Homework01VectorCalc

# Homework01VectorCalc - a 2 b. Curl (use formula from...

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ECE 3030: Electromagnetic Fields and Waves Fall 2009 Homework 1 SOLUTIONS for GRADER: Vector Calculus Reminder: These details are for the students to work out! Date: Tuesday 9/1 Problem 2.19 Note d = ˆxd x + ˆyd y + ˆ zd z , so for ~ A = xy ˆx + yz ˆy + xyz ˆ z, ~ A · d = xy d x + yz d y + xyz d z (1) Along path (0,0,0) to (0,0,1), x = 0, y = 0, d x = 0, and d y = 0, so R ~ A · d = 0 (2) Along path (0,0,1) to (0,2,2), x = 0, d x = 0, y = 2( z - 1), d y = 2d z , so R ~ A · d = R yz d y = 2 R 1 2 z ( z - 1)2d z = 2 R 1 (4 z 2 - 4 z )d z = (4 z 3 3 - 2 z 2 ) 2 1 = 32 3 - 8 - 4 3 + 2 = 10 3 x y z 1 2 3 4 Figure 1: The four paths for the line integral. (3) Along path from (0,2,2) to (1,1,0), x = 2 - y = 1 - z 2 d x = - d y = - 1 2 d z , and y = 2 - x , z = 2 - 2 x ), so ~ A · d = x (2 - x )d x - (2 - x )(2 - 2 x )d x - x (2 - x )(2 - 2 x )2d x = (2 x - x 2 - 4+6 x - 2 x 2 - 8 x +12 x 2 - 4 x 3 )d x = ( - 4 + 9 x 2 - 4 x 3 )d x Z ~ A · d = Z 1 0 ( - 4 + 9 x 2 - 4 x 3 )d x = ( - 4 x + 3 x 3 - x 4 ) | 1 0 = - 2 (4) Along path from (1,1,0) back to (0,0,0), z = 0, y = x , d y = d x , so Z ~ A · d = Z 0 1 x 2 d x = x 3 3 = - 1 3 Summing up all the pieces, we have I ~ A · d = 10 3 - 2 - 1 3 = 1 Problem 2.25 Contour integral around a circle. a. The contour is deﬁned by r = a , z = 0, and ~ d s = a d φ ˆ φ , so I ~ F · ~ d s = Z 2 π 0 a 2 d φ = 2
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Unformatted text preview: a 2 b. Curl (use formula from cylindrical coordinates): ~ ~ F = + 2 z. c. The elemental area is ~ d a = r d d r z, so x A ( ~ ~ F) ~ d a = Z a Z 2 2 r d r d = 2 a 2 d. From (a) and (c), we have shown that I ~ F ~ d s = x A ( ~ ~ F) ~ d a = 2 a 2 . e. For convenience, express the curl in spherical coordinates: ~ ~ F = +2cos r. On the spherical surface, the elemental area is ~ d a = a 2 sin d d r, so I ~ F ~ d s = Z / 2 Z 2 2 a 2 cos sin d d = 2 a 2 Although the surface is dierent than for part (c), the answer is the same! (Stokes Theorem.) CORNELL UNIVERSITY c WES SWARTZ (09/8/12) 11...
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## This note was uploaded on 01/16/2010 for the course ECE 3030 at Cornell.

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