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# HW1 - UNIVERSITY OF CALIFORNIA Santa Barbara Department of...

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UNIVERSITY OF CALIFORNIA Santa Barbara Department of Electrical and Computer Engineering Problem Set No. 1 Issued: October 1, 2008 Fall 2008 Due: October 8, 2008 ECE 201A 1. In this course we shall use many operator identities. All of them are derivable from the following relations of vector algebra. (a) In a triple scalar product A BC × one may interchange the and × without changing the product. An interchange of the order of the vector factors leaves the product unchanged, but changes the sign if two adjacent factors are interchanged. (b) () ( ) ( ) ABC A C B A B C ××=⋅ −⋅ . (c) The chain rule holds f gf g g f ⋅= + g and f can be replaced by vectors and the multiplication by a vector multiplication. Using these facts, prove 2 ( ) AA A ∇× ∇× =∇ ∇⋅ −∇ ( ) EH HE ∇⋅ × = ∇× − ∇× . Do not use decomposition into components in a coordinate system. 2. (a) Show that knowing the curl or divergence of a vector field alone is not enough to

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HW1 - UNIVERSITY OF CALIFORNIA Santa Barbara Department of...

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