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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 3, 2007 Fall 2007 Due: October 10, 2007 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 B C × 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) ( ) ( ) ( ) A B C A C B A B C × × = . (c) The chain rule holds ( ) f g f g g f = + g and f can be replaced by vectors and the multiplication by a vector multiplication. Using these facts, prove 2 ( ) ( ) A A A ∇× ∇× = ∇ ∇⋅ − ∇ ( ) ( ) ( ) E H E H H E ∇⋅ × = ∇× − ∇× . Do not use decomposition into components in a coordinate system. 2. A generalized coordinate system consists of three families of surfaces, whose equations in terms of

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