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Ch_03 - Vector I - Exam_Soln

Ch_03 - Vector I - Exam_Soln - Exam Problems of Vector...

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Exam Problems of Vector Differential Calculus
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Vector Differential Calculus Fundamentals Fundamentals 1. (05’) Prove (A) ( ) ( ) ( ) A B C B A C C A B (B) ( ) ( ) ( )( ) ( )( ) A B C D A C B D A D B C (C) ( ) ( ) ( ) ( ) ( ) ( ) A B C D A C D B B C D A A B D C A B C D Solution: (A) ( ) ( ) ( ) A B C B A C C A B ( ) ( ) [( ) ( ) ] ( ) ( ) A B C B C A B A C C A B B A C C A B (B) ( ) ( ) ( )( ) ( )( ) A B C D A C B D A D B C Let ( ) X A B , then ( ) ( ) X C D X C D ( ) ( ) {( ) } { ( ) ( )} ( )( ) ( )( ) A B C D A B C D B A C A B C D A C B D A D B C (C) ( ) ( ) ( ) ( ) ( ) ( ) A B C D A C D B B C D A A B D C A B C D By middle factor rule, ( ) ( ) ( ( )) ( ( )) A B C D A C D B B C D A A C D B B C D A or ( ) ( ) (( ) ) (( ) ) A B C D A B D C A B C D A B D C A B C D
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Vector Differential Calculus Fundamentals (Continued) (or for (A) ( ) ( ) ( ) A B C B A C C A B Set 1 2 3 a a a A i j k , 1 2 3 b b b B i j k , 1 2 3 c c c C i j k 1 2 3 1 2 3 1 2 3 2 3 1 3 1 2 1 2 3 2 3 1 3 1 2 1 2 3 2 3 3 2 1 3 3 1 1 2 2 1 LHS a a a b b b c c c b b b b b b a a a c c c c c c a a a b c b c b c b c b c b c i j k i j k i j k i j k i j k 2 1 2 2 2 1 3 1 3 3 3 1 1 1 2 1 2 1 3 2 3 3 3 2 1 1 3 1 3 1 2 2 3 2 3 2 LHS a b c a b c a b c a b c a b c a b c a b c a b c a b c a b c a b c a b c i j k 1 1 2 2 3 3 a c a c a c A C