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# hwsol_15 - Homework Assignment 15(12.5 Solutions Page 966 6...

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Homework Assignment 15 - (12.5) - Solutions Page 966 : 6*, 8*, 12*, 20*, 22*, 24*, 29* , 28 - for extra points 6. g ! t " ! f ! x ! t " , y ! t "" ! x 2 " y 2 , x ! t " ! sin t , y ! t " ! t 2 " 2 f x ! x x 2 " y 2 , f y ! y x 2 " y 2 , x # ! t " ! cos t , y # ! t " ! 2 t g # ! t " ! x x 2 " y 2 ! sin t " " y x 2 " y 2 ! 2 t " ! 1 x 2 " y 2 ! x sin t " 2 ty " 8. g ! u , v " ! f ! x ! u , v " , y ! u , v "" , f ! x , y " ! xy 3 ! 4 x 2 , x ! u , v " ! e u 2 , y ! u , v " ! v 2 " 1 sin u f x ! y 3 ! 8 x , f y ! 3 xy 2 , x u ! 2 ue u 2 , x v ! 0, y u ! v 2 " 1 cos u , y v ! v v 2 " 1 sin u g u ! f x x u " f y y u ! ! y 3 ! 8 x " 2 ue u 2 " 3 xy 2 v 2 " 1 cos u g v ! f x x v " f y y v ! ! y 3 ! 8 x "! 0 " " 3 xy 2 v v 2 " 1 sin u ! 3 xy 2 v v 2 " 1 sin u 12. g ! u , v , w " ! g x ! u , v , w " , y ! u , v , w " , z ! u , v , w " g u ! g x x u " g y y u " g z z u , g v ! g x x v " g y y v " g z z v , g w ! g x x w " g y y w " g z z w , 20. g ! t " ! f x ! t " , y ! t " g # ! t " ! f x x # ! t " " f y y # ! t " g ## ! t " ! ! f xx x # ! t " " f xy y # ! t "" x # ! t " " f x x ## ! t " " ! f yx x # ! t " " f yy y # ! t "" y # ! t " " f y y ## ! t " ! f xx ! x # ! t "" 2 " 2 f xy x # ! t " y # ! t " " f yy ! y # ! t "" 2 " f x x ## ! t " " f y y ## ! t " 22. g ! u , v " ! f ! x ! u , v " , y ! u , v "" g u ! f x x u " f y y u g uv ! ! f xx x v " f xy y v " x u " f x x uv " ! f yx y v " f yy y v " y u " f y y uv ! f xx x u x v " f xy y v ! x u " y u " " f yy y u y v " f x x uv " f y y uv 24. xyz ! 4 y 2 z 2 " cos ! xy " ! 0, let F ! x , y , z " ! xyz ! 4 y 2 z 2 " cos ! xy " . F x ! yz ! y sin ! xy " , F y ! xz ! 8 yz 2 ! x sin ! xy " , F z ! ! 8 y 2 z z x ! ! F x F z ! ! y ! z ! sin ! xy "" ! 8 y 2 z ! z ! sin ! xy " 8 yz z y ! ! F y F z ! ! xz ! 8 yz 2 ! x sin ! xy " ! 8 y 2 z ! xz ! 8 yz 2 ! x sin ! xy " 8 y 2 z 29. f ! x , y " , x ! r cos ! , y ! r sin ! f r ! f x cos ! " f y sin ! , f rr ! f xx cos 2 ! " 2 f xy cos ! sin ! " f yy sin 2 ! f ! ! ! f x r sin ! " f y r cos ! , f !! ! f xx r 2 sin 2 ! ! 2 f xy r 2 cos ! sin ! " f yy r 2 cos 2 ! ! f x r cos ! ! f y r sin ! 1

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f rr " 1 r f r " 1 r 2 f !! ! f xx cos 2 ! " 2 f xy cos ! sin ! " f yy sin 2 ! " 1 r ! f x cos ! " f y sin ! " " 1 r 2 f xx r 2 sin 2 ! ! 2 f xy r 2 cos ! sin ! " f yy r 2 cos 2 ! ! f x r cos ! ! f y r sin ! ! f xx " f yy 28. f ! x , y " , x ! r cos ! , y ! r sin ! x ! ! ! r sin ! , x !! ! ! r cos ! , y ! ! r cos ! , y !! ! ! r sin !
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hwsol_15 - Homework Assignment 15(12.5 Solutions Page 966 6...

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