Unformatted text preview: Problem 1. Show that for any vector field A,
Vx (VxA) =V(VA)—V2A Hlnt: Prove it for the x com ponent and the others will follow from circular permutation of indices. Orb = 92” vaawﬂﬁr
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[I fermmfﬂ'ﬂcb c'f "Adi/“ar— Problem 2. Prove thisB identity for the scalar triple product:
A (B x C): B(C x A) @ ghaw A (BKC>= 6 (6X12)
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= exfcgﬂr Mg) + 6%wa ‘CXA%>*@%<0‘42 ’ W“) = wax?) / Problem 3. Show that the curl of the gradient of any scalar ﬁeld is zero. l\ (\ ’\
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\ «JP—f 553;) *Mam 91> *7? MK 933+ r 0 M A f; +w.‘ce, Cowh'wuomlj Wﬁﬂévémﬁltzé/e. Problem 4. Find an expression for the electric ﬁeld of a plane wave with these properties: (0 it is
linearly polarized. (ii) It propagates in a direction that makes equal angles to the x, y, and z
axes, (iii) It has zero electric ﬁeld in the x direction. A "0 ,.. a 1. 4 q 4 A
(")E = Ea 005(IC'V‘W'6) I Ea = axx+€agj+Eﬂgi
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l<== 4m 0 El! C/‘JC, Val/K211 4': LP WP Q: “,4! us.
i lef ClL'Ow. ll EOX+Eag+Eae = 0
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 Fall '17
 Vector Calculus, Vector field, Gradient, scalar triple product, 0+L18V JIWANIOMI ppHaw

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