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Unformatted text preview: 2010년 2학기 공수(4) 과제-CSE 홈페이지 과제 문제들을 다음 일정과 같이 운영. 1차과제: 9.8,9.9,10.1,10.4,10.6,10.7 제출일자: 9월 13일(월) 2차과제: 10.8,10.9,13.1,13.2,13.3 제출일자: 9월 27일(월) 3차과제: 13.4,13.5,13.6,13.7 제출일자: 10월 4일(월) 4차과제: 14.1,14.2,14.3,14.4 제출일자: 10월 14일(목) 5차과제: 11.1,11.2,11.3,11.4 제출일자: 11월 1일(월) 6차과제: 11.6,11.7,11.9 제출일자: 11월 12일(금) 7차과제: 12.1,12.3,12.4,12.5,12.6 제출일자: 11월 25일(목) 8차과제: 24.1,24.3,24.5,24.6,24.7 제출일자: 12월 9일(목) 9차과제: 24.8,25.2,25.3 제출일자: 12월 16일(목)-과제풀이는 CSE홈페이지( http://web.yonsei.ac.kr/CSEgrad/index.htm ) Lecture 란에 제출일자 오후 7시에 또는 다음 날 오전에 제공 Homework 1 § 9 . 8- § 9 . 9 1. Find the divergence and the curl of the following vector functions. (a) F ( x, y, z ) = e x i + ye- x j +2 z sinh x k (b) F ( x, y, z ) = ( x 2 + y 2 + z 2 )- 3 / 2 ( x i + y j + z k ) 2. Show that the flow with velocity vector v = y i is incompressible. Show that the particles that at time t = 0 are in the cube whose faces are portions of the planes x = 0 , x = 1 , y = 0 , y = 1 , z = 0 , z = 1 occupy at t = 1 the volume 1. 3. The velocity vector v ( x, y, z ) of an incompressible fluid rotating in a cylindri- cal vessel is of the form v = w × r , where w is the constant rotation vector and r = [ x, y, z ]. Show that div v = 0. 4. Assuming sufficient differentiability, show that (a) div ( f ∇ g )- div ( g ∇ f ) = f ∇ 2 g- g ∇ 2 f (b) div (curl v ) = 0 (c) div( u × v ) = v • curl u- u • curl v . 5. Let v be the velocity vector of a steady fluid flow. Is the flow ir-rotational? Incompressible? Find the streamlines(the paths of the particles). (a) v = [ x,- y, z ] (b) v = [ y 3 ,- x 3 , 0] § 10 . 1 1. Calculate R C F ( r ) • d r , F ( x, y, z ) = [ z, x, y ] along the following path C . (a) C : r ( t ) = [cos t, sin t, t ] from (1 , , 0) to (1 , , 4 π ) (b) C : the straight segment from (1 , , 0) to (1 , , 4 π ). 2. Find the work done by the force F ( x, y, z ) = z i + y j- x k in the displacement along C the intersection y 2 + z 2 = 1 and y = x + 1, oriented counterclockwise as seen by a person standing at the origin. 3. Consider the integral R C F ( r ) • d r , where F ( x, y ) = xy i- y 2 j . (a) Find the value of the integral when r ( t ) = [cos t, sin t ] , ≤ t ≤ π/ 2. Show that the value remains the same if you set t =- p or t = p 2 . (b) Evaluate the integral when C : y = x n , ≤ x ≤ 1 , n = 1 , 2 , ··· . What is the limit as n → ∞ ? § 10 . 4 1. Evaluate R C F ( r ) • d r counterclockwise around the boundary curve C of the region R , where Computational Science & Engineering (CSE) C. K. Ko Homework 2 (a) F ( x, y ) = [ xy 4 , x 4 y ], R the rectangle with vertices (0 , 0) , (2 , 0) , (2 , 1) , (0 , 1)....
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This note was uploaded on 11/08/2011 for the course EE 111 taught by Professor Kim during the Spring '11 term at Korea University.

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