533 n1 k1 we need to determine the ank s and the bnk

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Unformatted text preview: usually quite diﬃcult to determine the eigenvalues λ1 , λ2 , · · · , λn , · · · explicitly for a given region in the plane. There are two important cases in which the eigenvalues can be explicitly determined. The ﬁrst is the case where D is the rectangle, D = {(x, y ) : 0 ≤ x ≤ a, 0 ≤ y ≤ b}. We saw how to solve the eigenvalue problem ∂2f ∂2f + 2 = λf, 2 ∂x ∂y f (x, y ) = 0 for (x, y ) ∈ ∂D, when we discussed the heat and wave equations for a rectangular region. The nontrivial solutions are λmn = − πm a 2 − πn b 2 , fmn (x, y ) = sin π ny π mx sin , a b where m and n are positive integers. A second case in which the eigenvalue problem can be solved explicitly is that of the disk, D = {(x, y ) : x2 + y 2 ≤ a2 }, where a is a positive number, as we will see in the next section.3 Exercises: 5.7.1. Show that among all rectangular vibrating membranes of area one, the square has the lowest fundamental frequency of vibration by minimizing the function π2 π2 f (a, b) = + a b subject to the constraints...
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This document was uploaded on 01/12/2014.

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