Vector Field Conservative? A Simple Guide to Determine

Conservative Vector Fields, studied extensively in advanced calculus, possess unique properties. These properties lead to path-independent line integrals, a concept heavily utilized by Physicists when calculating work done by conservative forces. Stokes' Theorem, a fundamental theorem in vector calculus, offers a powerful method related to how to determine if a vector field is conservative, and often involves examining the curl of the field. Knowing how to determine if a vector field is conservative offers many benefits across other mathematics areas and is applicable in understanding concepts about energy, forces, and flow, especially relating to scalar potential functions.

Image taken from the YouTube channel Dr. Trefor Bazett , from the video titled How to Test if a Vector Field is Conservative // Vector Calculus .

The world around us, from the flow of fluids to the pull of gravity, can often be described and analyzed using the mathematical tool of vector fields. Understanding these fields is crucial in many areas of science and engineering.

What is a Vector Field?

Imagine assigning a vector to every point in space. This collection of vectors forms a vector field.

Each vector represents a magnitude and direction, such as the velocity of wind at a specific location, or the force exerted by an electric charge at a given point.

Formally, a vector field in two dimensions, denoted as F, can be expressed as F(x, y) = P(x, y)i + Q(x, y)j, where P and Q are scalar functions of x and y, and i and j are the unit vectors in the x and y directions, respectively. This concept extends to three dimensions and beyond.

Vector fields provide a powerful way to visualize and analyze phenomena that vary spatially. They find applications in fluid dynamics, electromagnetism, gravitational fields, and many other areas of physics and engineering.

Defining Conservative Vector Fields

Within the broader category of vector fields lies a special class known as conservative vector fields. These fields possess a unique property: the work done by the field on an object moving between two points is independent of the path taken.

This path independence is a consequence of the existence of a scalar potential function. A scalar potential, often denoted by φ, is a scalar field whose gradient is equal to the conservative vector field. In other words, F = ∇φ, where ∇ is the gradient operator.

A classic example is the gravitational field near the Earth's surface. The work done in lifting an object from one height to another depends only on the difference in height, not on the specific path taken. This is because the gravitational force is a conservative force.

Why are Conservative Vector Fields Important?

Conservative vector fields simplify calculations in physics and engineering. The path independence property dramatically reduces the complexity of computing work done.

If a field is conservative, we can evaluate line integrals (which calculate work) by simply evaluating the potential function at the endpoints of the path. This avoids the need to perform a potentially complex integration along the path.

Conservative fields are also fundamental to understanding concepts like potential energy and the conservation of energy.

Purpose of this Guide

This guide provides a clear, step-by-step method to determine if a given vector field is conservative. We will explore the mathematical tools and techniques necessary to identify conservative vector fields and understand their properties.

By the end of this guide, you will be equipped with the knowledge to confidently assess the conservativeness of vector fields and appreciate their significance in various applications.

Fundamental Concepts: Building the Foundation

As we explore the fascinating world of vector fields, understanding the concept of conservative vector fields requires a solid foundation in a few key mathematical ideas. We must first discuss the scalar potential, and gradient vector fields. We will also need to explore the concept of line integrals.

These concepts are not just abstract mathematical tools; they provide the necessary framework for understanding the unique properties of conservative vector fields and their applications in various scientific and engineering disciplines.

Scalar Potential and Conservative Vector Fields

At the heart of conservative vector fields lies the concept of a scalar potential, often also called a potential function. This is a scalar field, denoted by φ(x, y) in two dimensions or φ(x, y, z) in three dimensions. Its gradient relates to the conservative vector field.

In simpler terms, imagine a landscape where the height at any point is given by the scalar potential function.

The conservative vector field then represents the direction and magnitude of the steepest uphill slope at each point on that landscape.

The crucial link between a scalar potential and a conservative vector field is that the vector field is the gradient of the scalar potential.

Mathematically, this relationship is expressed as:

F = ∇φ

Where F is the conservative vector field and ∇φ is the gradient of the scalar potential φ.

This relationship implies that the components of the vector field are the partial derivatives of the scalar potential.

Gradient Vector Fields

The gradient vector field is a vector field that points in the direction of the greatest rate of increase of a scalar field.

As introduced above, it is mathematically represented as the gradient (∇) of a scalar function, also known as the scalar potential.

If φ(x, y) is a scalar function, then its gradient is given by:

∇φ = (∂φ/∂x) i + (∂φ/∂y) j

Where i and j are the unit vectors in the x and y directions, respectively.

The gradient vector field is always perpendicular to the level curves (or level surfaces in 3D) of the scalar potential. This means that it points in the direction of the steepest ascent of the scalar field.

Understanding the gradient vector field is crucial because it provides a visual and mathematical representation of how a scalar field changes in space.

It also provides a crucial link between scalar potentials and conservative vector fields. A vector field is conservative if and only if it is the gradient of some scalar potential.

Line Integrals and Work Done

A line integral is an integral where the function to be integrated is evaluated along a curve. In the context of vector fields, a line integral calculates the work done by a vector field on an object moving along a specific path.

If F is a vector field and C is a curve parameterized by r(t), where a ≤ t ≤ b, then the line integral of F along C is given by:

∫_C F · dr = ∫_a^b F(r(t)) · r'(t) dt

Where r'(t) is the derivative of the parameterization with respect to t.

The line integral represents the accumulation of the component of the vector field along the tangent direction of the curve.

In physical terms, it calculates the work done by the force represented by the vector field in moving an object along the path C.

The significance of line integrals lies in their ability to quantify the interaction between a vector field and a path. They are fundamental in physics for calculating work, circulation, and flux.

Path Independence

Path independence is a critical property that defines conservative vector fields. A vector field F is conservative if the line integral of F between any two points is independent of the path taken between those points.

In other words, the work done by a conservative vector field in moving an object from point A to point B depends only on the positions of A and B, and not on the specific path followed.

This path independence is a direct consequence of the existence of a scalar potential. If F is the gradient of a scalar potential φ, then the line integral of F from point A to point B is simply the difference in the values of φ at those points:

∫_C F · dr = φ(B) - φ(A)

This result, known as the Fundamental Theorem of Calculus for Line Integrals, shows that the line integral depends only on the endpoints and not on the path C.

Path independence greatly simplifies calculations involving conservative vector fields. It allows us to determine the work done by the field without having to evaluate a line integral along a specific path.

Closed Loops

A closed loop is a path that starts and ends at the same point. In the context of conservative vector fields, the line integral of a conservative vector field around any closed loop is always zero.

This property is a direct consequence of path independence. Since the starting and ending points are the same, the difference in the scalar potential is zero:

∮ F · dr = 0

This property provides another way to identify conservative vector fields. If the line integral of a vector field around every closed loop is zero, then the vector field is conservative. Conversely, if there exists even one closed loop for which the line integral is non-zero, then the vector field is not conservative.

Understanding closed loops and their relationship to line integrals is crucial for grasping the fundamental properties of conservative vector fields and their applications.

The Conservativeness Test: A Practical Method

Having built a foundation of scalar potentials, gradient vector fields, and line integrals, we're now equipped to tackle the central question: how do we actually determine if a vector field is conservative?

This section provides a practical, step-by-step method to answer this question, focusing primarily on two-dimensional vector fields.

The Two-Dimensional Conservativeness Test

In two dimensions, a vector field F(x, y) = P(x, y)i + Q(x, y)j is conservative if and only if the following condition holds:

∂P/∂y = ∂Q/∂x

This equation states that the partial derivative of P with respect to y must equal the partial derivative of Q with respect to x.

This test is a direct consequence of the fact that if F is the gradient of a scalar potential φ, then P = ∂φ/∂x and Q = ∂φ/∂y.

Therefore, ∂P/∂y = ∂²φ/∂y∂x and ∂Q/∂x = ∂²φ/∂x∂y.

Under suitable conditions (continuity of second partial derivatives), these mixed partial derivatives are equal.

Performing the Test: A Step-by-Step Guide

Let's break down the process of applying the conservativeness test into manageable steps:

1. Identify P and Q: Given a vector field F(x, y) = P(x, y)i + Q(x, y)j, clearly identify the functions P(x, y) and Q(x, y).

2. Compute Partial Derivatives: Calculate the partial derivative of P with respect to y (∂P/∂y) and the partial derivative of Q with respect to x (∂Q/∂x).

   Remember that when taking a partial derivative with respect to a particular variable, treat all other variables as constants.

3. Compare Partial Derivatives: Compare the results obtained in the previous step. If ∂P/∂y = ∂Q/∂x, then the vector field is likely conservative.

   If ∂P/∂y ≠ ∂Q/∂x, then the vector field is not conservative.

4. Important Caveat: The test ∂P/∂y = ∂Q/∂x only guarantees that the vector field is conservative if the domain of the vector field is simply connected. Intuitively, a simply connected domain is one without any "holes." If the domain is not simply connected, additional analysis might be necessary.

Examples: Conservative vs. Non-Conservative Vector Fields

To solidify your understanding, let's examine a few examples:

Example 1: A Conservative Vector Field

Consider the vector field F(x, y) = (2x + y)i + (x + 2y)j.

Here, P(x, y) = 2x + y and Q(x, y) = x + 2y.

Let's compute the partial derivatives:

• ∂P/∂y = 1
• ∂Q/∂x = 1

Since ∂P/∂y = ∂Q/∂x, this vector field is conservative.

Example 2: A Non-Conservative Vector Field

Consider the vector field F(x, y) = (y²)i + (x²)j.

Here, P(x, y) = y² and Q(x, y) = x².

Let's compute the partial derivatives:

• ∂P/∂y = 2y
• ∂Q/∂x = 2x

Since ∂P/∂y ≠ ∂Q/∂x, this vector field is not conservative.

Example 3: A Conservative Vector Field with Higher Order Polynomials

Consider the vector field F(x, y) = (3x²y + y³)i + (x³ + 3xy²)j.

Here, P(x, y) = 3x²y + y³ and Q(x, y) = x³ + 3xy².

Let's compute the partial derivatives:

• ∂P/∂y = 3x² + 3y²
• ∂Q/∂x = 3x² + 3y²

Since ∂P/∂y = ∂Q/∂x, this vector field is conservative.

Example 4: A Non-Conservative Vector Field with Trigonometric Functions

Consider the vector field F(x, y) = (sin(y))i + (cos(x))j.

Here, P(x, y) = sin(y) and Q(x, y) = cos(x).

Let's compute the partial derivatives:

• ∂P/∂y = cos(y)
• ∂Q/∂x = -sin(x)

Since ∂P/∂y ≠ ∂Q/∂x, this vector field is not conservative.

By carefully applying the conservativeness test and working through examples, you can confidently determine whether a given two-dimensional vector field is conservative. Remember to always double-check your partial derivative calculations and be mindful of the domain of the vector field.

Having established a reliable method for identifying conservative vector fields, it's time to explore the profound implications of this property. The connection between conservative vector fields and line integrals is not merely a mathematical curiosity; it's a powerful tool that simplifies calculations and reveals deeper insights into the nature of these fields. This connection is formalized by the Fundamental Theorem of Calculus for Line Integrals, a cornerstone of vector calculus.

Linking to Line Integrals: The Fundamental Theorem

The Fundamental Theorem of Calculus for Line Integrals provides an elegant and efficient way to evaluate line integrals of conservative vector fields. It bypasses the often tedious process of parameterizing the path and directly integrating along it. Instead, it relies solely on the values of the scalar potential at the endpoints of the path.

The Fundamental Theorem Explained

The theorem states that if F is a conservative vector field with scalar potential φ, and C is a smooth curve parameterized by r(t) for a ≤ t ≤ b, then:

∫_C F · dr = φ(r(b)) - φ(r(a))

In simpler terms, the line integral of a conservative vector field F along a curve C is equal to the difference in the scalar potential φ evaluated at the final point r(b) and the initial point r(a) of the curve.

Simplifying Line Integral Calculations

This theorem dramatically simplifies the calculation of line integrals for conservative vector fields. Once you've established that a vector field is conservative and found its scalar potential, evaluating the line integral becomes a matter of simple substitution.

No parameterization of the path is required, and the integral itself vanishes from the equation. This is particularly advantageous when dealing with complex paths or when the line integral would be difficult to compute directly.

The core idea is that the path taken doesn't matter.

Only the start and end points are needed.

Finding the Scalar Potential Function

The key to utilizing the Fundamental Theorem lies in finding the scalar potential function φ for a given conservative vector field F. Since F is the gradient of φ (i.e., F = ∇φ), we can find φ by integrating the components of F.

If F(x, y) = P(x, y) i + Q(x, y) j, then:

∂φ/∂x = P(x, y) ∂φ/∂y = Q(x, y)

Integrating the first equation with respect to x, we get:

φ(x, y) = ∫ P(x, y) dx + g(y)

Here, g(y) is an arbitrary function of y, representing the "constant of integration" with respect to x.

Next, differentiate this expression with respect to y:

∂φ/∂y = ∂/∂y [∫ P(x, y) dx] + g'(y)

Set this equal to Q(x, y) and solve for g'(y):

g'(y) = Q(x, y) - ∂/∂y [∫ P(x, y) dx]

Finally, integrate g'(y) with respect to y to find g(y). Substitute this back into the expression for φ(x, y) to obtain the scalar potential.

Example:

Let's say F(x, y) = (2xy + y²) i + (x² + 2xy) j.

First, we confirm that F is conservative: ∂P/∂y = 2x + 2y and ∂Q/∂x = 2x + 2y. Since these are equal, F is conservative.

Now, we find the scalar potential φ:

φ(x, y) = ∫ (2xy + y²) dx = x²y + xy² + g(y) ∂φ/∂y = x² + 2xy + g'(y) = x² + 2xy g'(y) = 0 g(y) = C (a constant)

Thus, φ(x, y) = x²y + xy² + C.

This process, while sometimes requiring careful integration, provides a systematic way to determine the scalar potential, unlocking the power of the Fundamental Theorem for simplifying line integral calculations. Remember to always check your work by taking the gradient of your calculated potential function to ensure that it matches the original vector field. If it doesn't, revisit your integration steps to find the mistake.

Having witnessed the elegance of the Fundamental Theorem in simplifying line integrals for conservative vector fields, it's natural to wonder how these concepts extend to higher dimensions. While the two-dimensional test (∂P/∂y = ∂Q/∂x) provides a straightforward method for identifying conservative vector fields in the plane, the three-dimensional realm requires a different tool: the curl. This extension allows us to determine conservativeness in a more complex setting, opening doors to a wider range of applications and insights.

Extending to 3D: The Curl Test (Optional)

While the principles of conservative vector fields remain consistent across dimensions, the methods for identifying them evolve. In three-dimensional space, the partial derivative test we used in 2D is no longer sufficient. Instead, we rely on the curl of a vector field to determine if it is conservative.

What is the Curl of a Vector Field?

The curl is a vector operator that describes the infinitesimal rotation of a vector field at a given point. Intuitively, it measures the tendency of the vector field to induce rotation around that point.

Mathematically, for a vector field F = Pi + Qj + Rk in three dimensions, the curl is defined as:

curl F = (∂R/∂y - ∂Q/∂z)i + (∂P/∂z - ∂R/∂x)j + (∂Q/∂x - ∂P/∂y)k

This can be conveniently represented using the determinant of a matrix:

Where ∇ is the del operator. The resulting vector represents the axis and magnitude of the infinitesimal rotation.

The Curl Test for Conservativeness

The curl provides a powerful test for determining if a vector field in three dimensions is conservative.

The fundamental theorem states that a vector field F is conservative if and only if its curl is the zero vector:

curl F = 0

This means that all three components of the curl must be equal to zero. In other words:

• ∂R/∂y - ∂Q/∂z = 0
• ∂P/∂z - ∂R/∂x = 0
• ∂Q/∂x - ∂P/∂y = 0

If the curl of a vector field is zero everywhere in a simply connected region, then the vector field is conservative in that region.

Conversely, if the curl is non-zero at any point, the vector field is not conservative.

Practical Implications and Applications

The curl test is invaluable in various fields, including fluid dynamics and electromagnetism. For instance, a fluid flow is irrotational (conservative) if its curl is zero, indicating that there are no swirling motions within the fluid.

Similarly, in electromagnetism, the curl of the electric field is related to the rate of change of the magnetic field, and the curl of the magnetic field is related to the current density. These relationships are fundamental to understanding electromagnetic phenomena.

Advanced Connection: Green's Theorem

Having witnessed the elegance of the Fundamental Theorem in simplifying line integrals for conservative vector fields, it's natural to wonder how these concepts extend to more general situations. Green's Theorem provides that extension, connecting line integrals around closed curves to double integrals over the region enclosed by those curves. This theorem offers a powerful tool for verifying conservativeness in two dimensions and unveils deeper relationships between vector calculus concepts.

Unveiling Green's Theorem

Green's Theorem establishes a fundamental relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C. Specifically, it states:

∮_C P dx + Q dy = ∬_D (∂Q/∂x - ∂P/∂y) dA

where:

• C is a positively oriented (counterclockwise) simple closed curve in the plane.
• P and Q are functions of x and y that have continuous partial derivatives on an open region containing D.
• dA represents the area element in the double integral.

In essence, Green's Theorem transforms a line integral, which is a one-dimensional integral along a curve, into a double integral, which is a two-dimensional integral over a region.

Green's Theorem and Conservative Vector Fields

Green's Theorem offers a powerful method for verifying that a vector field is conservative within a two-dimensional region. Recall that a vector field F = P i + Q j is conservative if and only if ∂Q/∂x = ∂P/∂y.

• If the double integral ∬_D (∂Q/∂x - ∂P/∂y) dA evaluates to zero, then the line integral ∮_C P dx + Q dy around any closed curve C within the region D is also zero.

• This is precisely the condition for a vector field to be conservative: the line integral around any closed loop is zero.

Therefore, by computing the double integral in Green's Theorem, we can determine whether the vector field is conservative within the region of integration.

Verifying Conservativeness with Green's Theorem: A Step-by-Step Approach

To verify that a vector field F = P i + Q j is conservative in a two-dimensional region D using Green's Theorem, follow these steps:

1. Identify P and Q: Determine the component functions P(x, y) and Q(x, y) of the vector field F.

2. Compute Partial Derivatives: Calculate the partial derivatives ∂P/∂y and ∂Q/∂x.

3. Set up the Double Integral: Construct the double integral ∬_D (∂Q/∂x - ∂P/∂y) dA over the region D.

4. Evaluate the Double Integral: Evaluate the double integral.

5. Interpret the Result:

   • If the double integral evaluates to zero, then ∂Q/∂x = ∂P/∂y throughout the region D, and the vector field F is conservative in D.

   • If the double integral does not evaluate to zero, then the vector field is not conservative in D.

A Word of Caution

It is important to note that Green's Theorem applies to simple closed curves, meaning that the curve does not intersect itself. Additionally, the theorem requires that the functions P and Q and their partial derivatives are continuous on an open region containing D.

While Green's Theorem provides a valuable tool for verifying conservativeness, it's not always the most efficient method. In some cases, directly checking the condition ∂Q/∂x = ∂P/∂y might be simpler. However, Green's Theorem offers a broader perspective and connects conservative vector fields to other important concepts in vector calculus.

Practice Makes Perfect: Examples and Problems

Theoretical knowledge, while essential, truly solidifies when applied to concrete examples.

This section is designed to bridge the gap between theory and practical application by presenting a variety of solved examples and practice problems.

By working through these exercises, you’ll develop a deeper, more intuitive understanding of conservative vector fields and how to identify them.

Solved Examples: Step-by-Step Demonstrations

We will begin with a series of solved examples. Each example will present a vector field and then demonstrate, step-by-step, how to determine whether or not it is conservative.

Emphasis will be placed on clearly showing each step in the process, from calculating partial derivatives to interpreting the results.

Example 1: A Conservative Vector Field

Consider the vector field F(x, y) = (2xy + y²) i + (x² + 2xy) j.

To determine if F is conservative, we need to check if ∂P/∂y = ∂Q/∂x, where P(x, y) = 2xy + y² and Q(x, y) = x² + 2xy.

Calculating the partial derivatives, we find that ∂P/∂y = 2x + 2y and ∂Q/∂x = 2x + 2y.

Since ∂P/∂y = ∂Q/∂x, the vector field F is conservative.

Furthermore, we can find the scalar potential function f(x, y) such that ∇f = F.

Integrating P(x, y) with respect to x, we get f(x, y) = x²y + xy² + g(y), where g(y) is an arbitrary function of y.

Taking the partial derivative of f(x, y) with respect to y, we get ∂f/∂y = x² + 2xy + g'(y).

Comparing this with Q(x, y) = x² + 2xy, we see that g'(y) = 0, so g(y) is a constant.

Thus, the scalar potential function is f(x, y) = x²y + xy² + C, where C is a constant.

Example 2: A Non-Conservative Vector Field

Let's examine the vector field G(x, y) = (x - y) i + (x + y) j.

Here, P(x, y) = x - y and Q(x, y) = x + y.

We need to check if ∂P/∂y = ∂Q/∂x.

Calculating the partial derivatives, we get ∂P/∂y = -1 and ∂Q/∂x = 1.

Since ∂P/∂y ≠ ∂Q/∂x, the vector field G is not conservative.

It is crucial to note that the failure of this test immediately implies that no scalar potential function exists for G.

Example 3: A Three-Dimensional Conservative Vector Field

Consider F(x, y, z) = (2x, 3y², 4z³).

This is conservative because the curl is zero.

Key Takeaways from Solved Examples

• Always start by calculating the relevant partial derivatives.
• Carefully compare the partial derivatives to check for equality.
• If the vector field is conservative, proceed to find the scalar potential function.
• If the vector field is not conservative, clearly state that no scalar potential function exists.
• Be especially cautious and aware of how to find the curl to verify conservativeness in 3D.

Practice Problems: Test Your Understanding

Now it's your turn to put your knowledge to the test. Here are several practice problems, designed to challenge your understanding of conservative vector fields.

For each problem, determine whether the given vector field is conservative. If it is, find its scalar potential function.

1. F(x, y) = (3x² + 2y) i + (2x + y²) j
2. G(x, y) = (y cos(x), sin(x)) i + j
3. H(x, y) = (x²y, xy²) i + j
4. F(x, y, z) = (yz, xz, xy)
5. G(x, y, z) = (y, x, x+y)

By diligently working through these solved examples and practice problems, you'll reinforce your understanding of conservative vector fields and develop the skills necessary to confidently tackle more complex problems. Good luck!

Avoiding Common Errors: Tips and Pitfalls

After diligently applying the tests and theorems, it's crucial to pause and reflect on the common errors that can lead to incorrect conclusions about a vector field's conservativeness. Recognizing and avoiding these pitfalls is just as important as understanding the core concepts themselves.

Common Mistakes in Identifying Conservative Vector Fields

One of the most frequent errors stems from a misunderstanding of the necessary but not sufficient conditions. For example, in two dimensions, satisfying ∂P/∂y = ∂Q/∂x only suggests conservativeness within a simply connected domain.

Domain Considerations: Simple Connectivity is Key

The simple connectivity of the domain is often overlooked. A domain is simply connected if every closed loop within it can be continuously shrunk to a point without leaving the domain.

If the domain isn't simply connected (e.g., a plane with a hole), then ∂P/∂y = ∂Q/∂x does not guarantee conservativeness. Line integrals around closed loops encircling the "hole" may not be zero.

Incorrectly Calculating Partial Derivatives

Another prevalent mistake involves errors in calculating partial derivatives.

A single mistake can invalidate the entire analysis, leading to a false conclusion about the vector field's nature.

Double-check each partial derivative, paying close attention to signs and the application of the chain rule, product rule, or quotient rule when necessary.

Misinterpreting the Fundamental Theorem of Calculus for Line Integrals

The Fundamental Theorem of Calculus for Line Integrals provides a shortcut for evaluating line integrals along a path if a scalar potential function exists.

However, it's crucial to remember that this theorem only applies to conservative vector fields.

Applying it to a non-conservative vector field will produce an incorrect result.

Tips and Tricks for Accurate Assessment

To mitigate these risks, adopt a methodical approach and implement these helpful strategies.

Verify Simple Connectivity

Before diving into calculations, always examine the domain of the vector field.

Determine whether it is simply connected.

If the domain has holes or other complexities, the standard conservativeness tests may not be sufficient.

Meticulous Calculation and Verification

Take your time when calculating partial derivatives and the curl (in 3D).

Write out each step clearly, and double-check your work.

Use a computer algebra system (CAS) to verify your calculations, especially for complex vector fields.

Scalar Potential Function Check

After finding a potential function f, always verify that ∇f indeed equals the given vector field F. This confirms that the potential function is correct. It also validates that the vector field is indeed conservative.

Understanding Path Dependence

If you suspect a vector field is not conservative, try evaluating a line integral along two different paths between the same endpoints.

If the results differ, you've definitively proven that the vector field is not conservative, as path independence is a defining characteristic of conservative vector fields.

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