Wu Chong-shi Wu Chong-shi a0a1a2a3 a4a5a6a7a8a9a10a4a5a11a12 a131 a14 a15a16a17a18 a19a20a21a22a23a24a25a19a20a26a27 a28a29a30a31a32a33a34a35a36a37a38a30a31a29a39a40a41a42a43a44a45a46a29a47a48a49a46a29a50a51a52a53a54a55a56a57a32a33a34a58 a59a60a61a62a63a64a58a65a54a66a57a32a33a47a56a57a66a57a32a33a67a36a56a57a32a33a68a69a70a34 ? a71a72a73 a67a74a75 a73a76a77 a54 Laplacea32a33a78Poissona32a33 ? a79 a54a80a81a51 a76a77 a54 a79a82 a32a33 ? a83 a80a84a85a86a67a87a88a85a86a50a54 a83 a80a84a32a33 ? a89a90a91a92 a75a29a50a54 Navier–Stockesa32a33a93a67 Eulera32a33a93 ? a94a95a72a96a97a98a82a99a100 a54 Maxwella32a33a93 ? a101a102 a56a103a30 a92a98a82a104a105a106a107 a54 Schr¨odingera32a33a67Diraca32a33 ? a108a109 a75a29a50a54 Saint-Venanta32a33a93 a110a110a68a111a112a32a33(a93) a113a114a115a116a117a109 a55a56a57a32a33 (a93)a68a51a118a34 a105a119 a33a120a121a50a122a123a124a125a126a127a54 a115a116a117 a109 a55a56a57a32a33a68 a105a128 a38a129a112a30a31a85a86a84a130a129a112a126a127a54 a115a116a117a109 a55a56a57a32a33a68a118a131a122a123a111a112a32a33a54a129a132 a109a92 a39 a133a134a68 Wu Chong-shi §13.1 a135a136a137a138a139 a8a9 a132 a14 §13.1 a140a141a142a143a144a145a146 a147a148a149a150a151a152a153a154a155a156a157a34a158a159a160a161a162a163a164a34a165a166a167a168a169a170a171a172a173a34a174a157a175a176a148a149a160a177a178a179 a180a181a182a68a183a184a157a154a185a181a182a170a186a68 a187a188a189a190a191a192a193a194 x a195 a34a196a197a198a199a200a201a194 x = 0 a202x = la68 a203u(x,t) a204 a200a201a194xa189a188a205a206a199a207t a208a209 a189( a210a211)a192a212a68a207a188a205a213a214a215a216a194 dxa189a206a217a218(a188 a219)a68a188a219a189a188a216a220a221a217a34a222a223a224a225a222a226a227a228a229 a204a230 a199a68 a231a232a188a219a233a234a235a227a207a236a237a198a199 x a238x + dxa239 a233a240a241a234a189a242a243a68 a24413.1 a245a246a247a248a249 tanθ1 = parenleftbigg?u ?x parenrightbigg x , tanθ2 = parenleftbigg?u ?x parenrightbigg x+dx a250 a102a251a114 a150a151a152a153a54a34a252a253a254a255a0a1a2a75 a3a75T a54 a101a4 a34a5a6a58a134a1a2a75a68a7 a59a34a8a9a10a11a75a54 a101a4 a68 a12a13a14 (T sinθ)x+dx ?(T sinθ)x = dm ? 2u ?t2 , (T cosθ)x+dx ?(T cosθ)x = 0. a180a181a182a15a16a235 x + dx a63 x a17a18a19a20 a129a59a21a22a1a23a24a25a26 u(x + dx,t) ?u(x,t) a34a63 dx a27a28 a114 a129a29a30a31a34a32 |?u/?x| lessmuch 1. a207a217a33a34a35a36a37a34 sinθ ≈ tanθ = ?u?x parenleftbigga38a39a40?u ?x a189a41a42a43 parenrightbigg , cosθ ≈ 1 parenleftbigga38a39a40?u ?x a189a44a42a43 parenrightbigg . Wu Chong-shi a0a1a2a3 a4a5a6a7a8a9a10a4a5a11a12 a133 a14 a45a46a34a47a14 (T)x+dx ?(T)x = 0 a48 (T)x+dx = (T)x, a49a50T a51a52xa53a54 a34a157a55a56a57a154a58a59a60a61a68a224 a204 a34 ρdx? 2u ?t2 = T bracketleftBiggparenleftbigg ?u ?x parenrightbigg x+dx ? parenleftbigg?u ?x parenrightbigg x bracketrightBigg = T ? 2u ?x2dx, a48 ρ? 2u ?t2 ?T ?2u ?x2 = 0, a62a63ρ a204 a188a189a64a65a66 ( a67 a192a216a66a189 a230a68) a68a69a70 a = radicalBigg T ρ , a71a72a73a225a222a74a229 ?2u ?t2 ?a 2?2u ?x2 = 0. aa47a204 a188a189a33a34a75a76a77a66a68 a78a79a118a80a81a235a82a30a83 a82a84a85a86 a34a58a59T a87ta88a89 a68a111 a114 a250 a102a251a90 a54a91a92 ds?dx = radicalbig du2 + dx2 ?dx = ? ? radicalBigg 1 + parenleftbigg?u ?x parenrightbigg2 ?1 ? ?dx = O parenleftbiggparenleftbigg?u ?x parenrightbigg2 parenrightbigg , a51a118a34a82a93a94a255?u/?xa54a129a95a96(a32a30a83 a82a84a85)a54a97a98a86 a34 a251a90 a54a92a99a100a101a59 a19a99a100 a68 a250a64a34a102a103Hooke a104a107 a34T a60a100a101a59 a19a99a100 a68 a105a106a107a108 a109a80a81a110a34 T a60a100a101x a99a100 a34a51a118T a111 a148a149a112a113a68 a114a115a188a207 a210a211(a48 a192a212ua189a116 a211)a205a117a233a240a118a234a189a242a243a34a203a67 a192a216a66a119a233a189a118a234a194 f a34a71a120a121 a122a123a189a124a125a34a14 ρdx? 2u ?t2 = T ?2u ?x2dx + fdx. a12a13a34 ?2u ?t2 ?a 2?2u ?x2 = f ρ, a62a63a189a126a127a128a43 f/ρ a204a67 a192 a230a68 a119a233a189a118a234a68 Wu Chong-shi §13.2 a129a136a130a138a139 a8a9 a134 a14 §13.2 a131a141a132a143a144a145a146 a133a134a148a155a156a135a136a34a158a136a137a170a138a179a180a181a182a68a139a140a175a141a161a136a137a170a138a154a142a148a143a177a178a56a57a154a181a182a144a145 (a146a147a148)a150a151a60a176a68 star a114a14913.2a34a187a150a216a72a211 a194x a195 a72 a211 a34a151a152a224a150a216a72 a211 a189 a153a154a123a155a243a227a189a190a191a192a193 xa201a156a68 star a207a157a206a208a209ta34a13a154a123a158a159a224a190a191a192a193a189a192a212a194 u(x,t)a68 star a207a150a63a213a214a215a206a217a218 (x, x + dx)a34a231a232a233a234a235 ? a160a161 a154a123 xa34a233a240a162a163a234 P(x,t)S a189a242a243 ? a160a161 a154a123 x + dxa233a240a162a163a234P(x + dx,t)S a189a242a243 P(x,t)a194a67 a192a123a164a119a233a189a162a163a234 ( a165 a234)a34a166xa72 a211 a194a116a68 a24413.2 a167a246a168a248a249 a169a170a171a169a172 a12a13a34a173a174Newton a175 a44a69a176a34a47a177a240 dm? 2u ?t2 = [P(x + dx,t)?P(x,t)]S. a178a150a189a65a66a194 ρa34a71dm = ρdx·S a34 ρ? 2u ?t2 = ?P ?x . a114a115 a38a39 a151a152a150a216a72 a211 a189a179a180a34a173a174 Hookea69a176a34 a165 a234P a202a165 a180?u/?xa229a116a181 P = E?u ?x, a181a182a183a184E a185 a194a150a189Young a186a68 a34a227 a204 a206a237a187 a230a188 a184a68a45a46a34a47a177a240 a40 a150a189a189a33a34a72a73 ?2u ?t2 ?a 2?2u ?x2 = 0, a62a63 a = radicalBigg E ρ . a150a189a189a33a34 a202 a188a189 a210 a33a34a190a191a192a193a194a195a158a196a34a197a227a198a199a220a189a200a201a231a72a73a189a179a202a203a194a195a206a46a68 a45a206a204a72a73a205 a185 a194a206a182a170a186a68 a207a206a208a209a34a207a41a210a211a212a63a189a213a34a72a73 a204 ?2u ?t2 ?a 2?2u = 0, a62a63?2 ≡ ? 2 ?x2 + ?2 ?y2 + ?2 ?z2 a185 a194Laplace a214a215 a34 ?2 = ?·? a48 ?2u = ?·(?u). Wu Chong-shi a0a1a2a3 a4a5a6a7a8a9a10a4a5a11a12 a135 a14 §13.3 a216 a217 a218 a145 a146 a219a84 a83 a80a84a32a33a51 a4 a54a28a29a32a134a67a220a106a54a221a222a27a7a68a100a7a25a223a82a224a225a226a54a30a31 a106a107 a100 a7a68a111a227 a4 a255a54 a114a83 a29a32a106a54 a17 a29 a104a105a106a107 a34a32 a228a229a230a231a232a233 a67 a234a235a236a154 Fourier a232a233a68 a234a235a236a154 Fourier a232a233 a203a14a206a237a238a239a240 a230 a68a187a69a206a69a200a201a183a34a192a243 u(x,y,z,t) a241a242 a240 a230a243 a211a212a200a201a194(x,y,z)a189a206a199a207t a208a209 a189a244a66a68a178a166xa72 a211 a14a206a69a189a244a66a245a34a207xa72 a211a246 a47a206a69a14a247 a68 a189a75a248a68a249a250a251a205a228a34a252 a147a253a254a255a0a1 a141a161xa170a138a154a252 a147 a177a2a154a234a229q a87a3a4 a154a5 a254a53a54a6a7 a8a9a34 a48 q = ?k?u?x, q a185 a194a247a10a65a66a34 k a185 a194a125a247a11a68 k a63a91a92 a54 a92a12 a58a65a34a5a13a34a14a15a16a17a34a63a18a99u a19a20a21a22a23 a24a25a26a18a27a28a29a30a31a32a33 a34a35a36a37a38a39a40a41a42 k a43a44a45ua46a21a23 a47a48a49a50a51a52a53a54a55a56a57a58a59a60a61a62a63a64a65a66a59a60a61a67a68a69a70a35a71a57a72a73a74a63a58a61a75a63 a23 a76a77a78a79a80a81a82a83a84a85a51a52a86a87a88a35a89a84a85a51a78a90a91a81a47a92a93a89a94a95a96a35a97a98 qx = ?k?u?x, qy = ?k?u?y, qz = ?k?u?z, a99 q = ?k?u, a100a86a101a102a95a103a104q a105 a94a95a106a95?u a107a108a109a23 a110a111Fourier a112a113a114a115 a104a116a117 a112a113a118 a88a119a120a80a81a82a83a84a85a51a52a86a87a88a91a121 a23 a89a84a85a122a123a124a125a126a127a90a128a129a130a48a131 ( a132a13313.3)a35a130a90a48a92a114a134a135a48a136a137a23 Wu Chong-shi §13.3 a138a139a140a141a142 a1436a144 a13313.3 a86a87a88a91a121 a145a146 (x,y,z)a147 a52a148a130a48a131 star ?ta149a150 a122a151xa91a81a101a152a130a48a131a52a86a104 bracketleftbig(q x)x ?(qx)x+dx bracketrightbig?y?z?t =bracketleftBigparenleftbiggk?u ?x parenrightbigg x+dx ? parenleftbigg k?u?x parenrightbigg x bracketrightBig ?y?z?t =k? 2u ?x2?x?y?z?t. star ?ta149a150 a122a151y a91a81a101a152a130a48a131a52a86a104 bracketleftBig (qy)y ?(qy)y+dy bracketrightBig ?x?z?t = k? 2u ?y2?x?y?z?t, star a89?ta149a150 a122a151z a91a81a101a152a130a48a131a52a86a104 bracketleftbig(q z)z ?(qz)z+dz bracketrightbig?x?y?t = k?2u ?z2?x?y?z?t. a153a154a130a48a131a122a155a98a156a157a86a104a158a159a99a160a161a35a97a110a111 a115 a104a116a117 a112a113 a35a162a58a163a59a57a72a164a165a166a167a168a169a170a171 a172a173a174a63a64a175a74a176a177a178a59a57a72a35 k parenleftbigg?2u ?x2 + ?2u ?y2 + ?2u ?z2 parenrightbigg ?x?y?z?t = ρ?x?y?z ·c·?u. a179a180 ?u ?t ? k ρc? 2u = 0, a156a51ρ a181 a84a85a52a102a95a35 c a181a109 a86a182 a23 Wu Chong-shi a183a184a185a186 a187a188a189a190 a141a142a191 a187a188a192a193 a1437a144 a194κ = k/ρca35a97a98 ?u ?t ?κ? 2u = 0, a156a51κ a195a196a197a198a199 a35a99a94a95a87a88 a199a23 a153a154a89a84a85a122a98a86a104a200a201( a202 a153a35a98a203a204a205a206a207a201a35a99a208a98a209a101a35······)a35a210a145 a149a150 a122a210a145a131 a211a84a85a51a200a201a52a86a104 a196 F(x,y,z,t) a35a97a98 k?2u?x?y?z?t + F(x,y,z,t)?x?y?z?t = ρ?x?y?z ·c·?u, ?u ?t ?κ? 2u = 1 ρcF(x,y,z,t) = f(x,y,z,t). a153a154a84a85a212a119a120a35a97a88a86 a199 k a105a134a135(x,y,z) a98a213 a23a214a149 a35a86a87a88a91a121a215a216 a196 ρc?u?t ??·(k?u) = F(x,y,z,t). a57a217a218a60a219a59a220a221a222a223a224 a194j = ρcua35 a195a196 a86a101( a225 a95)a35a97 ?j ?t +?·q = F(x,y,z,t). a226a90a91a121a227 a195a196a228a229a230 a60a219 a23 a153a154 a181 a80a81a231a83a84a85a35a97 Fourier a112a113 a206a232a233 a107 q = ?K·?u. a234a235a28K a236a237a2383×3a239a240 a35a241a242?u a243a239a240a244a245 a28a246a36a247 a244a23 a248a206a249a35a86a87a88a91a121a216 a196 ρc?u?t ??·(K ·?u) = F(x,y,z,t). a250a251a252a253a254a52a255a95a0a35a94a95a52a1a2 a181 a251a252a86a253a254a3a4a121a95a52a205a5 a23 a251a252a86a253a254a52a212a128a6a35a208 a7a8a9a10a11 a115 a104a35a89a12a13a47a215a14a15 a196 a86a104a52a87a16 a23 a17a180a18a19a35a153a154a84a85a122a93a89a20a21a212a119a120a22a23a35 a202 a153a24a85a25a95a52a212a119a120a35a208a7a251a252a52a253a254a26a27a207a201a24a85a52a10a11a35a226a89a12a13a47a215a14a15 a196 a251a252a52 a197a198a23 a226a21a89a28a13a29a30a47a52a248a31a83a35a215a32 a112a33a197a198 a91a121 a114 a86a87a88a91a121a98a248a82a52a34a50a35 ?u ?t ?D? 2u = f(x,y,z,t), a156a51a52u(x,y,z,t) a35 a14a251a252a25a95a35 D a181a197a198a199 a35f(x,y,z,t)a97 a181 a210a145 a149a150 a122a89a210a145a131a211a51a36a21a251 a252a52a200 a199a23 Wu Chong-shi §13.4 a37a38a39a40 a1438a144 §13.4 a41 a42 a43 a44 a45a46a63a64a47a48 a89a127 a112a49a50a51 a35a24a131a52a94a95a52a53a54 a112a55 a100a212a56 a149a150 a216a203 a149 a35a97a94a95a251a57a58a59 Poissona91a121 ?2u = ?fκ. a60a20 a181 a35a153a154 f = 0a35a97a98Laplacea91a121a35 ?2u = 0. a226a61a21a91a121a62a233a52 a181 a52a53a54a117a52a24a30a22a63 a23 a64a65a66a59a65a67 a153a154a68a254a91a121a51u a26a212a56 a149a150 a216a203a35 a202 a153a69a209a70a52a209a71 u(x,y,z) a35a26a58a59 Poissona91a121 ?2u = ? ρε 0 , a156a51ρ a181 a209a72a102a95a35ε 0 a195a196a73a74 a209a182 a199 (a73a74 a84a209a227a75) a23 a153a154a209a72a102a95 ρ ≡ 0a35a97a69a209a70a52a209a71 a58a59Laplacea91a121 ?2u(x,y,z) = 0. a76a77a78 a153a154a68a254a91a121 ?2u ?t2 ?a 2?2u = 0 a51a35u(x,y,z,t)a56 a149a150a79a80 a249a216a203a35a81 a199a196 ω a35 u(x,y,z,t) = v(x,y,z)e?iωt, a97v(x,y,z)a58a59Helmholtza91a121 ?2v(x,y,z) + k2v(x,y,z) = 0, a156a51k = ω/a a195a196 a68a75 a23 a180a47a52a82a21a83a84a52a85a28a251a91a121a35a250a24a30a47a0a35a98 stara86a87a88a89a90a91 a28 a88a89a92a91 stara86a87a93a94a90a91 a28a95a96a97 a92a91 stara86a87a98a99a100a101 a28 Poisson a92a91 a242Laplace a92a91 a250a75a204a47a0a35a226a26 a108a102 a248a206a249a251 a196 a78a103a104 stara88a89a92a91 a35a105a106a107a108a109a110a111a112a113 a92a91 stara95a96a97a92a91 a35a105a106a107a108a109a110a114a115a113 a92a91 star Poissona92a91 a242Laplace a92a91 a35a105a106a107a108a109a110a116a117a113 a92a91 a226a78a103a91a121a52a118a119a120a121a35a122 a181 a84a123a121a52a51a124a125a126 a23