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How to determine if a vector field is conservative

A conservative vector field(also called a path-independent vector field) is a vector field whose line integral

over any curve depends only on the endpoints of . The integral is independent of the path that takes going from

its starting point to its ending point. The below applet illustrates the two-dimensional conservative vector field

.

[Broken applet]

What are some ways to determine if a vector field is conservative? Directly checking to see if a line integral doesn't

depend on the path is obviously impossible, as you would have to check an infinite number of paths between any pair of

points. But, if you found two paths that gave different values of the integral, you could conclude the vector field waspath-dependent.

Here are some options that could be useful under different circumstances.

As mentioned in the context of the gradient theorem, a vector field is conservative if and only if it has a potential

function with . Therefore, if you are given a potential function or if you can find one, and that potential

function is defined everywhere, then there is nothing more to do. You know that is a conservative vector field, and

you don't need to worry about the other tests we mention here. Similarly, if you can demonstrate that it is impossible

to find a function that satisfies , then you can likewise conclude that is non-conservative, or

path-dependent.

For this reason, you could skip this discussion about testing for path-dependence and go directly to the procedure

for finding the potential function. If this procedure works or if it breaks down, you've found your answer as to

whether or not is conservative. However, if you are like many of us and are prone to make a mistake or two in a

multi-step procedure, you'd probably benefit from other tests that could quickly determine path-independence. That

way, you could avoid looking for a potential function when it doesn't exist and benefit from tests that confirm your

calculations.

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Another possible test involves the link between path-independenceand circulation. One can show that a

conservative vector field will have no circulationaround any closed curve , meaning that its integral

around must be zero. If you could somehow show that for every closed curve (difficult since there

are an infinite number of these), then you could conclude that is conservative. Or, if you can find one closed curve

where the integral is non-zero, then you've shown that it is path-dependent.

Although checking for circulation may not be a practical test for path-independence, the fact that path-independence

implies no circulation around any closed curve is a central to what it means for a vector field to be conservative.

2.

If is a three-dimensional vector field, (confused?), then we can derive another condition. This condition

is based on the fact that a vector field is conservative if and only if for some potential function. We can

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to determine if a vector field is conservative - Math Insight http://mathinsight.org/conservative_vector_field_d

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calculate that the curl of a gradient is zero, , for any twice continuously differentiable .

Therefore, if is conservative, then its curl must be zero, as .

For a continuously differentiable two-dimensional vector field, , we can similarly conclude that if the

vector field is conservative, then the scalar curl must be zero,

We have to be careful here. The valid statement is that if is conservative, then its curl must be zero. Without

additional conditions on the vector field, the converse may not be true, so we cannotconclude that is

conservative just from its curl being zero. There are path-dependent vector fields with zero curl. On the other hand,

we can conclude that if the curl of is non-zero, then must be path-dependent.

Can we obtain another test that allows us to determine for sure that a vector field is conservative? We can by linking

the previous two tests (tests 2 and 3). Test 2 states that the lack of macroscopic circulation is sufficient to

determine path-independence, but the problem is that lack of circulation around any closed curve is difficult to

check directly. Test 3 says that a conservative vector field has no microscopic circulation as captured by the curl.

It's easy to test for lack of curl, but the problem is that lack of curl is not sufficient to determine path-independence.

What we need way to link the definite test of zero macroscopic circulation with the easy-to-check test of zero

microscopic circulation. This link is exactly what both Green's theoremand Stokes' theoremprovide. Don't worry

if you haven't learned both these theorems yet. The basic idea is simple enough: the macroscopic circulation

around a closed curve is equal to the total microscopic circulation in the planar region inside the curve (for two

dimensions, Green's theorem) or in a surface whose boundary is the curve (for three dimensions, Stokes' theorem).

Let's examine the case of a two-dimensional vector field whose scalar curl is zero. If we have a closed

curve where is defined everywhere inside it, then we can apply Green's theorem to conclude that the

macroscopic circulation around is equal to the total microscopic circulation inside . We can indeed

conclude that the macroscopic circulation is zero from the fact that the microscopic circulation is zero

everywhere inside .

According to test 2, to conclude that is conservative, we need to be zero around everyclosed curve . If

the vector field is defined inside every closed curve and the microscopic circulation is zero everywhere inside

each curve, then Green's theorem gives us exactly that condition. We can conclude that around every

closed curve and the vector field is conservative.

The only way we could run into troubleis if there are some closed curves where is not defined for some points

inside the curve. In other words, if the region where is defined has some holes in it, then we cannot apply Green's

theorem for every closed curve . In this case, we cannot be certain that zero microscopic circulation implies zero

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macroscopic circulation and hence path-independence. Such a hole in the domain of definition of was exactly

what caused in the problem in our counterexample of a path-dependent field with zero curl.

On the other hand, we know we are safe if the region where is defined is simply connected, i.e., the region has no

holes through it. In this case, we know is defined inside every closed curve and nothing tricky can happen. We

can summarize our test for path-dependence of two-dimensional vector fields as follows.

If a vector field is continuously differentiable in a simply connected domain and its curl iszero, i.e.,

everywhere in , then is conservative within the domain .

It turns out the result for three-dimensions is essentially the same. If a vector field is continuously

differentiable in a simply connected domain and its curl is zero, i.e., , everywhere in , then

is conservative within the domain .

One subtle difference between two and three dimensions is what it means for a region to be simply connected. Any

hole in a two-dimensional domain is enough to make it non-simply connected. But, in three-dimensions, a simply-

connected domain can have a hole in the center, as long as the hole doesn't go all the way through the domain, as

illustrated in this figure.

The reason a hole in the center of a domain is not a problem in three dimensions is that we have more room to move

around in 3D. If we have a curl-free vector field (i.e., with no microscopic circulation), we can use Stokes'

theoremto infer the absence of macroscopic circulation around any closed curve . To use Stokes' theorem, we

just need to find a surface whose boundary is . If the domain of is simply connected, even if it has a hole that

doesn't go all the way through the domain, we can always find such a surface. The surface can just go around any

hole that's in the middle of the domain. With such a surface along which , we can use Stokes' theorem to

show that the circulation around is zero. Since we can do this for any closed curve, we can conclude that

is conservative.

The flexiblity we have in three dimensions to find multiple surfaces whose boundary is a given closed curve is

illustrated in this applet that we use to introduce Stokes' theorem.


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Macroscopic and microscopic circulation in three dimensions.The relationship between the macroscopic

circulation of a vector field around a curve (red boundary of surface) and the microscopic circulation of

(illustrated by small green circles) along a surface in three dimensions must hold for any surface whose

boundary is the curve. No matter which surface you choose (change by dragging the green point on the top

slider), the total microscopic circulation of along the surface must equal the circulation of around the

curve. (We assume that the vector field is defined everywhere on the surface.) You can change the curve to a

more complicated shape by dragging the blue point on the bottom slider, and the relationship between the

macroscopic and total microscopic circulation still holds. The surface is oriented by the shown normal vector

(moveable cyan arrow on surface), and the curve is oriented by the red arrow.

More information about applet.

Of course, if the region is not simply connected, but has a hole going all the way through it, then is not

a sufficient condition for path-independence. In this case, if is a curve that goes around the hole, then we cannot

find a surface that stays inside that domain whose boundary is . Without such a surface, we cannot use Stokes'theorem to conclude that the circulation around is zero.

See alsoA conservative vector field has no circulation

A path-dependent vector field with zero curl

Testing if three-dimensional vector fields are conservative


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Finding a potential function for conservative vector fields

Finding a potential function for three-dimensional conservative vector fields

Lighten upAn introduction to conservative vector fields

Cite this as

Nykamp DQ, How to determine if a vector field is conservative. FromMath Insight. http://mathinsight.org

/conservative_vector_field_determine

Keywords: conservative, gradient, gradient theorem, path independent, vector field

How to determine if a vector field is conservative by Duane Q. Nykampis licensed under a Creative Commons

Attribution-Noncommercial-ShareAlike 4.0 License. For permissions beyond the scope of this license, please contact us.


how to determine if a vector field is conservative - math insight.pdf

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