Conservative Vector Fields

Consider a vector field . This vector field is said to be conservative if it can be written as the gradient of a scalar function . Conservative vector fields have quite a few unique properties that make them interesting to study.

Table of Contents

• Table of Contents
• All Gradient Fields are Conservative Fields
• All Conservative Fields are Gradient Fields
  • The Actual Proof (Optional Section)
• Conservative Fields in Closed Paths
• Potential Functions
  • Example Problem: Electric Potential Energy
• Summary and Next Steps

All Gradient Fields are Conservative Fields

Previously, we discussed the gradient theorem, an extension of the fundamental theorem of calculus to line integrals. One of the key takeaways from the gradient theorem is that the line integral of a gradient field is path-independent:

This property is a direct consequence of the fundamental theorem of calculus. It should also make intuitive sense (even though it initially appears very surprising): imagine the function as a hill. Wouldn't it make sense that the total change in height as you walk from point to point is the same regardless of the path you take?

Let's shift our focus by redefining some things. Define . Then, the line integral of along a curve is path-independent, which makes a conservative vector field.

Generally, a vector field is conservative if one of the following conditions is met:

1. is the gradient of a scalar function ; .
2. The line integral of along any closed path is .
3. The line integral of along any path is path-independent.
4. The curl of is if it is continuously differentiable. We will discuss this in more detail in the future.

We will see that these conditions are logically equivalent to each other, and they all define the same fundamental property of a vector field.

All Conservative Fields are Gradient Fields

One important question one would start wondering is: all gradient fields are conservative, but can all conservative fields be represented as gradient fields? We can't just assume this to be true, or else we would be succumbing to a logical fallacy (the fancy term for this is affirming the consequent).

Start with the gradient theorem:

Suppose we choose an arbitrary such that . Then, the line integral simplifies to:

Meaning, for any point , the value of the function at that point is the line integral of the gradient field from the to that point.

If we then denote , we can write the line integral as:

Before doing the proof, let's try to understand this from a high-level perspective:

• We are trying to show that all conservative fields are gradient fields.

• We know that the line integral of a gradient field is path-independent.

• From Equation , we know that the line integral of a conservative field is the value of the function at the end point.

• By the gradient theorem on , we can set the following:

• This implies that .

We did it, right? This is a good way to understand gradient fields and conservative fields, but it's not a rigorous proof. Below is a more rigorous proof that all conservative fields are gradient fields.

The Actual Proof (Optional Section)

The following proof is a bit more rigorous, and it will look a bit similar to how we consider accumulation functions in single-variable calculus. In this proof, the following is an assumption:

The statement we want to prove is:

For any that is conservative, there exists a scalar function such that .

Next, consider any nonzero vector in the same space as . If we take the directional derivative of in the above equation with respect to , we get, by the definition of the directional derivative:

Plug in Equation into this equation:

Let's digest this equation a bit. After plugging in the definition of into the directional derivative, we get an expression that looks like a difference of two integrals.

We end up with a limit of a single integral over a path from to . As approaches , the length of the path approaches .

One consequence of this is that the path from to becomes arbitrarily close to a straight line. Thus, we can parameterize this path as a straight line, i.e. for .

Rewrite the integral in terms of :

We need to use a clever trick to simplify this expression. Consider the line integral inside this limit. At , the path is a single point, and the integral evaluates to . To help with this, let's define a new function as follows:

The limit we are interested in is then:

Which is just the definition of the derivative of at :

By the fundamental theorem of calculus, the derivative of an integral is the integrand evaluated at the upper limit. Thus, this becomes:

Finally (and this is the key step), since was arbitrary, this equation holds for all vectors .

Recall the definition of the directional derivative (which is what we did):

Equating Equations and , we get:

Thus, we have shown that all conservative fields are gradient fields.

Conservative Fields in Closed Paths

A closed path is a path that starts and ends at the same point. For a conservative field, the line integral along a closed path is always :

This little circle on the integral sign denotes a closed path.

There are two ways to show this, both of which are quite interesting.

1. Using the Gradient Theorem - Since is conservative, we can write it as . Then, by the gradient theorem, the line integral of along a closed path is .

2. Using properties of line integrals - The left hand side is a line integral for a vector field. Consider a closed path and its reverse path . Recall that a line integral over a reversed path is the negative of the original path:

   However, we also know that the line integral of is path independent:

   Combining these two equations, we get:

   Which implies that the line integral is .

Potential Functions

Previously, we discussed work done by forces as a line integral:

A force is conservative if the work done by the force is path-independent. This means that the work done by the force is only dependent on the initial and final positions, not the path taken. The gravitational force, electric force, and spring force are all examples of conservative forces, while friction and air resistance are non-conservative forces.

If is conservative, by definition, there exists a scalar function such that . This function is called the potential function of the vector .

Conservative forces get its name from the fact that they "conserve" mechanical energy (kinetic and potential energy). Since they're path-independent, simply going back to the initial position will result in the same energy as the initial position. Thus, all processes involving conservative forces are reversible.

One might hear "potential function" and think of the function as the potential energy. In reality, the actual definition of the potential energy is the negative of the potential function; .

The work done by a force is equivalent to a change in the kinetic energy of the system:

By the conservation of energy, the total kinetic and potential energy of the system is constant (assuming all forces are conservative):

This means that a change in potential energy is equal to the negative of the work done by the force to uphold the conservation law:

Example Problem: Electric Potential Energy

A point charge is placed somewhere in space. Another test charge is placed at a distance from the charge. The electric force between the two charges is given by Coulomb's law:

The electric force is conservative. Write an expression for the potential energy of the test charge , .

Before doing anything, notice that there is only one dimension in this problem, the radial direction. So any vector can be written as a scalar times the unit vector in the radial direction.

Sine (from the definition of potential energy), we can write:

This implies that:

By the fundamental theorem of calculus, we can integrate this to find :

Here, we have introduced a constant to account for the fact that potential energy is defined up to an arbitrary constant. By convention, we set the potential energy to be at infinity to eliminate the term. Then;

Summary and Next Steps

This was a large section, and we covered a lot of content. We introduced the concept of conservative vector fields, as well as their properties and implications.

Here are the key points to remember:

• A vector field is conservative if it can be written as the gradient of a scalar function .
• The converse is also true: all conservative fields are gradient fields.
• Conservative fields have the property that the line integral of the field along any path is path-independent.
• The line integral of a conservative field along a closed path is always .
• The potential function of a conservative field is a scalar function such that .
• The potential energy of a system is defined as .

In the next section, we will introduce the concept of flux, briefly mentioning Gauss's law.