclopediaofmath.org/wiki/Potential_field)</sup> Where a conservative field has no sources, its potential satisfies [Laplace's equation](https://www.edgepedia.org/laplaces-equation), \( \Delta u = \operatorname{div} \operatorname{grad} u = -\operatorname{div} \mathbf{v} = 0 \); the gravitational potential in a region free of attracting masses and the electrostatic potential in a charge-free region are examples. In the presence of sources, the right-hand side becomes proportional to the source density and the equation becomes [Poisson's equation](https://www.edgepedia.org/poissons-equation).<sup>[12](https://encyclopediaofmath.org/wiki/Laplace_equation)</sup>

One boundary of the concept is worth noting. Physicists customarily limit the term conservative to velocity-independent forces expressible as the gradient of a potential; the magnetic force cannot be written as a gradient, yet it conserves energy because it can be expressed through an Euler-Lagrange equation, so some formalisms use a deliberately less restrictive definition.<sup>[13](https://link.aps.org/accepted/10.1103/PhysRevE.104.055302)</sup> Research also continues on whether certain physical forces are conservative at all: whether the force on an ion in the presence of an electric current can be written as the derivative of a potential remained an open question in a 2004 [Physical Review Letters](https://www.edgepedia.org/physical-review-letters) analysis, which proposed mechanisms and first-principles calculations bearing on it without a formal resolution.<sup>[14](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.92.176803)</sup>

## References

1. [Section 8.1 Gradient Vector Fields and Potentials, University of Sydney](https://www.maths.usyd.edu.au/u/daners/publ/vector-calculus/part-line-integrals-3-3.html)
2. [Calculus III: Fundamental Theorem for Line Integrals, Paul Dawkins, Lamar University](https://tutorial.math.lamar.edu/Classes/CalcIII/FundThmLineIntegrals.aspx)
3. [Potential Function, Wolfram MathWorld](https://mathworld.wolfram.com/PotentialFunction.html)
4. [Calculus Volume 3, Section 6.3: Conservative Vector Fields, OpenStax](https://openstax.org/books/calculus-volume-3/pages/6-3-conservative-vector-fields)
5. [Refresher: Vector Calculus, Introduction to Mathematical Modeling, University of Arizona](https://opentextbooks.library.arizona.edu/mathematicalmodeling/chapter/appendix-vector-calculus/)
6. [Conservative Vector Fields, Millersville University](https://sites.millersville.edu/bikenaga/calculus3/conservative-fields/conservative-fields.html)
7. [8: Gradient and Conservative Fields, MIT 18.04 notes (Orloff)](https://math.mit.edu/~dunkel/Teach/18.04_2019S/review/orloff-greenstheorem.pdf)
8. [Potential field, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Potential_field)
9. [Vector analysis, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Vector_analysis)
10. [V14: Some Topological Questions, MIT 18.03 supplementary notes (Jorloff)](https://math.mit.edu/~jorloff/suppnotes/suppnotes02/v14.pdf)
11. [Forces and System Analysis Lesson Notes, The Physics Classroom](https://www.physicsclassroom.com/getattachment/Physics-Video-Tutorial/Work-Energy-and-Power/Forces-and-System-Analysis/Lecture-Notes/LessonNotes.pdf?lang=en-US)
12. [Laplace equation, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Laplace_equation)
13. [Conservative and non-conservative forces in the Lagrangian neural network formalism, Physical Review E 104, 055302 (2021)](https://link.aps.org/accepted/10.1103/PhysRevE.104.055302)
14. [M. Di Ventra, Y.-C. Chen, T. N. Todorov (2004). Are Current-Induced Forces Conservative? Physical Review Letters 92, 176803](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.92.176803)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Analysis and mathematical models › Multivariable and vector calculus*

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