On Monday we will investigate some of the questions raised by the lab activity on Friday. In particular we will look at the Fundamental Theorem of Calculus in the setting of complex line integrals.
On Wednesday we will come to the central theorem in the first part of the course -- Cauchy's Theorem. This is really another version of Green's Theorem, and we will approach it this way -- in contrast to the approach in the text. For this you should review the notion of line integral of a vector field and Green's Theorem. Use your calculus text or our text. In our text these ideas are discussed in Sections 13.1 and 13.2.
Cauchy's Theorem is the key to unlocking the properties of differentiable complex functions. We will investigate the first consequences on Friday. Note that there is no lab this week.
Also review the concept of the line integral of a vector field and Green's Theorem in your calculus book or in Sections 13.1 and 13.2 of our text.
You should calculate these by hand. Then you might want to use Maple to check your answers.
You may need to use the Cauchy Integral Formula, the Cauchy Integral Theorem, Theorem 18.4, or the definition. Consider the various possibilities before you start to calculate. You may use Maple for any extensive integral computations.
Last modified: February 8, 1999