Table of Content

• What is Conservative Vector Field?
• Properties
• How to Determine Vector Field is Conservative?
• Line Integrals in Conservative Vector Fields
• Difference Between Conservative and Non-conservative Vector Fields
• Applications
• Solved Examples
• Practice Problems

Characteristics	Conservative Vector Field	Non-conservative Vector Field
Definition	Can be expressed as the gradient of a scalar potential function F = ∇ Φ	Cannot be expressed as the gradient of any scalar potential function
Path Dependence	Line integral is independent of the path taken	Line integral depends on the specific path taken
Curl Condition	Zero curl [Tex](∇ × F = 0)[/Tex]	Generally, has non-zero curl [Tex](∇ × F ≠ 0)[/Tex]
Potential Function	Exists (φ such that F = ∇ Φ)	Does not exist
Work Done	Zero work done around any closed loop	Non-zero work done around a closed loop
Energy Conservation	Energy is conserved in a conservative field	Energy is not conserved; it can be dissipated as heat, sound, etc.
Examples	Gravitational field, electrostatic field	Magnetic field, frictional forces
Mathematical Representation	Can be represented by a scalar potential function	Cannot be represented by a single scalar potential function
Physical Interpretation	Represents conservative forces like gravity, electrostatics	Represents nonconservative forces like friction, air resistance
Closed Path Integral	Integral over a closed path is zero	Integral over a closed path is non-zero
Dependence on Initial and Final Points	Depends only on initial and final points of the path	Depends on the entire path taken between points

Applications of Conservative Vector Fields

Conservative vector field have numerous applications in various fields as following:

• Mechanics: In classical mechanics, conservative forces arise from potential energy function. The work done by such force depends only on the initial and final positions making conservative vector field valuable for analyzing motion .
• Electrostatics: The electric field generated by a static charge distribution is conservative . The electric potential, a scalar quantity , relates directly to the electric field through its negative gradient.
• Gravitation :The gravitational field due to mass distribution is also conservative. The gravitational potential energy depends solely on the position of object relative to mass distribution.

Related Articles:

• Calculus
• Vector Algebra
• Partial Derivative

Examples on Conservative Vector Fields

Example 1: Verifying Conservative Vector Field Given the vector field F = (2xy, x² – y²) determine if it is conservative or not.

Solution:

Compute the curl of F:

∇ × F = [[Tex]\frac{\partial}{\partial y}[/Tex](x² – y²) + [Tex]\frac{\partial}{\partial x}[/Tex]2xy] k

∇ × F = (2y + 2y) k

∇ × F = 0

Since the curl is zero and the domain is simply connected, F is conservative.

Example 2: Find the potential function φ for the conservative vector field F = (y, x).

Solution:

Integrate the first component with respect to x:

[Tex]\frac{\partial \phi}{\partial x} [/Tex]= y ⇒ Φ = xy + g(y)

Integrate second component with respect to y:

[Tex] \frac{\partial \phi}{\partial y} [/Tex]= x ⇒ Φ = xy + C

Thus, the potential function is Φ = xy.

Example 3: Consider a two-dimensional force field that represents the force exerted on a particle: F (x, y) = (2x, 3y). What is work done by this force if particle travels along any path from point A (1, 2) to point B (3, 4)?

Solution:

The work done depends only on the beginning and finishing positions since the force field is conservative (verification left as an exercise; check for path independence, for example). For F, we may determine a potential function φ.

Now let’s determine Φ:

Φ_x = 2x (conforms to F_x)

After integrating Φ_x

Φ(x, y) = x² + C₁

Φ_y = 3y (conforms to F_y)

After integrating Φ_y

Φ(x, y) = 3y²/2 + C₂

By solving for C and equating both formulas for φ, we obtain a particular potential function, such as Φ(x, y) = x² + 3y²/2 – 1

Work done W is given by difference in potential energy between point A and B:

W = Φ(B) – Φ(A)

W = [{3² + (6/2)} – {1² + (6/2)}]

W = 4

It follows that, whichever direction is chosen, the force field will work for 4 units.

Practice Problems on Conservative Vector Fields

Q1. Determine if the vector field F (x, y) = (2x, y²) is conservative.

Q2. Find a potential function for the vector field F (x, y) = (y, -x)

Q3. (3D) Is the vector field F (x, y, z) = (x², 2xy, z³) conservative in all regions?

Q4. Consider the vector field F=(−y,x)\mathbf{F} = (-y, x)F=(−y,x). Verify if this vector field is conservative. If it is, find the potential function ϕ\phiϕ.

Q5. Let F = (e^xsin⁡y, e^xcos⁡y). Show that F is conservative and find the potential function ϕ.

FAQs on Conservative Vector Fields

What is Conservative Vector Fields in Math?

A vector field that is gradient of some function is called as conservative vector fields.

Why are Conservative Vector Fields Called Conservative?

The conservative vector fields are called conservative because these vector fields force energy is conserved.

What is an Example of a Conservative Vector Field in Real Life?

Some examples of conservative vector fields include gravitational fields and electronic fields around a point particle.

Can Conservative Vector Field be Zero?

The curl of conservative vector field is zero vector.