Explanation Given information: The given vector function is. F → = ( y + z ) i ^ + x j ^ + ( z − x ) k ^ …. ( 1 ) The surface S is the triangular region of the plane x + 2 y = 3 in the first octant. Calculation: The stokes’ theorem states that: Let S is an oriented surface with unit normal vector field N , and let S be bounded by a smooth curve C compatible with S . If F is a vector field. ∮ C F → . d R → = ∬ S ( c u r l F → . N ^ ) d S The surface S is the triangle with vertices ( 3 , 0 , 0 ) , ( 0 , 3 2 , 0 ) , ( 0 , 0 , 3 ) . The three edges of the boundary curve C are. E 1 : x + 2 y = 3 , z = 0 E 2 : x + z = 3 , y = 0 E 3 : 2 y + z = 3 , x = 0 For Edge E 1 . The parameters are x = t , y = 3 − t 2 , z = 0 for 0 ≤ t ≤ 3 . R → ( t ) = t i ^ + ( 3 − t 2 ) j ^ d R → = d t i ^ − 1 2 d t j ^ From equation ( 1 ) . F → ( t ) = ( 3 − t 2 ) i ^ + t j ^ − t k ^ ∫ E 1 F → . d R → = ∫ 3 0 ( 3 − t 2 ) d t = 1 2 ( 3 t − t 2 ) 3 0 = 0 For edge E 2 . The parameters are x = s , z = 3 − s , y = 0 for 0 ≤ s ≤ 3 . R → ( t ) = s i ^ + ( 3 − s ) k ^ d R → = d s i ^ − d s k ^ = d s ( i ^ − k ^ ) From equation ( 1 ) . F → ( t ) = ( 3 − s ) i ^ + s j ^ − ( 3 − 2 s ) k ^ ∫ E 2 F → . d R → = ∫ 0 3 s d s = ( s 2 2 ) 3 0 = 9 2 For edge E 3 . The parameters are x = 0 , z = 3 − 2 r , y = r for 0 ≤ s ≤ 3 2 . R → ( t ) = r j ^ + ( 3 − 2 s ) k ^ d R → = d r j ^ − 2 d r k ^ = d r ( j ^ − 2 k ^ ) From equation ( 1 )

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ISBN: 9781524916817

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 13.6, Problem 4PS

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Calculus

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