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MATH 381 HOMEWORK 2 SOLUTIONS 3 Question 6 (p106 #23) (a) Prove the expression given in the text for the nth derivative of f(t) = t t21 = Re 1 t−i (Note t ∈ R) (b) Find similar expressions for the nth derivative of f(t) = 1 t21 = Im 1. V N R G E T A P B R L S O T D G K G K N A N E W S P A P E R Y V F I X W P C X G A W Q U L L A B N S M S E H S I D L S B C W Q K C Z B L K O H R E A D I N G O F Find the following words in the puzzle ball flying riding ball house signing bicycle kicking song. Handout 8 Problem Set 1 Solutions 5 The precondition is that we are given a unimodal array A1n The postcondition is that Aa is the maximum element of A1n.

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Solution We proceed by induction on k For k = 1, there are two cliques K 1 is just a single point, which is trivially a bipartite graph K 2 is also a single bipartite graph (each vertex in its own group) Now, for the inductive step assume the claim holds for k = m. Ŋ ̃C X g ^ A ې ̃z y W ł B ̉ߋ ̍ i ₽ ̂ C X g f ڂ Ă ܂ B eonetnejp WWW g b v y W L y W V N ȗ 01 L y. IZ Xdci Zci d iZ GZedgi gZbV^ch iZ hda Z gZhedch^W^a^in d iZ >ci ZgcVi^dcVa meZgi ccd "kVi^dc VcY >ci Zaa ZXijVa Eg deZgin VcY YdZh cdi cZX Zh hVg^a n gZeg ZhZci.

T/ 罹 7 J u qQېQC c ϡ2ⷠ d l ~W u !l 8 y P p R d @ F' u gR _d @ Z= X 6 6{/& c g n ~ h J B t ;. Electromagnetic energy flow the rate at which energy flows through a surface S is GG given by P = ∫ S da ⋅ S Useful Math Cartesian Gradient ∇t = ∂ t xˆ ∂ t yˆ ∂ t zˆ ∂ x ∂ y ∂ z Divergence ∇ ⋅ v z G = ∂ vx ∂ vy ∂ v ∂ x ∂ y ∂ z G ∂ vy ∂ v Curl ∇ × v = (x x ∂ vz ∂ vy ) xˆ ( ∂ v ∂ vz ) yˆ ( − )zˆ ∂ y ∂ z ∂ z ∂ x ∂ x. (ii) Recall that if Gacts linearly on a vector space V, then the natural action on V nis given by g(e i 1 e i 2 e i n) = ge i 1 ge i 2 ge i n This action restricts to the Ginvariant subspaces S nV and V V as follows g(e i 1 e i n) = 1 n!.

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X ˙2S n sgn(˙)ge ˙(1) ge n. V n C b N X CR01 C h E C X o M ` v CC01 SkyToyR/C d. Vn kvnk Then w1,w2,,wn is an orthonormal basis for V Theorem Any finitedimensional vector space with an inner product has an orthonormal basis Remark An infinitedimensional vector space with an inner product may or may not have an orthonormal basis Title.

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