Wu Chong-shi a0a1a2a3a4 Green a5a6 (a7) §28.1 a8a9a10a11a12a13a14 δ a15a11 star a16a17a18a19a20a21 a22 a23a24a25 a26 δ a23a24a27 star δ a23a24a28a29a30a31a32a33P. A. M. Diraca34a35a36a37a21 a26 a38a39a40a30a31a32a41a42a43a44a45 a21a46a47 a27a48a49a50 a47a51a52a53 a30a31a32a41 a21a18a54a55a56 a26a57a58 a55a59a56a60a55a61a62a60a63a64a65a66 a26 a30a31a67a68a69a70a27 star a38a24a32a71a26δ a23a24a49a50a72a73a74a75a23a24a18a76a37a77a78a79 a26a58 a37a77a80a81a82a83a81a84a85 a26 a86a87 a46a47a51 a88a89 a80a81a90a91 a26a92a93a94a95 a21a96a97a82 a30a31 a96a98 a28 a18a99a21 a27 star a78a47δ a23a24a26a49a50a100a101a102a103a31a42a104a21 a24a32a30a31a105a106 a26a107a108a109a110 a21a111a112 a27 star δ a23a24a28a18a113a22 a114a115a25 a21 a23a24 a26a116a117 a22 a118a119a25 a21 a24a32a120a121a28a122a123a31a89 a21 a27a48 a21a124a125 a24a32a31 a98 a26a126a127a128 a45a23 a81a129a21a130a131 a27 star a132a133a134a135 a30a31a32 a21a136a137a138a139 a26 a36a37 δ a23a24a21 a120a121 a26 a16a17 a48 a21a140a141a132a21a130a131 a128a142a143a144 a46a47 a27 a73a100δ a23a24 a21 a30a31a145a146 a26 a35a147a98a55a65a60 a57a58 a55a61a62a21a61a62a81a148a149a150 a23a24 a21 a24a32a151a152a27a100 a153a154a155a156 a26a157a126 a147a98a18a158a159a160 a27 a161a16228.1 a26 a163a164a165a166a167a168a1690 < x < l a170a171a172a173a174a175a176a177 a178a179a180 a26 a181 a177a182a183 1a184a185a186 a26 a164 a170a171a187 a165 a177 a178 a26 a188 a177 a178a189a190a191 a192 a183 δl(x) = ? ??? ? ??? ?? 0, a193x < ?l2; 1 l , a193 ? l 2 < x < l 2; 0, a193x > l2. a194a195a196a197a198 a184 a164?l/2 < x < l/2 a172a199a200a176 a191a192 f(x) a26a201a202a203 a204a205a206 a26 a173 a20728.1 a208a209a210a211a212a213a211a212a214a215 integraldisplay ∞ ?∞ f(x)δl(x)dx = f(θl), ?1/2 ≤ θ ≤ 1/2. a216a217a169 a26a218 a179a219a220a198a205a221±∞a27a222a223a < ?l/2, b > l/2 a26a224 a173integraldisplay b a f(x)δ(x)dx = f(θl), ?1/2 ≤ θ ≤ 1/2. Wu Chong-shi §28.1 a225a226a227a228a229a230a231 δ a232a228 a2332a234 a235 a183a236 a219a237a238 a26 a193 l → 0a239 a26a224a240a241a242 a177 a178 a176 a189a190a191a192 a26a243 a183 δ(x) = lim l→0 δl(x) = ?? ?? ?? 0, a193x < 0; ∞, a193x = 0; 0, a193x > 0. a244a245 a26 a194a195a196a197a198 a184 a164 x = 0 a242 a199a200a176 a191a192 f(x) a26 a173integraldisplay ∞ ?∞ f(x)δ(x)dx = f(0). a216a217a169 a26a218 a179a219a220a198a205a221±∞a27a222a223a < 0, b > 0 a26a224 a173integraldisplay b a f(x)δ(x)dx = f(0). a246a247 a26 a248a49a50a249a250a251a252 a21a61a62a81a148 a23a24a253a254a100 a142a255a0a1a2a3 a23a24 a26 a4a5a6a71a7 a21a147a98 a27 a73a100a48a102 a21 a109a8 a159a160 a26 a101a102a9a10 a94a95a11 a76a21a96a97 a27 star δa191a192a26a12a220a221a13a14a197a15a16a176a191a192a17a18a12a19a173a20a21a191a192a22a23a24a182a25a171a176a194a26a27a28a26a29a30a31a26 a18 a20a21a176 a194a26a27a28 δ(x) = braceleftBigg 0, a193x negationslash= 0; ∞, a193x = 0 a164a13a14a197a15a16a221a19 a173 a197a15 a176 a27 star δ a191a192a32a33a176 a221(a196a197a34a35a36) a191a192a37a38 a176a236 a219a27a18a39 a20a21a176 a22 a191a192a204a25a222a221a164 a218 a179a40a41 a203 a42 a173 a197a15a27 integraldisplay ∞ ?∞ f(x)δ(x)dx = f(0), a43a44 a221 integraldisplay ∞ ?∞ δ(x)dx = 1. star a45a184 a218 a179a26a46a206a47 a183 integraldisplay ∞ ?∞ f(x)δ(x)dx = lim l→0 integraldisplay ∞ ?∞ f(x)δl(x)dx. a48a49a41 a176a50 a190a51a52 a26a53a54δ a191a192 a176a55a176 a26a56 a164a195a57a58a194a191a192a37a38 a54a59 a36 a218 a179a49a41 a60 a244a60a61 a236 a219 a176a62a63 a27a64a195a191a192a37a38a221a64a65 a173a66a67a68a176a199a200a69a70a176 a191a192a71a72 a176 a26 a39a73 a26 a164a49a41 a203 a35a73a74δ a191 a192 a193 a235(a196a197a34) a199a200 a35a36 a176 a191a192a75a206 a26 a76a77a35a73a205a15 δ a191a192 a176a78 a192δprime(x)a17a194a195a164x = 0 a242 a199a200 a12 a173a199a200a78 a192 a176 a196a197a191a192 f(x) a26 a173integraldisplay ∞ ?∞ f(x)δprime(x)dx = f(x)δ(x) vextendsinglevextendsingle vextendsingle ∞ ?∞ ? integraldisplay ∞ ?∞ fprime(x)δ(x)dx = ?fprime(0). a45a79 a26a224 a74δ a191a192 a193 a235a80a13 a176a199a200 a191a192a198a81 a54a59 a179a82 a218 a179a27 star δ a191a192a12a220a221a20a21 a80a13 a176 a192a204 a25a171a176 a194a26a27a28 a26 a83a12a220a84a80a13a191a192a85a81a65 a173a86 a198 a60 a87a205 a176 a32a88a89a27 Wu Chong-shi a90a91a92a93a94 Green a232a228(a95) a2333a234 star a96 a221a65 a173 lim l→0 integraldisplay ∞ ?∞ f(x)δl(x)dx = f(0) a69a70a176 a191a192a37a38 δ l(x)a26 a29a221a65 a173 limn→∞ integraldisplay ∞ ?∞ f(x)δn(x)dx = f(0) a69a70a176 a191a192a37a38 δ n(x)a26 a18a97 a176a236 a219a98a221 δ a191a192a27 (a) n√piexpbracketleftbig?n2x2bracketrightbig (b) npi 11 + n2x2 (c) sinnxpix a20728.2 δ a99a100a213a101a102a103a104a105a106 star a107 a81 a26 a173 a27 δ a191a192 a176a108 a89 a26 a83a26 a193 a48 a218 a179a197a15a16a109a206a47a27a161 xδ(x)=0 a26a206a47a183 integraldisplay ∞ ?∞ f(x)xδ(x)dx=0, δ(?x)=δ(x) a26a206a47a183 integraldisplay ∞ ?∞ f(x)δ(?x)dx= integraldisplay ∞ ?∞ f(x)δ(x)dx, δprime(?x)=?δprime(x) a26a206a47a183 integraldisplay ∞ ?∞ f(x)δprime(?x)dx=? integraldisplay ∞ ?∞ f(x)δprime(x)dx, δ(ax)= 1|a|δ(x) a26a206a47a183 integraldisplay ∞ ?∞ f(x)δ(ax)dx= integraldisplay ∞ ?∞ f(x) bracketleftbigg 1 |a|δ(x) bracketrightbigg dx, g(x)δ(x)=g(0)δ(x) a26a206a47a183 integraldisplay ∞ ?∞ f(x)g(x)δ(x)dx= integraldisplay ∞ ?∞ f(x)bracketleftbigg(0)δ(x)bracketrightbigdx. star δ a191a192a110a35a73a32a33a72a111a108 a191a192 a176 a36a112a27a64a195 integraldisplay x ?∞ δ(x)dx = η(x), a113a114 a26 δ(x) = dη(x)dx . Wu Chong-shi §28.1 a225a226a227a228a229a230a231 δ a232a228 a2334a234 star a83a35a73a194δ a191a192a235Laplacea24a115a26 δ(t?t0)equaldotleftright integraldisplay ∞ 0 δ(t?t0)e?ptdt = e?pt0, t0 > 0. star δ a191a192a83a35a73a32a33a72a111a108a191a192a176 Fouriera218a179a27a113a183integraldisplay ∞ ?∞ δ(x)e?ikxdx = 1, a39a73 a26a201a202 Fourier a24a115 a176a116a117a118 a89 a26 a173 δ(x) = 12pi integraldisplay ∞ ?∞ eikxdk. a119a120a121a122a120 δ a123a124 star a164a125a126a169 (x0, y0)a242a75a173a198a184a185a186a242a177a178a26a85a127a26a189a190a179a180a191a192a224a221 δ(x?x0)δ(y ?y0)a27 star a164a128a129a130a171(x0, y0, z0)a75a173 a198 a184a185a186 a242 a177 a178 a26 a189a190a179a180a191a192 a224 a221 δ(x?x 0)δ(y?y0)δ(z?z0) a27 star a48a128a129a130a171 a51a52 a26 a39a131a198a129 a242 a177 a178a26a46a221a128a129a130 a171a172a176 a126 a177 a178a132 a133 a129 a242 a177 a178 a224 a221a128a129a130 a171a172 a176 a168 a177 a178a27 a134 28.1 a135a136 ?21r = ?4piδ(r), a137 a203r = radicalbigx2 + y2 + z2, δ(r) = δ(x)δ(y)δ(z) a27 a138 a139a84a140a126a141 a21a176 a26 a96 a221a142a143 δ a191a192 a176a108 a89a98a26a46a48 a218 a179a197a15a16a109a206a47 a26a56 a26a46a109 a135a136 integraldisplayintegraldisplayintegraldisplay V ?21rdxdydz = braceleftBigg 0, a193r = 0 negationslash∈ V ; ?4pi, a193r = 0 ∈ V. a193r negationslash= 0a239 a26 a167a144a36a112a35 a240 ? ?x 1radicalbig x2 + y2 + z2 = ? x (x2 + y2 + z2)3/2 , ?2 ?x2 1radicalbig x2 + y2 + z2 = 3x2 ?parenleftbigx2 + y2 + z2parenrightbig (x2 + y2 + z2)5/2 . a107 a206 a26 ?2 ?y2 1radicalbig x2 + y2 + z2 = 3y2 ?parenleftbigx2 + y2 + z2parenrightbig (x2 + y2 + z2)5/2 , ?2 ?z2 1radicalbig x2 + y2 + z2 = 3z2 ?parenleftbigx2 + y2 + z2parenrightbig (x2 + y2 + z2)5/2 . a128a89a145a146 a26a56a240 ?21r = 0, r negationslash= 0. a45 a81 a224 a135 a240 a17 a193 a218 a179a147 a218 V a172 a220a148a149a150 a242r = 0 a239 a26a218 a179a151 a1830a27 Wu Chong-shi a90a91a92a93a94 Green a232a228(a95) a2335a234 a193 a218 a179a147 a218V a172 a148a149a150 a242r = 0 a239 a26 a64a195a191a1921/ra164r = 0 a242 a220a35 a78 a26 a169a126 a176a152a153 a220a72a154a27 a45 a239 a220a155a156V a224 a61 a183a157a184(a128a129)a130a171a27a158a159a240a241integraldisplayintegraldisplayintegraldisplay ?21rdxdydz = lima→0 integraldisplayintegraldisplayintegraldisplay ?2 1√r2 + a2dxdydz = ? lima→0 integraldisplayintegraldisplayintegraldisplay 3a2 (r2 + a2)5/2 r2drsinθdθdφ = ?12pi lima→0 integraldisplay ∞ 0 a2 (r2 + a2)5/2 r2dr, a160r = atanθ a26a56 a35 a135a136 a169a126 a176 a218 a179a22 aa165a27 a26 a245 integraldisplayintegraldisplayintegraldisplay ?21rdxdydz = ?12pi integraldisplay pi/2 0 tan2θparenleftbig 1 + tan2θparenrightbig3/2 dθ = ?12pi integraldisplay pi/2 0 sin2θcosθdθ = ?12pi· 13sin3θ vextendsinglevextendsingle vextendsinglevextendsingle pi/2 0 = ?4pi. square Wu Chong-shi §28.2 Greena232a228a227a161a162 a2336a234 §28.2 Green a15a11a10a163a164 a165a166a198 a184a167a177a168a176a169a170 a27 a163a164a165a171a130 a171 a203 a173 a198a205 a176a177 a178a179a180 a26 a177 a178a189a190 a183ρ(r)a27a45 a81 a26 a164a172a173 a183rprime = (xprime,yprime,zprime)a176 a147a174 drprime a172a176a177a182 a56 a183 ρ(rprime)drprime a26a18a164a130a171r = (x,y,z)a242a176a177a175 a221 1 4piε0 ρ(rprime) |r?rprime|dr prime, a201a202 a177a175a176 a146a150a206 a26 a74a130 a171 a203 a176a177 a82 a177 a178a178a179 a176a177a175a176 a146a180a51 a26a224a240a241 a164 r a242 a176 a181 a177a175a183 φ(r) = 14piε 0 integraldisplayintegraldisplayintegraldisplay ρ(rprime) |r?rprime|dr prime. a45a184a152a153 a31 a136 a26 a222a223a181a182a183 a185a186 a242 a177 a178a164a130 a171a176a177a175 a179a180 a26 a85a127 a26 a13 a62a177 a178 a176 a179a184 a22 a176 a146 a26a224 a35a73 a240a241 a196a197 a177 a178a179a180 a239a176a177a175 a27 a45a185a186a187 a222a220 a62 a221a188a189a183a190a36a179a191 a63a176 a168 a69a69a70 a27 ? a161a153a221a173a171a130a171a26a150a188a169a192a193a35a73a74a130a171a172a176a177a178a165a219a179a184a27 ? a64a195a194a171a195a196a176a197a198 a26 a164a194a171a126a169a83a199 a173 a198a205 a176(a185a200 a29a201 a236a200a176)a202 a179a126 a177 a178a179a180 a26 a83a203a223 a156 a45a204 a126 a177 a178a165a219a179a184a27 ? a183 a183 a86 a198a205a87a205 ( a173 a171a130 a171a172) a242a177 a178 a176a177a175 a26 a83a203a223a141a205a206 a193a176 a194a171a195a196a27 a164 a173 a171a130 a171a176 a237a238a16 a26 a207a208 a224 a221a17a209a210a211a212( a213a214a215a216a217a218a219a220)a221a222a223a222a224a220a225a226 a26 a227a228a229a230a231 a222a223a232a233a234 a230a231 a215a216a217a218a235a220a222a224 a27 a45 a224 a221a31 a26 a223a189a205a47a207a208 ?2G(r;rprime) = ? 1ε 0 δ(r?rprime), r,rprime ∈ V a206 a193a176 a194a171a195a196 a176 a47G(r;rprime) a176 a146 a21 ?2u(r) = ? 1ε 0 ρ(r), r ∈ V uvextendsinglevextendsingleΣ = f(Σ) a176 a47u(r) a26a56 a74u(r)a189ρ(r), f(Σ)a73a143G(r;rprime)a32a33 a21 a51a27 ? a192a236a237a65a17 Greena238 a133 a118 a89(a29a57a239Green a118 a89) integraldisplayintegraldisplayintegraldisplay V bracketleftBig u(r)?2v(r)?v(r)?2u(r) bracketrightBig dr = integraldisplayintegraldisplay Σ bracketleftBig u?v ?v?u bracketrightBig ·dΣ, a137 a203f(r) ≡ f(x,y,z), dr = dxdydz, Σ a221V a176 a194a171a126 a26 a12a245a240a205 a187a187 a168a191a241 a183 a139a27 Wu Chong-shi a90a91a92a93a94 Green a232a228(a95) a2337a234 a191 a187 a17a156G(r;rprime) a242u(r)a243a66a176 a191 a63 a179 a44a244 a73 u(r) a242G(r;rprime)a26a145a245a26a246a164a130a171 V a172 a218 a179 a26integraldisplayintegraldisplayintegraldisplay V bracketleftbigu(r)?2G(r;rprime)?G(r;rprime)?2u(r)bracketrightbigdr = ? 1 ε0 integraldisplayintegraldisplayintegraldisplay V bracketleftbigu(r)δ(r?rprime)?G(r;rprime)ρ(r)bracketrightbigdr = ? 1ε 0 bracketleftBigg u(rprime)? integraldisplayintegraldisplayintegraldisplay V G(r;rprime)ρ(r)dr bracketrightBigg . a201a202Green a118 a89 a26 a35a73a156a169a89a247a248 a176 a147 a218 a179a58 a183 a126 a218 a179 integraldisplayintegraldisplay Σ bracketleftbigu(r)?G(r;rprime)?G(r;rprime)?u(r)bracketrightbig·dΣa27 a249 a62a250a251a60a157 a206 a26a224 a173 u(rprime) = integraldisplayintegraldisplayintegraldisplay V G(r;rprime)ρ(r)dr?ε0 integraldisplayintegraldisplay Σ bracketleftbigu(r)?G(r;rprime)?G(r;rprime)?u(r)bracketrightbig·dΣ. a164a169a126 a176 a126 a218 a179 a203a26 ? a238 a198 a251u(r)a164a194a171a126Σ a169a176 a192a204a64a194a171a195a196 a20a21 a26 a221a252a181 a176 a132 G(r;rprime)a35a64a205a47a207a208a253a21 a26a254 a244a18 a176a255 a190 ?G(r;rprime)a143a137a164a194a171a126a169 a176 a192a204 a193 a193a83a35a253a132 ? a238 a133 a251 a203a26?u(r) a164a194a171a126a169 a176 a192a204a0a181 a26 a39a73 a26 a183 a183a223a1 a67 a74u(r)a189ρ(r), f(Σ)a73a143G(r;rprime)a32a33 a21 a51 a26 a2a3a194G(r;rprime)a146a169a4a5a194a171a195a196 G(r;rprime)vextendsinglevextendsingleΣ = 0. a195a221 a26a6 a60 a224a240a241 u(rprime) = integraldisplayintegraldisplayintegraldisplay V G(r;rprime)ρ(r)dr?ε0 integraldisplayintegraldisplay Σ f(Σ)?G(r;rprime)vextendsinglevextendsingleΣ ·dΣ, a29a30a74r a242rprime a194a115a198a16a26 u(r) = integraldisplayintegraldisplayintegraldisplay V prime G(rprime;r)ρ(rprime)drprime ?ε0 integraldisplayintegraldisplay Σprime f(Σprime)?primeG(rprime;r)vextendsinglevextendsingleΣprime ·dΣprime = integraldisplayintegraldisplayintegraldisplay V prime G(rprime;r)ρ(rprime)drprime ?ε0 integraldisplayintegraldisplay Σprime f(Σprime) ?G(r prime;r) ?nprime vextendsinglevextendsingle vextendsinglevextendsingle Σprime dΣprime, a137 a203 a176?prime a242?/?nprime a32a33a194a23a24a182rprime a36a112a26Vprime a242Σprime a110a221a150a51a176a130a171a170a7a242a18a176a194a171a126a26a222a220a62 a221a74a18a97 a176 a172a173a24 a182a8 a72a183rprime a27 a9 a10 ? G(r;rprime)a164r = rprime a242a220a199a200 a26a201a11 a220a1a26a189 Green a118 a89a132a169a126 a240a241 a176a152a153 a221a12a139a87a13 ? a183 a183a14a15 a45 a198a16a17 a26 a35a73a156 G(r;rprime)a39 a243a66a176 a191 a63a18a8a183 ?2Gn(r;rprime) = ? 1ε 0 δn(r?rprime), r,rprime ∈ V. a19a248 a176a177 a178a189a190a191a192 δ n(r?rprime) a221 a66a67a68a176a199a200 a191a192 a26 a164rprime a20a21 a198a205a22a23a24a25a26a27a28 0 a29 a30 a31a32a33a281 a34a35a36a37a38n →∞a39δn(r?rprime) → δ(r?rprime)a37 Wu Chong-shi §28.2 Greena40a41a42a43a44 a458a46 ? a47a48a49a50a51a52a53 Greena54a55a37 ? a56a57a58a59a60a61a62a29a63a64a65a66 n →∞a37 ? a67a68δ a69a70a60a71a72a73a73a49a74a75a76a77a47a78a79a80a81a82a29a73a73a49a74a75a50a51a83δ a69a70a38a84a85a86a69a70a87 a72a88a37 ? a89a90a29a58a59a91a92a60a93a94a95a96a97a60a29a95a98a99a60a37 ? a100a101a102a78a96a97a60a61a62a95a83a103 a32a104a105 a74a60 rprime a103a60a106 a21a107 a77a102a34a108a109a110a29a74a47a34a111a60a112a113a114a115 a116 a52a53 Greena54a55(a117a118a119a120a29a121a74a60a122a123a59a124a125a126a87a60 Σ a127a101a29a128a129a74rprime a103a72a60a123a59) a29 a63a64a65a66a47a34a108a109a110a130a75 0a37 a51a58a131a81a132 a32a133 a60a134a135a67a68a125 Poissona136a82a74a137a102a138a122a123a139a140a141 (a142a143 Poissona136a82 a60a137a102a122a144a145a146) a60Greena69a70a37 a142a147a127a29 a105a148Green a69a70a49a95a35a36a103 a32a104 a74a149a150a122a123a139a140a141a60 a32a151 a37 a152 a75a153a154a138a155a60a122a123a139a140a29a126a156a58a157a50a51a138a158a159a160a161a37 a162 a70a163a58a164a29 a27a165 a39a113(a166a167a145a146)a60a168a169a170a136a82(Laplacea136a82a29Poissona136a82a29Helmholtza136 a82······)a74a102a167a122a123a139a140a141a60 Greena69a70a49a50a51a167a171 a28 a102a34a172a173a60a167a174a145a146a60a174a175 ? a176a177a178a179a180a181a182a183a184a185a176a177a186a187a29a188a189a190a191a192a193a194a195 δ a196a197(a198a199)a200 ? a201a202a203a204a185a191a192a205a206a207a208a37 a209 a95a29a74a210a211a172a173a212a213a141a29a47a48a167a171a60 Greena69a70a214a215a50a216a217a174a37a135a218 a152 a75a58a59a60 Poissona136a82a167 a174a145a146a29a219a122a123a139a140a220 a28 ?u(r) ?n vextendsinglevextendsingle vextendsingle Σ = f(Σ)a29a156a221a222a58a59a60a160a161a29Greena69a70G(r;rprime)a74a122a123a59a58 a117a118a223a224a149a150a60a137a225a138a122a123a139a140 ?G(r;rprime) ?n vextendsinglevextendsingle vextendsingle Σ = 0. (#) a74Greena54a55 a116 a66u(r) = 1, v(r) = G(r;rprime)a29a52a226a129integraldisplayintegraldisplayintegraldisplay V ?2G(r;rprime)dr = integraldisplayintegraldisplay Σ ?G(r;rprime)·dΣ = integraldisplayintegraldisplay Σ ?G(r;rprime) ?n dΣ, a227 a136a82a110a170a29a49a91a92 integraldisplayintegraldisplayintegraldisplay V ?2G(r;rprime)dr = ? 1ε 0 . a47a48a29Greena69a70G(r;rprime)a74a122a123a59a58a60a59a110a170a117a118a223a224 (Gaussa167a88)integraldisplayintegraldisplay Σ ?G(r;rprime) ?n dΣ = ? 1 ε0 negationslash= 0. a26 a63a228a122a123a139a140(#)a229a230a37a47a164 a25 a29a74a149a150a60a137a225a138a122a123a139a140(#)a141a29a136a82a102a167a217a174a29 a231a232a233 a164a29 a47a48a60Greena69a70a102a167 a27a234 a74a37a74a47a78a212a213a141a29a235a236a67a237a238a171a60 Greena69a70a37