The Fundamental Theorem for Line Integrals

Questions:  How do you determine if a vector field is conservative?  What special properties do conservative vector fields have? To answer these questions some preliminary definitions are needed: Closed, simple Closed, not simple Simple, not closed Not simple, not closed

The Fundamental Theorem for Line Integrals Types of Regions: Suppose D is an open region ( D does not contain any of its boundary points).  D is connected if any two points in D can be joined by an arc in D  D is simply connected if it is connected and every simple closed curve in D can be contracted to a point without leaving D. If the region is in the plane, simply connected means that it has no hole and cannot consist of two separate pieces. Simply connected Connected but not simply connected D Not connected D

The Fundamental Theorem for Line Integrals NOTE: The definition of independence of path and circulation can be generalized to a three-dimensional vector space. However, the definition of irrotational is different in the 3D case. C

The Fundamental Theorem for Line Integrals C

Proof: (one direction: F conservative implies path independent) (Chain Rule)

The Fundamental Theorem for Line Integrals Why are the Theorems important? Strategy 1. is the most common, so now we are left with the question: How do we find a potential function for a conservative vector field? Theorem 1 gives an easy way to check whether a vector field is conservative, using the irrotational property.

Check: The Fundamental Theorem for Line Integrals – Example 1 (K arbitrary constant)

The Fundamental Theorem for Line Integrals – Example 1 continued A direct evaluation of the line integral requires a different parameterizations for each of the three line segments of the curve C, and therefore three different line integrals. A B

The Fundamental Theorem for Line Integrals – Example 2

The Fundamental Theorem for Line Integrals – Example 3 (3D) Check:

The Fundamental Theorem for Line Integrals – Example 3 continued Fundamental Theorem for Line Integrals: c. Evaluate the line integral directly As expected, the answers to b. and c. are the same.

Determining graphically whether a vector field is conservative: (a) Draw an arbitrary closed path. Since the circulation is not zero, the field is not conservative. Along the red path the work is positive since along the segment AB the force is greater in magnitude than along the segment CD. The Fundamental Theorem for Line Integrals – Example 4 C

Determine graphically whether the vector field is conservative. (b) Along the path shown, the circulation is positive, since the vectors always point in the same direction as the path. The vector field appears to be conservative. (c) No matter the shape of the closed path we draw, the circulation is always zero, since all the positive contributions are balanced by the negative ones. Since the circulation is not zero, the vector field is not conservative. The Fundamental Theorem for Line Integrals – Example 4 continued