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**Unformatted text preview: **MATH 412: Midterm 2
Thursday, April 18, 2008 NAME: This exam consists of 3 problems, with points as 1abeﬂed. Please write each answer Clearly
and legibly. Don’t forget to show all of your work! fill M ﬂ- (3 1. (32 pts) For the function h(33) = cc + 1 with a: E (—1,1), ﬁnd the ﬁrst two non—zero
terms in the full Fourier series or 2. (32 pts) Consider the following general statement of Fourier decomposition (the rep—
resentation of a function as an inﬁnite sum of orthogonal eigenfunctions X j) fm=2aam (a) Using an inner product (f, g) but Without Specifying its particular form, derive
an abstract expression for Ag; in terms of f and the Xj. (b) By taking the inner product of f with itself, derive the relation with its Fourier
coefﬁcients known as Parseval’s identity. (c) NOW take the inner product of f with another function 9(a"), and derive another
relation involving the Fourier coefﬁcients of both functions. 222% <><,»¢> an //»¢//é 55’. “2 04:7/ ﬁ/ Lyn/8'15 “SQ L95 4€f>:4§4%2%~6> [2M Oweaé/A/V/W/Q 3. (36 pts} Find the spatial eigenfunctions and eigenvalues, and the general series solution
for the following PDE (which is like the diffusion equation with an increasing diffusion
constant), then answer the questions that follow {at = Kt2uw, a: 6 (0,12), t > 0 Zyﬁimt) :0" 4?“ My [Z 7L) :0 What is the largest eigenvalue? The smallest? What is it about the PDE that allows
for a series solution? we was-772:3 a) XT’:L¢PX”7’ 31;”: X4 WA
F/Tt; ‘ - leRT 7: (>0
LE”; “Tit/th; 1&3 A ) SPQh‘q/ 9B3: {—3 C5 [6:45 7“» :14 A17€AK7L5 Some Useful Formulae and Identities: 1. Green’s identity: For 9 C IR“ fwAudV=—/Vu°del/+/ w(Vu-ﬁ)dA
o 52 so Where it is the outward normal to 59. 2. Divergence Theorem: Let 13 be a smooth vector ﬁeld deﬁned in 9. Then fv-FdV:/13.ndA
9 an where n is the outward normal to 89. 3. Deﬁnitions of Fourier Series coefﬁcients: The full Fourier series for the function
ﬁr) on the interval (—3, E) is ﬁr) : % + Z An cos(mr;z:/€) + Bn sin(mr:r:/€)
1121
where l 3
A0 = 2
An = élfﬂw) cosmms/E) dr
and 1 e ‘
Bn 2 E Sin(mT:t/€)d$ 4. gonservation Law in IR”: Let p be the density for some conserved quantity M , and
F be a vector ﬁeld representing the ﬂux associated with M , both deﬁned in S2. Then at each point in Q —; ...

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