( : December 27, 2015) CONTENTS I. 1 II. 2 III. 2 IV. 3 V. 5 VI. 6 VII. 7 VIII. 9 I. 1 f(x) f (x) y = f(x) x ϕ(r) (gradient) ϕ(r) (gradϕ(r) ) ( ) ϕ(r)

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( : December 27, 215 CONTENTS I. 1 II. 2 III. 2 IV. 3 V. 5 VI. 6 VII. 7 VIII. 9 I. 1 f(x f (x y f(x x ϕ(r (gradient ϕ(r (gradϕ(r ( ϕ(r r ϕ r xi + yj + zk ϕ(r ϕ(r x i + ϕ(r y j + ϕ(r z k (1.1 ϕ(r ϕ(r i x + j y + k z (1.2 ϕ(r ϕ(r ϕ(r x ϕ(r i (1.3 ϕ(r x y z x ϕ(r i (x ( ϕ(r x 2 ϕ(r 2xi (1.4 ϕ x ϕ y y y z x x > x x < x x y x 2 ϕ(r dy/dx ( x 1. p.42 [!] ϕ(r ϕ(r ϕ (2 p.42[ 5.2] harada(atartsci.kyushu-u.ac.jp ϕ(r x 2 + y 2 + z 2 ϕ(r 2xi + 2yj + 2zk 2r (1.5 ϕ(r 2 ϕ(r ϕ r ϕ(r 2 (! ϕ(r r

2 II. A(r (divergence A(r (diva(r A A x i + A y j + A z k 1.1 1. A(r A x(r x + A y(r y + A z(r z (2.1.9 A(r A(r A(r.8 A(r ai + bj + ck (a, b, c A(r (2.2 ( A(r xi + yj + zk r A(r 3 (2.3 A(r ( A > 1 2 r r r r + r ( r 2 A A(r r A(r + r r + r r 2 r (r + r 2 (r + r (r + r 3 r 3 12πr 2 r (2.4 ( r 2 r 2 r 3 A 3 1.4.5.6.7 2. ( 2 3 A(r yi xj A(r (2.5 A(r z 3 E p.28 p.283 III. rotation A(r (rota(r curla(r ( Az (r A(r A y(r i y z ( Ax (r + A z(r j z x ( Ay (r + A x(r k (3.1 x y i j k A(r x y z A x (r A y (r A z (r (3.2 A(r A(r

3 1.1 1..9.8.4.5.6.7 3. A(r A(r ai + bj + ck (a, b, c A(r xi + yj + zk r A(r (3.3 A(r (3.4 A(r A(r yi xj A(r 2k (3.5 z A(r r z 4. F z 3 2 z θ ( (r r + r A (r + r (r + r θ (r + r A r r θ r (r + r 2 θ + r 2 θ 2r r θ (3.6 r r θ 2 A 2k IV. p.39 p.44 3 F (r Γ A B 4 W F (r dr (4.1 Γ F l F l cos θ 2 3 m mg

A(r dr (ai + bj + ck dyj bdy OBC 1 1 A(r dr bdy + adx b + a (4.4 OAC 1 OB 2 BC OAC OBC 2 O A p.43 p.44 p.277 p.279 A(r 2 A(r 4 5. 4 Γ F i dr i (θ Γ 5 i F i dr i F i dr i i F i dr i (4.1 (4.1 Γ p.4 [ 5.1] III A(r yi xj Γ c Γ c A(r dr (4.2 A(r ai + bj + ck (a, b, c O(,, A(1,, B(, 1, C(1, 1, OAC 1 1 A(r dr adx + bdy a + b (4.3 OAC 1 OA 2 AC OA dr dxi A(r dr (ai+ bj + ck dxi adx AC dr dyj A(r xi + yj + zk r III OAC 1 1 A(r dr xdx + ydy 1 2 + 1 2 1 (4.5 OAC BC OC C OC r t r(t t(i + j ( t 1 dr (i + jdt OC A(r dr 1 1 A(r yi xj (ti + tj + k (i + jdt 2tdt 1 (4.6 A(r 1 1 A(r dr (i xj dxi + (yi 1j dyj OAC A(r dr OBC 1 (4.7 1 (yi j dyj + 1 (1i xj dxi 1 (4.8

ds 6 A A A(r ai + bj + ck (a, b, c 5 A(r ϕ(r 6. B A ϕ(r dr ϕ(r A ϕ(r B (4.9 p.43 b a df(x dx f(b f(a (4.1 dx A(r ϕ(r A(r A(r ai + bj + ck ϕ(x, y, z ax + by + cz A(r r ϕ(r 1 2 r 2 A(r yi xj V. A(r S A(r ds (5.1 S A(r ds A(r ds z OACB z x 1 y 1 x y ds dxdyk 1 dx 1 dy(ai + bj + ck k c (5.2 1 z ( p.98 [ 11.1] r e r sin θ cos ϕi + sin θ sin ϕj + cos θk (5.3 e r r r r 1 r ds ds sin θdθdϕe r 4 A(r ds π/2 π/2 2πc sin θdθ sin θdθ π/2 2π 2π sin θ cos θdθ dϕ(ai + bj + ck e r dϕ(a sin θ cos ϕ + b sin θ sin ϕ + c cos θ πc (5.4 ( 1 c OACB (1 1 c ( (z ( c 2 S 1 4 R ds R 2 sin θdθdϕ e r

6 S 2 S 2 S 2 A(r ds S 2 A(r ds S 2 (5.5 S S 1 S 2 S S A(r ds A(r ds + A(r ds (5.6 S 1 S 2 A(r ds A(r ds (5.7 S 1 S 2 7. VI. A(r xi + yj + zk r z 1 z 2 (x r cos ϕ, y r sin ϕ ds rdrdϕk 1 A(r ds rdr 2π dϕ(r cos ϕi + r sin ϕj + k k (5.8 z z 1 A(r e r A(r ds π/2 sin θdθ 2π dϕ e r e r 2π (5.9 A(r 3 3 z 2π/3 (2π/3 3 2π (5.8 (5.9 3 ( ( 1 p.98 p.114 a dv dv dxdydz dv d 3 r d 3 r V dv a a dx dy a dz a 3 (6.1 ( 1 3 1 R dv r 2 sin θdrdθdϕ 7

7 V R dv r 2 dr π sin θdθ 2π dϕ R3 3 2 2π 3 R3 (6.2 1 z R 2 z 2 z z + dz r z R 2 z 2 dz dz y 2 rz 2 y 2 2 R 2 y 2 z 2 y y + dy 2 R 2 y 2 z 2 dy dy ( R 2 y 2 z 2 + R 2 y 2 z 2 x (1 V R dz (z ( R dz R 2 z 2 R 2 z 2 dy ( y ( R dz R 2 z R 2 2 y 2 z 2 dy R 2 z 2 R 2 y 2 z 2 dx (6.3 x y z y z z V 2 R R R 2 z 2 dz dz R 2 y 2 z 2 dy R 2 z 2 R 2 z 2 R 2 y 2 z 2 dx R 2 z 2 dy R 2 y 2 z 2 (y R 2 z 2 sin t R π 2 2 dz R2 z 2 cos θdθ 2 π 2 R 2 z 2 1 sin 2 θ R dz(r 2 z 2 π 2 π 2 cos 2 θdθ 2 4 3 R3 π 2 3 R3 (6.4 13 VII. A(r S V A(r ds A(r dv (7.1 S p.281 p.283 S ( ( ( ρ(r E(r E(r 1 ϵ V ρ(r (r r r r 3 d 3 r (7.2 E(r r r E(r 1 ϵ ρ(r ( (r r r r 3 d 3 r (7.3 r r ( r r (7.4 [ r r 2 + ϵ 2 ] 3 2 ϵ ( x x x [(x x 2 + 2 ] 3 2 1 (x x 2 3 [(x x 2 + 2 ] 3 2 [(x x 2 + 2 ] 5 2 ( r r [ r r 2 + ϵ 2 ] 3 2 3 3 r r 2 [ r r 2 + ϵ 2 ] 3 2 3ϵ 2 (7.5 (7.6

8 r r ϵ r r ϵ 3/ϵ 3 r r ρ(r ρ(r ( (r r r r 3 d 3 r 3ϵ 2 d 3 r (7.7 5 ϵ R r r 3ϵ 2 d 3 r 12πϵ 2 R 2 dr (7.8 [R 2 + ϵ 2 ] 5 2 R ϵ tan t dr ϵdt/ cos 2 t ϵ 2 R 2 ϵ 4 tan 2 t [R 2 + ϵ 2 ] 5 2 ϵ [ 5 1 + tan 2 t ] 1 5 2 ϵ tan2 t cos 5 t ϵ 2 1 ϵ sin2 t cos 3 t (7.9 R 2 dr [R 2 + ϵ 2 ] 5 2 25 2 15 1 5 1 3 π 2 cos t dt sin 2 t -1 1 2 3 (7.1 8. (7.7 5 8 ϵ 1 3 y 3ϵ 2 /[(x 1 2 + ϵ 2 ] 5 2 y x 2 + 2x + 4 ( 7 ( ϵ ϵ E(r ρ(r ϵ (7.11 ( ( ( E(r ds 1 ρ(rd 3 r (7.12 ϵ S V Q E(r ds Q (7.13 ϵ S p.142 ( ( λ z r E(r z z z ( E(r E(r E(r ( r z E(r E r (re r e r r E(r z ( L a E(r E(r ds (7.14 E(r (E r (a E(r ds 2πaL E r (a (7.15 2πaL E(r ds 2πrL E r (a (7.16 V

9 Q λl 2πaL E r (a λl ϵ (7.17 E r (a λ/(2πϵ a VIII. E(r λ 2πϵ r e r (7.18 A(r Γ c ( S A(r dr ( A(r ds (8.1 Γ c S p.277 p.279 ( ( z I - ( p.177 p.179 B(r µ Ids (r r C r r 3 (8.2 r xi + yj + zk B(r 6 B(r µ µ I ( yi + xj Idz k (xi + yj + (z z k [x 2 + y 2 + (z z 2 ] 3 2 µ I (x 2 + y 2 ( yi + xj dz [x 2 + y 2 + (z z 2 ] 3 2 π 2 π 2 cos θdθ µ I 2π (x 2 + y 2 ( yi + xj (8.3 B(r k r i z z a z B(r dr (8.4 b(r dr B(r dr µ I 2πa 2π adϕ µ I (8.5 z (8.3 x y ϵ B(r µ I 2π (x 2 + y 2 + ϵ 2 ( yi + xj (8.6 B z B x B y z ( By B(r x B x k y µ I 2π 2ϵ 2 (x 2 + y 2 + ϵ 2 2 k (8.7 x 2 + y 2 ϵ x 2 + y 2 ϵ a ds rdrdϕk a ( B(r ds µ I a 2 2ϵ 2 rdr (r 2 + ϵ 2 2 µ I a 2 + ϵ 2 (8.8 ϵ µ I i e (r 7 r i e (r i e (r 6 B 7 i(r i e

1 ( i e (r - B(r µ ie (r (r r r r 3 d 3 r (8.9 i e (r 1 2Iϵ 2 2π (x 2 + y 2 + ϵ 2 2 k (8.1 (I > z z x-y x-y i e (r ds 2Iϵ 2 rdr (r 2 + ϵ 2 2 I (8.11 ϵ z (8.9 i e (r ϵ x y B(r µ ie (r (r r r r 3 d 3 r µ I ϵ 2 2 (x 2 + y 2 + ϵ 2 2 k (xi + yj + (z z k dx dy dz [x 2 + y 2 + (z z 2 ] 3 2 µ I dz ( yi + xj 2 [x 2 + y 2 + (z z 2 ] 3 2 ϵ 2 (x 2 + y 2 + ϵ 2 2 dx dy µ I ( yi + xj µ I ( yi + xj dz [x 2 + y 2 + (z z 2 ] 3 2 dz [ x2 + y 2 + z 2 ] 3 2 µ I 2π(x 2 + y 2 ( yi + xj (8.12 (8.9 B(r ie (r (r r r r 3 d 3 r ie (r r r d3 r (8.13 x [ie (r ] z y r r d3 r [ie (r ] y z r r d3 r [i e (r 1 ] z y r r d3 r [i e (r 1 ] y z r r d3 r [i e (r (y y ] z r r 3 d3 r [i e (r (z z ] y r r 3 d3 r [ie (r ] y (z z [i e (r ] z (y y r r 3 d 3 r (8.14 x y z [ µ ie (r ] B(r r r d3 r (8.15 8 ( X ( X 2 X (8.16 B(r ( X(r 2 X(r (8.17 X(r µ ie (r r r d3 r (8.18 2 2 x 2 + 2 y 2 + 2 z 2 (8.19 r X(r µ 1 r r 1 r r µ µ i e (r 1 r r d3 r i e (r 1 r r d3 r ( i e (r (8.2 1 r r d3 r (8.21 8 3 ( p.8 X

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