Lecture 30, Friday, 11/7/08: Section 14.3: The Fundamental Theorem for Line Integrals

Topics

Fundamental theorem for line integrals
Independence of path property
Loop integral zero property
Connected regions and simply connected regions
The "mixed partials test" for "conservativeness" (2 dimensional case)

Read

Section 14.3.

Homework

Section 14.3: 1, 5, 15, 21, 23, 25, 31-34, 37. Solutions.

Notes

Conservativeness test for 3-dimensional fields: The analog of the mixed partials test for 3D fields is the "curl test", which is given in Section 14.5.

The assigned problems 21 and 23 in this section involve 3D fields, but in both cases, a potential function exists and can be found by the method of 14.1. One could have performed the curl test in those cases, but this would simply have confirmed that the fields are conservative and that, therefore, one should go ahead and find a potential function.

The main use of such "conservativeness tests" is to quickly spot cases where the field is not conservative. In those cases, a potential does not exists, and knowing this one doesn't have to spend the time and effort in trying to find a potential.

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Last modified Sun 09 Nov 2008 02:25:06 PM CST