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Scanned by CamScanner L j s t a 4 C a c u 1o 1 I I A 20 15 2 1 1 U n i v e r s id a d e d e r a l F l u m i n e n s e L I S T A 4 2 0 1 5 2 E G ] \ ( I n s t i t u t o d e M a t e m á t i c a e E s t a t ís t i c a I n t e g r a l de L i n h a d e C a m p o V e t o r i a l G M A D e p a r t a m e n t o d e M a t c m 1i t i c a A p l i c a d a T e o r e m a d e G r e e n C a m p o s c o n s e r v a t i v o s 1 C a l c u l e J . c d Æ + a 2 dy d e ( 1 , 0) a (1 , 0) a o l o n g o (a ) d o e i x o Æ (b ) d c c j (t ) = ( c o s t , s e n t i , O < t < ?r (c ) d a p o l i go n a l d e v ér t i c e s ( 1 , 0) , (0 , 1) , (1 , 1) , (1 , 0) 2 C a l c u l e [ c y dn + z d y , a o l o n g o d o s s e g u i n t e s c a i n i n h o s f e c h a d o s , o r i e Ä a d o s n o s e n t i d o a n t i h o r h i o (a ) c i r c u n fe r ên c i a d e c e n t r o n a o r i g e m e r a i o 2 (b ) e l i p s e z 2 + 3 6y 2 3 6 (c ) t r i ân g u l o d e v ér t i c e s (0 , 0) , (1 , 0) · (0 , 1) 3 C a l c u l e Jf (3 a + 2y ) d z + (3 z y ) Cl yde (0 , 0 ) · (1 , 1) , a o l o n g o (a ) d o s e g m e n t o d e r e t a (b ) d o a r c o d e p a r áb o l a y - z 2 (c ) d o a r c o d e c i r c u n f e r ên c i a z 2 + (y 1) 2 _ 1 , o r i e n t a d o n o s e n t i d o a n t i h o r in i o 4 C a l c u l e j 3 z z d z + 4 y z d y + 2 × y d z , d o p o n t o A (0 , 0 , 0 ) a D p o n t o B (1 , 1 , 2 ) , a o l o n g o d o s s e g u i n t e s c a m i n h o s (a ) s e g m e n t o d e r e t a A B (b ) i n t e r s e ç ão d a s s u p e r f ic i e s z - z 2 + y 2 e = y 5 C a l c u l e j . P d r + q d y + R d z , o n d e (a ) _ (P , Q , R ) (y , z , £ ) e Céai nt er seçi udassuper f i ci es = + y 2 e c2 + y 2 + z 2 _ 2 (T + y ) , p e r c o r r i d a n o s e n t i d o a n t i h o r á r i o q u a n d o v i s t a d a o r ig e m (b ) i (P , Q , R ) = ( 2y , z , Æ) e ¡ é a i n t e r s e ção d a s s u p e r f i c i e s 4 c2 + y 2 1 e y 2 + z 2 _ 1 , c o m z > O e z > O , p e r c o r r i d a u m a v e z d o p o n t o (O, 1 , 0 ) a o p o n t o (D, 1 , 0 ) (c ) _ (P , Q , R ) - (z , z , y ) c ¡ é a i n t e r s e ç ão d a s s u p e r f í c i e s z - 2 + y 2 e 4 c + 2 y + z = 1 o r i e n t a d a de m o d o q u e s u a p r o j e gão n o p l a n o z y s e j a p e r c o r r i d a u m a v e z n o s e n t id o h o r á r i o 6 Se j a ¡ a i nt er seçi o d o c i l i n d r o a 2 + y 2 4 c o m o s e m i p l a n o ¥ + z 0 , y · 0 , p e r c o r r i a d e m o d o q u e s u a pr oj e n o p l a n o r y t e n h a s e n t id o t i h o r á r i o e s e ja F (T , y , z ) = T 1 ï + y 4 + z 4 C a l c u l e a \n Ae g r a 1 Jc Ê d i 7 0 c p o d e fo r ga s F (z , y ) · ) = (a 2 , y 2 , z 4 n 4 ) a t u a s o b r e u m a p a r t íc u l a t r a n s l a d a n d o a a o l o n g o d a c u r v a i n t e r s e g o d a s s u p e r f ic i e s z - z 2 + y 2 e z 4 » + 4 y 4 , o r i e n t a d a d e m o d o q u e s u a p r o j e ç ão n o p l a n o z y s e j a p e r c o r r i d a u m a v e z , n o s e n t i d o h o r á r i o C a l c u l e o t r a b a l h o r e a l i z a do p o r 8 C a l c u l e o t r a b a lh o r e a l i z a d o p e l a f o r ça F _ (z , y , z ) p a r a d e s l o c a r u m a p a r t íc u l a a o l o n g o d a c u r v a i n t e r s e ç ão d a s s u p e r f íc i e s = + z 5 e z 4 y 2 , o r i e n t a d a d o p o n t o (5 , 2 , 0) · (5 , 2 , 0) g A c h a r o t r a b a l h o d e u m a fo r ça v a r i áv e l , d i r i g i d a p a r a a o r i ge m d a s c o o r d e n a d a s , c u j a g r a n d e z a é p r o p o r c i o n a l a o a f a s t a m e n t o d o p o n t o e m rel aç9o à o r ig e m d a s c o o r d e n a d a s , s e o p o n t o d e a p l i c a ç ão d e s t a fo r qa d e s c r e v e , " " t i d o a n t i h o r i i r i o , ° p a r t e d a e l i p s e + = 1 n o p r i m e i r o q u a d r a n t e Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Q - - 1 ( - - 1 ^ · . ^ o ( 0 ( - . f l ) l L g . 1 . ;· 1 + r E . 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