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HW1Solution - ECE 201A Homework 1 solution 1(a A"B(b A C...

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ECE 201A Homework 1 solution 1. (a) 2 ( ) ( ) ( ) ( ) ( ) ( ) A A A A A A B C B A C A B C ! " ! " # ! ! $ % ! $ ! # ! !$ % ! & & & & & & & & & " " # $ $ (b) ( ) ( ) ( ) c c E H E H E H !$ " # !$ " ’ !$ " This is the application of the chain rule. Subscript c means that in the differentiation the term with that subscript will be treated as a constant. Then since doesn't rule #1 rule #2 operate on ( ) ( ) ( ) ( ) c c c c E E H H E H E H E ! !$ " # %!$ " # %!" $ # % !" $ since doesn't rule #1 operate on ( ) ( ) c c c H E H E H E H ! !$ " # !" $ # !" $ Combining the two ( ) ( ) ( ) E H E H H E !$ " # !" $ % !" $ 2. (a) Let’s assume that the vector field is ( ) A r ! ! and ( ) ( ) A r B r !" # ! ! ! ! and ( ) ( ) A r r * ! + # ! ! ! . ( ) r ! is a shorthand for coordinates of a point say ( ) , , x y z in the Cartesian coordinate system. If we know ( ) A r !" ! ! only we can form infinitely many vector fields ( ) Z r ! ! by adding the gradient of a scalar function to ( ) A r ! ! such that ( ) ( ) ( ) Z r A r r , # ’ ! ! ! ! ! ! . Note that ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Z r A r r A r r A r B r , , !" # !" ’ ! # !" ’ !"! # !" # ! ! ! ! ! ! ! ! ! ! ! ! since ( ) 0 r , !"! - ! . Hence there are infinitely many vector fields having the same curl and knowing the curl of a vector field we cannot uniquely find the field. If we know ( ) A r ! + ! ! only we can form infinitely many vector fields ( ) Z r !
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