Intro Complex Analysis, Lec 20, Invariance of Laplace's Eq, Real & Im Parts of Complex Integrals

Lecture 20. (0:00) Exam 2 next week (advising/assessment day this week). (0:48) Plan for the lecture. (1:42) Invariance of Laplace's equation (basic set up). (7:24) The proof that psi (the newly-defined function) is harmonic. (16:54) Example of the invariance of Laplace's equation: a harmonic function gets mapped to another harmonic function (under the inversion mapping w = 1/z). Also look at the contour maps (level curves) of the original harmonic function and the transformed harmonic function. The boundary values also match. (29:49) Defining a harmonic function on the open unit disk whose limiting values are 1 (along the upper half of the unit circle) and -1 (along the lower half of the unit circle). (33:11) Introduction to contour integrals of vector fields over oriented smooth curves (arcs) (a.k.a. line integrals of vector fields over oriented smooth curves (arcs)). (42:23) Complex contour integrals of complex functions of a real variable. Determine the real and imaginary parts of the integral. They are line integrals in the sense of multivariable calculus (involving dot products). The integrands are the conjugate vector field and its corresponding orthogonal complement. (50:28) Generalization to piecewise smooth curves (arcs) that are "sums" of smooth curves. (52:28) These integrals can be related to applications: 1) work done by a force field along a curve, and 2) flux of a fluid flow across a membrane.

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