Problem Set 1 Solution

2 sin 2 k 2 cos 1 cos 2 2

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Unformatted text preview: 20 CHAPTER 1 ! a $ er % !! & r# 2 r ! " ! ! ! % & k # 2 cos # ' #! sin # & k !1 ' cos # " # 2 ! ( ) # 2 sin 2 # ! ! % & k *# 2 cos # ' ' !1 ' cos # " # 2 + 2 !1 ' cos # " * + , ( ) 1 & cos 2 # ! % & k# 2 * 2 cos # ' ' 1+ 2 !1 ' cos # " + * , 3! % & k# 2 !1 ' cos # " 2 (9) or, 3 v2 4k (10) 3 v 2 sin # 4 k 1 ' cos # (11) ! a $ er " 2 ' ! a $ e# " 2 (12) a $ er % & In a similar way, we find a $ e# % & From (10) and (11), we have a% or, 3 v2 4k MATRICES, VECTORS, AND VECTOR CALCULUS a% 1-27. Since 1-28. 2 1 ' cos # (13) 21 # grad ! ln r " $ % ! ln r " e i #x r . ! v . r " % ! r $i r " v i& ! r $ v " r (1) where we have Therefore, r$ x2 d d $ iv / r . ! v . r " 0 % dt / ! r $ r " v & ! r %" ri 0 dt (2) % !r $ r" a ' 2 !r $ v" v & !r $ v" v & ! v $ v" r & !r $ a" r # 1x ! ln r " $ ' ri $ 2 % r 2 a#x!i r $ v " v & rr v 2 % xia ' ! Thus, " x $ i2 d 2 r $ v" v & r r $ a ' v2 / r . ! v . r "0 % r a ' ! dt ! so that (1) i grad ! ln r " $ " (3) (2) ' 1& xe 2 (% i i+ )i * r (4) r r2 (5) or, grad ! ln r " $ 1-29. Let r 2 $ 9 describe the surface S1 and x , y , z 2 $ 1 describe the surface S2 . The angle between S1 and S2 at the point (2,–2,1) is the angle between the normals to these surfaces at the ! " r2 1-29. Let r 2 $ 9 describe the surface S1 and x , y , z 2 $ 1 describe the surface S2 . The angle between S1 and S2 at the point (2,–2,1) is the angle between the normals to these surfaces at the point. The normal to S1 is ! ! " grad ! S1 " $ grad r 2 . 9 $ grad x 2 , y 2 , z 2 . 9 22 In S2 , the normal is: CHAPTER 1 $ ! 2xe1 , 2 ye2 , 2ze3 " (1) x $ 2 , y $ 2 , z $1 $ 4grad4e21 " $ grad ! S2 " e1 . ! S , 2e3 cos # % grad ! S1 " grad ! S2 " ! or, " " grad ! S2 "e$ & 4e2 ' 2ey",!z 21 . 1 2 ' 2e3 " ! 4 1 grad x , 3 $ e ' e % 66 $ ! e1 , e2 , 2ze3 " x $ 2 , y $.2 , z $1 (3) (2) $ e1 , e2 , 2e3 4 cos # % 66...
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