laplace_eqn

laplace_eqn - MIT OpenCourseWare http:/ocw.mit.edu 18.02...

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MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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V7. Laplace's Equation and Harmonic Functions In this section, we will show how Green's theorem is closely connected with solutions to Laplace's partial differential equation in two dimensions: where w(x, y) is some unknown function of two variables, assumed to be twice differentiable. Equation (1) models a variety of physical situations, as we discussed in Section P of these notes, and shall briefly review. 1. The Laplace operator and harmonic functions. The two-dimensional Laplace operator, or laplacian as it is often called, is denoted by V2 or lap, and defined by The notation V2 comes from thinking of the operator as a sort of symbolic scalar product: In terms of this operator, Laplace's equation (1) reads simply Notice that the laplacian is a linear operator, that is it satisfies the two rules (3) v2(u + v) = v2u + v2v , v2 (cu) = c(v2u), for any two twice differentiable functions u(x, y) and v(x, y) and any constant c. Definition. A function w(x, y) which has continuous second partial derivatives and solves Laplace's equation (1) is called a harmonic function. In the sequel, we will use the Greek letters q5 and $ to denote harmonic functions; functions which aren't assumed to be harmonic will be denoted by Roman letters f, g, u, v, etc. . According to the definition, (4) 4(x, y) is harmonic H v2q5 = 0 . By combining (4) with the rules (3) for using Laplace operator, we see (5) q5 and $ harmonic + q5 + $ and cq5 are harmonic (c constant). Examples of harmonic functions. Here are some examples of harmonic functions. The verifications are left to the Exercises.
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2 V. VECTOR INTEGRAL CALCLUS A. Harmonic homogeneous polynomials1 in two variables. Degree 0: all constants c are harmonic. Degree 1: all linear polynomials ax + by are harmonic. Degree 2: the quadratic polynomials x2 - y2 and xy are harmonic; all other harmonic homogeneous quadratic polynomials are linear combinations of these: q5(x, y) = a(x2 - Y2) + bxy, (a, b constants). Degree n: the real and imaginary parts of the complex polynomial (x + are harmonic. (Check this against the above when n = 2.) B. Functions with radial symmetry. Letting r = dm, the function given by $(r) = In r is harmonic, and its constant multiples clnr are the only harmonic functions with radial symmetry, i.e., of the form f (r).
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laplace_eqn - MIT OpenCourseWare http:/ocw.mit.edu 18.02...

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