SUNY Geneseo Department of Mathematics

A Fundamental Theorem for Line Integrals

Tuesday, May 2

Math 223
Spring 2023
Prof. Doug Baldwin

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Line Integrals and Conservative Vector Fields

Based on “Curves and Regions” and “Fundamental Theorem for Line Integrals” in section 5.3 of the textbook.

Key Ideas

The equation from the Fundamental Theorem (a line integral of a conservative vector field is the difference of the potential function at the endpoints of the path — see the textbook or example below for a more concrete description).

Definitions of simple vs non-simple, and closed vs not closed, curves. The definition of connected regions.

Path independence & the path independence test for conservative fields.

Questions

Different kinds of curve and region?

Informally, simple curves don’t cross themselves; closed curves end at the same point where they started:

Curves: arch (simple, not closed), circle (simple, closed), eyeglasses (not simple, closed), curlycue (neither)

Connected regions are, well, connected, i.e., you can go from any point to any other without leaving the region. Simply connected regions are connected and have no holes (at least in 2 dimensions):

Ellipse is simply connected, donut connected but not simply, disjoint circles not connected

Examples

Path independence: vector field F(x, y) = ⟨ 2x, 2y ⟩ has potential function f(x,y) = x2 + y2, and so must be conservative. Consider integrals of the field along 2 paths between (-1,0) and (1,0). One path is semicircular, the other is a straight line:

Integrating vector 2 X comma 2 Y along semicircle and straight line both produce 0

The integral along both paths is 0. This is “path independence,” i.e., no matter what path you follow between 2 points, you get the same integral.

But there’s an even easier way to evaluate the integral, thanks to the Fundamental Theorem for Line Integrals — evaluate the potential function at the ends of the path and subtract:

Potential function X squared plus Y squared at point 1 0 minus at point negative 1 0 is 0

To see what the alternative to path independence looks like, consider a similar but not conservative vector field integrated along the same paths:

What is vector negative 2 Y comma 2 X integrated along semicircle and straight line between same points?

Work out these two integrals for tomorrow’s class. We’ll talk about them, and a bit more about path independence, then.

Next

Finish path independence/dependence.

Then, a connection between line integrals and area integrals (also another scaling up of the Fundamental Theorem): Green’s Theorem.

Please read “Extending the Fundamental Theorem of Calculus” and “Circulation Form of Green’s Theorem” in section 5.4 of the textbook.

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