Gradient Theorem

Todo

• Add some examples to demonstrate its usefulness.

Previously, we introduced the concept of line integrals, which are integrals of functions along curves. This is the first integral we have seen outside of single-variable calculus, so one natural question to ask is: how do we evaluate these integrals?

In single-variable calculus, we primarily rely on the fundamental theorem of calculus to evaluate integrals:

We will extend this theorem to line integrals in this section, a concept known as the fundamental theorem of line integrals, or the gradient theorem. Additionally, we will explore some implications of this theorem, especially as a building block to understand the concept of conservative vector fields.

This section combines a lot of concepts like the fundamental theorem, line integrals, the gradient, the directional derivative, the multivariable chain rule, parametric curves, and more. It's a lot to take in, but it's a crucial step in understanding the broader context of calculus in higher dimensions.

Table of Contents

• Table of Contents
• Fundamental Theorem of Calculus Review
• Intuition for the Gradient Theorem
• Proof By Multivariable Chain Rule
• Implications: Path Independence
• Summary and Next Steps

Fundamental Theorem of Calculus Review

To get an intuition behind the gradient theorem, we need to get a better understanding of the fundamental theorem of calculus in single-variable calculus.

Consider a function , which can represent anything. Let's think about moving along the -axis from to , and consider how much changes as we move from to .

1. Firstly, the change can simply be represented as .

2. Second, let's try to add up incremental changes as we move along the curve. Each time we move a small distance , we can think of the change in as . If we add up all these changes from to , we get the integral:

3. Both of these represent the same thing: the change in as we move from to . The first is a direct calculation, while the second sums up individual changes along the curve. Since they are equivalent, we can say:

   This is the fundamental theorem of calculus.

Intuition for the Gradient Theorem

Consider a scalar field . Just like in single-variable calculus, we can think of moving along a curve in .

Denote the starting point as and the ending point as , and the curve we move along as . We can think of the curve as a parametric curve , where is the parameter that moves us along the curve.

1. The change in as we move along the curve (from to ) can be represented as:

2. Alternatively, we can think of moving along the curve in small steps.

   In each step, we increment by a small amount , resulting in a small change in , denoted as . The change in as we move along this small step can be represented as a directional derivative:

   Next, we sum up all these changes along the curve:

3. Finally, we can say that both of these represent the same thing: the change in as we move along the curve. Since they are equivalent, we can say:

   This equation is known as the fundamental theorem of line integrals, or the gradient theorem.

We can also express it in terms of instead of . Denote and . Then, we can write the integral as:

Proof By Multivariable Chain Rule

To prove the gradient theorem, we will use the multivariable chain rule.

Consider the derivative of the composition . By the chain rule, this is:

Looks familiar? Plug this into the integral:

The right-hand side can be evaluated simply by the regular fundamental theorem:

Thus, we have proven the gradient theorem.

Implications: Path Independence

One of the most important implications of the gradient theorem is the concept of path independence.

Consider two curves and that start and end at the same points and .

By the gradient theorem, we have:

This means that the value of the line integral of along any curve from to is the same. In other words, the line integral of is path independent.

This should be a surprising result - it means that no matter where you start and end, the total change in is always the same.

This property is crucial in the study of conservative vector fields, which we will explore in the next section.

Summary and Next Steps

In this section, we introduced the fundamental theorem of line integrals, also known as the gradient theorem:

And in terms of :

Here are the key points to remember:

• The gradient theorem extends the fundamental theorem of calculus to line integrals.
• It states that the line integral of the gradient of a function is equal to the change in the function between the endpoints of the curve.
• This theorem has important implications, such as path independence.

In the next section, we will explore the concept of conservative vector fields, which are closely related to the gradient theorem.