Section 16.5

Fundamental Theorem for Line Integrals

Like the Fundamental Theorem of Calculus, but for Vector Fields.

Introduction

In Calc I, .

In Calc III, if a field is a gradient field (), then the work done depends only on the endpoints, not the path taken.

Let C be a smooth curve from point A to point B. If is continuous:

This implies Path Independence.

Find for from (0,0) to (3,4).

1. Find Potential f:

So works.

2. Apply Theorem:

No parameterization needed!

Verify is conservative and evaluate .

1. Find Potential:

So .

2. Evaluate:

Is conservative?

1. Component Test:

2. Conclusion:

. Not conservative.

Integral on closed loop (Circle) was (from previous lesson). Confirmed.