多变量函数的微分学

� 8 � ������������ � �� §8.1 �������� R2 ������� §8.1.1 ����������ff�fi�fl�ffi��! "�#�$�%�&�' M1(x1, y1),M2(x2, y2) (*),+�-�.�/ ρ(M1,M2) 0�1�243�5�6 ρ(M1,M2) = M1M2 = √ (x1 − x2)2 + (y1 − y2)2 7�8�9�:�;�<�= ρ(M1,M3)− ρ(M3,M2) ≤ ρ(M1,M2) ≤ ρ(M1,M3) + ρ(M3,M2). > r > 0, ? "@#@$@A M0 BDCFE@G r B@H@I + C +DJFK ( L '@M {M | ρ(M,M0) < r}) B M0 + r N�O G4P�Q O(M0, r). > I R J S %�&*T ) GVU!W : '!M D = {(x, y)| x ∈ I, y ∈ J} P�Q I × J , ? B�X &�Y�Z*T ),2 > E ⊂ R2. [�\�]�^�_�` R, a E ⊂ O(O,R) (O B�b�c�d ' ), U ? E S X & 6�e M G4f�U ? B�g e M 2 > E S "�#�$�h!i!'!M G sup{ρ(M ′,M ′′)| M ′,M ′′ ∈ E} ? B E +�j I�G4P Q diamE. diamE = +∞ k G E l�S g e M 2 > E ⊂ R2, R2 m ;�n�o E +�p�q ' + M�r G ? B E +�s M G4P�Q Ec. 3 5�6 G (Ec)c = E. "�#�$ + '�t�A!u!v!w!x E +�y�z�{ Q 8�| 2 1◦ M B E +*J '�} 2◦ M B E +�~ '�} 3◦ M B E +��e ' 2 E +�p�q��e ' + M!r ? B E +��e G P�Q ∂E. E +�p�q*J '�€ Q + M�r* E +�‚ G4P�Q E◦. > M ∈ R2, [�\�]�^�_�` r a O(M, r) ∩E = {M}( ƒ '�M ), U ? M S E + „�… ' 243�5 E + „�… '�† S E +��e ' 2 > M ∈ R2, [‡\‡ˆ‡‰‡Ł r > 0, O−(M, r) mŒ‹ 6 E m + ' (L O−(M, r)∩E 6= φ), U ? M B E +� '�Ž���' 2 ‘‡’ 8.1.1 > {Mn} S "‡#‡'‡“ 2”[�6 M0 ∈ R 2, a lim ρ(Mn,M0) = 0, U ? '�“ {Mn} B�•�– '�“ 2 M0 ? B '�“ + � 2 1 §8.1.2 —���˜*™,� > E ⊂ R2, š E m + ' ‹ S E +*J ' G4U ? E S X &�›�M 2 %�&�›�M +�œ M R� M!ž S ›!M 2 š Ec S ›�M G4U ? E S*Ÿ M 2 %�& Ÿ M +�œ M R� M!  ‹!¡ S*Ÿ M 2 ‘�¢ 8.1.1 E ⊂ R2 S ›�M +�£�{ †�¤�¥�¦ S ∂E ∩E = φ. ‘�¢ 8.1.2 E ⊂ R2 S*Ÿ M +�£�{ †�¤�¥�¦ S ∂E ⊂ E. ‘�¢ 8.1.3 E ⊂ R2 S*Ÿ M +�£�{ †�¤�¥�¦ S E §�¨�©�p�K� ' 2 §8.1.3 ª�«�� > 6 %�&�¬�­�® ` x = x(t), y = y(t), (α ≤ t ≤ β), (8.1.1) ? '�M L = {(x(t), y(t))| α ≤ t ≤ β} B‡X ¥‡"‡#°¯Œ± 2 (8.1.1) ? B L +‡²‡`‡³‡´‡2 X & ƒ '�M� ‡µ Q�X ¥ (¶‡·‡¸‡+ ) ¯Œ± 2F[‡\ x′(t), y′(t) ‹ ^ [α, β] ¬‡­ G œ‡¹ x′(t), y′(t) ;*º k B‡»‡GŒU ? L S X ¥‡¼‡½°¯Œ± 2¾[�\�ˆ‡‰�Ł α ≤ t1 < t2 < β, ‹ 6 (x(t1), y(t1)) 6= (x(t2), y(t2)), U ? L S X ¥�¿�À*¯,±�Ž Jordan ¯,± 24[�\ M(α) = (x(α), y(α)) = M(β) = (x(β), y(β)), U ? L S X ¥ Ÿ ¯,± 2 ‘!’ 8.1.2 > E ⊂ R2. [!\!‰!Á P,Q ∈ E, ‹ ]!^ "!#¯ñ L = {M(t) = (x(t), y(t))| α ≤ t ≤ β} ⊂ E, ¹ P = M(α) = (x(α), y(α)), Q = M(β) = (x(β), y(β)), U ? E S X &�¬�Ä�M 2 hÅi!¬!Äś!M ? B T O G T O x!w +!!eÅ+!œ M ? B Ÿ O�2 ‘‡¢ 8.1.4 > E S X &‡¬‡Ä‡M G [�\!6 E + Y‡&‡h‡i‡Æ�M A,B a A∩B = φ,A ∪B = E, U † 6 E m X ' M0, w S A,B +�Ç�È��e ' 2 É!Ê 8.1.1 > D S R2 m + T O G [!\!6 D m Y!&!›!M A,B a A ∩ B = φ,A ∪ B = D, U A = φ Ž B = φ. Ë�Ì�( G4X &*T O ;!Í { Q %!&!Î ;!Ï !+ h i�›�M 2 "�#*T O�{ Q À�¬�Ä +!R�Ð ¬!Ä + % | 2 2 [‡\ T O D JŒ‰ X ¥‡¿‡À Ÿ ¯,± +*J,K ¡ ^ D J GÑU ? D S À‡¬‡Ä +‡2ÓÒ ( G l�? D S�Ð ¬�Ä +�2 ‘!’ 8.1.3 1) > D S R2 m +!6!e T O!24[!\ Dc SÂÔ n(n ≥ 1) &!Î ; Ï �+ ¬�Ä Ÿ M�€ Q + G l!? D S X & n ¬�Ä O G n ,Õ D + ¬�Ä `�2 n = 1 k G ? D B À�¬�Ä O G n ≥ 2 k�? D B Ð (Ö ) ¬�Ä O�2 2) > D S R2 m + g e T O G [�\�]!^ R0 > 0, a‡× R > R0 k D ∩O(O,R) S n ¬�Ä O G4U ? D S n ¬�Ä O�2 §8.1.4 R2 ��Ø�Ù�Ú ‘‡¢ 8.1.5 > E S X & Ÿ M GÛU E m ‰‡Ü •‡– '‡“ + ��ž S E m + ' 2 ‘‡’ 8.1.4 > {Mn} S "‡#‡'‡“ 2F[�\�ˆ!‰�Á ε > 0, ]‡^ N ∈ N, a‡ˆ‡‰‡Ü m,n > N 6 ρ(Mn,Mm) < ε, U ? {Mn} B�Ý�Þ “�Ž Cauchy “ 2 ‘!¢ 8.1.6 (R2 ßÃà!á!â ) "!#!'!“ {Mn(xn, yn)} •!– +!£!{ †Å¤Å¥ã¦ S {Mn} B�Ý�Þ “ 2 ‘�¢ 8.1.7 (ä*å â ) 6�e '�“ {Mn(xn, yn)} † 6 •�– Æ�“ 2 ‘@¢ 8.1.8 ( æFç@è@é@ê ) > {En} S X “ 6@e°Ÿ M Gìë‡í‡î 1◦ En+1 ⊂ En(n = 1, 2, · · ·); 2◦ lim diamEn = 0. U 6�ï X ' M0 n�o�ð 6 En. ‘�¢ 8.1.9 [�\ E S "�#�$�h!i 6!eŸ M G E′ S "�#�$�h�i Ÿ M 2 U † 6 M0 ∈ E R M ′ 0 ∈ E ′ a ρ(M0,M ′ 0) = ρ(E,E ′). É@Ê 8.1.2 [@\ E S h@i 6@eDŸ M G E′ S h@i Ÿ M G E∩E′ = φ, U ρ(E,E ′) > 0. §8.2 ñ�ò���ó�ô�õ�� §8.2.1 ö�÷�ø4ù�ú�û�ü�ø4ý!þ!ß�û!ü��!�! > 6 %‡&‡M�r X,Y 7 X &�� U f . ‰‡Ł‡Á�� X m X &���� x, ‹ t‡A‡u‡v�� U f ��� Y m ï X ��� y( P‡Q y = f(x)) x x ˆ�� GFU ? f S X � Y + X &�� G4P�Q f : X → Y. X ? B � + ����O!2Œ[�\ y = f(x), U ? y S x ^ f ‡+�� G x S y + d � ( ��Ł î y + d � t�A ; � X & ). M�r f(X) = {f(x)| x ∈ X} �� + ��O�2 3�5 f(X) ⊂ Y . 3 [�\ � S X�X ˆ � G,U!X & � �!6 X & d ��2���k t�A ��� X & f(X) � X + � î f−1 : f(X) → X. w + � � U f−1 S G ‰�Á y ∈ f(X), † 6�ï X x ∈ X a y = f(x). � & x l�S y ^ f−1 �+ ��24L�S��Å[!\ y = f(x), U x = f−1(y). ��k�? f B t � � G f−1 ? B f + � � 2 w Ï × o X �!® `!+!Ò ® `!2 '�M E = {(x, y, f(x, y))| (x, y) ∈ D} S X � i ) ¯Ã# G ? B ® ` z = f(x, y) +�����2 Y ��® ` z = f(x, y) ff�0�+ ¯,# ? B 3 = ¯Ã# 2 > D ⊂ Rn, f : D → Rm  �t�A fi Q 6 fl�+ m & n ��® ` f :   y1 = f1(x1, · · · , xn), · · · · · · · · · · · · · · · ym = fm(x1, · · · , xn). (x1, · · · , xn) ∈ D f ffi D m + ' (x1, · · · , xn) � Q Rm m + ' (y1, · · · , ym). [‡\��� �ffi (y1, · · · , ym) ! Q R m m +#"%$ r. &�' G l‡? � f S����‡^ D $ +�"�$ � ® `�2 w ffi D m + ' � Q Rm m +�"�$�2 P�Q r = r(x1, · · · , xn) = (y1, · · · , ym) = (f1(x1, · · · , xn), · · · , fm(x1, · · · , xn)), © m + yi = fi(x1, · · · , xn)(i = 1, · · · ,m) ? B "�$ � ® `�+ ( i & { $ ® `�2 §8.2.2 ù�ú�û�ü�� ) * ‘‡’ 8.2.1 > f S����‡^‡^ "�#�'�M D $ + Y��‡® ` G M0 S D +‡ ' G [�\�]�^ +�` a, a�ˆ�‰�Ü ε > 0, ]�^ δ > 0, × 0 < ρ(M,M0) < δ ¹ M ∈ D k G 6 |f(M)− a| < ε, l ��× M , o M0 k f(M) A a B � G4P�Q lim M→M0 f(M) = a. 4 §8.2.3 ù�ú�û�ü���ª -�Ú ‘�’ 8.2.2 > f ^ "�#�'�M D 6 � � G M0 ∈ D. [�\�ˆ�‰�Ł ε > 0, ]�^ δ > 0, × M ∈ D ¹ ρ(M,M0) < δ k ‹ 6 |f(M)− f(M0)| < ε, l‡? f ^ M0 ¬‡­ 2 ‘�’ 8.2.3 [�\ f ^ D + . X &�'�¬�­ G l!? f ^ D ¬�­ 2 ‘�¢ 8.2.1 (/ 0 1�ê ) > f(M) ^ ¬�Ä�M E m ¬�­ G M1,M2 ∈ E. U f ^ E m�2 � f(M1) R f(M2) (*),+ ð 6 ��2 ‘‡¢ 8.2.2 D S R2 m +‡6‡e°Ÿ M G f ^ D $‡¬‡­ GÑU f ^ D $ 2 ��3�4 ��R 3 5 ��2 ‘Å’ 8.2.4 > f ^ "Å#Å'ÅM D 66�6�Å24[Å\ÅˆÅ‰ÅŁ ε > 0, 6 δ > 0, × M,M ′ ∈ D ¹ ρ(M,M ′) < δ k ‹ 6 |f(M)− f(M ′)| < ε. l�? f ^ D X 7 ¬�­ 2 ‘�¢ 8.2.3 > D S R2 m +�6�e*Ÿ M G f ^ D ¬�­ G4U f † ^ D X 7 ¬ ­ 2 §8.2.4 ö�÷�� ) * 8�ª -�Ú > D ⊂ Rn, f : D → Rm. 9 q :�Ì G > 6 f :   y1 = f1(x1, · · · , xn), · · · · · · · · · · · · ym = fm(x1, · · · , xn), (x1, · · · , xn) ∈ D Ž r = r(x1, · · · , xn) = (f1(x1, · · · , xn), · · · , fm(x1, · · · , xn)), (x1, · · · , xn) ∈ D. f( Ž r) ^ ' M0 6 � (a1, · · · , am) l�S ; fi(i = 1, · · · ,m) ^ M0 6 � ai. f( Ž r) ¬�­ G l�S ; fi(i = 1, · · · ,m) ¬�­ 2 f( Ž r) X 7 ¬�­ G l�S ; fi(i = 1, · · · ,m) X 7 ¬�­ 2 5 §8.3 ������������ � =<=>� =? §8.3.1 ù @�þ�û�ü�� A B ‘@’ 8.3.1 > z = f(x, y) ^ M0(x0, y0) +@N@O m 6C�C� G P ρ = √ ∆x2 + ∆y2, [�\�]�^ D�` A,B, a�× ρ→ 0 k�6 f(x0 + ∆x, y0 + ∆y)− f(x0, y0) = A∆x+B∆y + o(ρ), U ? z = f(x, y) ^ M0 t E G œ�? A∆x+B∆y B f(x, y) ^ M0 + E { G4P�Q dz = df(x0, y0) = A∆x+B∆y. ‘�¢ 8.3.1 [�\ f(x, y) ^ M0(x0, y0) t E G4U f(x, y) ^ M0 ¬�­ 2 §8.3.2 ù�ú�û�ü�� F G�ü ‘�’ 8.3.2 > z = f(x, y) ^ M0(x0, y0) +�N�O m 6 � � G [!\ lim ∆x→0 f(x0 + ∆x, y0)− f(x0, y0) ∆x ]Å^ G4U ? w B z = f(x, y) ^ M0 y o x +6H6IÅ` ( Ž H E6J ), PÅQ ∂f ∂x ∣∣∣ M0 , ∂f ∂x ∣∣∣ (x0,y0) , ∂z ∂x ∣∣∣ M0 , ∂z ∂x ∣∣∣ (x0,y0) , f ′x(M0), f ′ x(x0, y0) < 2 K L ? lim ∆y→0 f(x0, y0 + ∆y)− f(x0, y0) ∆y B f(x, y) ^ M0 y o y + H I�` ( Ž H E J ), P�Q ∂f ∂y ∣∣∣ M0 , · · · < 2 H I�`�6�M,3�+ N�Ü�Ł���2 ‘Å¢ 8.3.2 [Å\ z = f(x, y) + %Å& H6IÅ`ã^ M0(x0, y0) ‹ S ¬Å­ + GVU f(x, y) ^ M0 t E 2 §8.3.3 O P F A Q X R H E J + H E J G l ,Y R H E�J 2 K L t�A � � S T R +�H E�J 2 ‘�¢ 8.3.3 š z = f(x, y) ^�O D m 6 � � G ∂2f ∂x∂y R ∂2f ∂y∂x ‹ ¬�­ G4U ^ D m ∂2f ∂x∂y = ∂2f ∂y∂x . 6 §8.4 UWV������� � =X §8.4.1 Y Z�û�ü [ G�� \ ] ^�_ ‘‡¢ 8.4.1 > z = f(x, y) t�E G x = ϕ(r, s) R y = ψ(r, s) 6 X�R H�I‡` G U z ˆ r R s 6 H I�` G œ�6 ∂z ∂r = ∂z ∂x ∂x ∂r + ∂z ∂y ∂y ∂r , ∂z ∂s = ∂z ∂x ∂x ∂s + ∂z ∂y ∂y ∂s . X `�G > y = f(x1, x2, · · · , xn) t E G : xi = ϕi(ξ1, ξ2, · · · , ξm) (i = 1, 2, · · · , n) 6 X R H I�` G4U ∂y ∂ξj = ∂y ∂x1 ∂x1 ∂ξj + ∂y ∂x2 ∂x2 ∂ξj + · · ·+ ∂y ∂xn ∂xn ∂ξj (j = 1, 2, · · · ,m). ‘Å¢ 8.4.2 (a6bdce06fhg ) > z = f(x, y) ^ÅO D t6E Gji6k (x0, y0) R (x0 + h, y0 + k) + ± l ^ D m G4U † 6 0 < θ < 1, a ‘�¢ 8.4.3 > ^�O D m ∂f ∂x = ∂f ∂y = 0, U f(x, y) ≡ c ( D�` ). §8.4.2* Jacobian > y = f(x1, x2, · · · , xn) 6 X R H I�` G ?�"�$ ( ∂f ∂x1 , ∂f ∂x2 , · · · , ∂f ∂xn ) B y = f(x1, x2, · · · , xn)( ˆ x = (x1, x2, · · · , xn)) + Jacobian, P�Q Jf, Jxf, Jy, Jxy < 2 > 6 n � � y = (y1, y2, · · · , ym) = (f1(x1, x2, · · · , xn), f2(x1, x2, · · · , xn), · · · , fm(x1, x2, · · · , xn)) = (f1(x), f2(x), · · · , fm(x)) = f(x). 7 U ? W�m   ∂f1 ∂x1 ∂f1 ∂x2 · · · ∂f1 ∂xn ∂f2 ∂x1 ∂f2 ∂x2 · · · ∂f2 ∂xn · · · · · · · · · · · · ∂fm ∂x1 ∂fm ∂x2 · · · ∂fm ∂xn   B � y = f(x) + Jacobian, P�Q Jy Ž Jxy. n�o G Ö r�® ` p I!+ q = r U t�A�fi Q Jξy = JxyJξx. §8.4.3 s�ý G�ü Gjt u > u = f(x, y, z) ^�O D $�t E G l = cosαi + cos βj + cos γk S X &�À v " $�2jw x u = f(x, y, z) ^ l ³�" $ + y�· z�2 Á � M0(x0, y0, z0) ∈ D, { M0 : " | o l +�j ± l +�²�`�³�´ B x = x0 + t cosα, y = y0 + t cos β, z = z0 + t cos γ (t ∈ R). š ffi ® ` }�³�" l + y�· z P�Q ∂f ∂l , U ∂f ∂l = lim t→0 f(x0 + t cosα, y0 + t cos β, z0 + t cos γ)− f(x0, y0, z0) t . Ô,Ö r�® `�+ p I r U t ~ ∂f ∂l = ∂f ∂x cosα+ ∂f ∂y cos β + ∂f ∂z cos γ = Jf · l = |Jf | cos θ, (1) © m θ S Jf R l +  9 2 ? ∂f ∂l ( Ž ∂u ∂l ) B u = f(x, y, z) }�³�" l +�³�"�I�` Ž ³�" E J 2 Ô (1) t�~ G × θ = 0 L l x Jf = ( ∂f ∂x , ∂f ∂y , ∂f ∂z ) º "Œk G ³�"�I�` ∂f ∂l 2 � 3 4 ��24^ €  m G ?‚"�$ Jf = ( ∂f ∂x , ∂f ∂y , ∂f ∂z ) B u = f(x, y, z) + ƒ „ G4P�Q grad f Ž gradu. 8 p ® `�+ ƒ „�S X | … ��+ E { † µ G w�‡�ˆ�A �† µ r U 2 1◦ grad (c1u1 + c2u2) = c1gradu1 + c2grad u2, © m c1, c2 S�‰�Ł D�`�2 2◦ grad (u1u2) = u1gradu2 + u2gradu1. 3◦ grad f(u) = f ′(u)grad u. §8.4.4 ff P A B�� ‰ ] Ł @�Ú > z = f(x, y), × x, y SŒ‹y $�k dz = ∂f ∂x dx+ ∂f ∂y dy. [�\ x, y S m )�y $ î x = ϕ(r, s), y = ψ(r, s), U 6 dz = ∂z ∂r dr + ∂z ∂s ds = ( ∂z ∂x ∂x ∂r + ∂z ∂y ∂y ∂r ) dr + ( ∂z ∂x ∂x ∂s + ∂z ∂y ∂y ∂s ) ds = ∂z ∂x ( ∂x ∂r dr + ∂x ∂s ds ) + ∂z ∂y ( ∂y ∂r dr + ∂y ∂s ds ) = ∂z ∂x dx+ ∂z ∂y dy. §8.5 Ž������� � WX §8.5.1 ù�ú s   ‘ ‘ � ’�û!ü!� “�” ‘!¢ ‘�¢ 8.5.1 > T O D ⊂ R2, M0(x0, y0) ∈ D. [�\ F (x, y) ^ D m 6 � ��œ ë�í�î 1◦ F (x, y) ∈ C(1)(D); 2◦ F (x0, y0) = 0; 3 ◦ F ′y(x0, y0) 6= 0. U ]�^ M0 +�N�O I × J ⊂ D, a î 1◦ ˆ I m ‰‡Ł x, 6 J m ï X y a F (x, y) = 0; 2◦ Ô 1◦ ð�• �‡+ ® ` y = f(x) ∈ C(1)(I)( Ô,ï X – t ~�† 6 y0 = f(x0)). ‘�¢ 8.5.2 > T O D ⊂ R3, M0(x0, y0, z0) ∈ D. [�\ F (x, y, z) ^ D m 6 � � G œ ë�í�î 1◦ F (x, y, z) ∈ C(1)(D); 2◦ F (x0, y0, z0) = 0; 3 ◦ F ′z(x0, y0, z0) 6= 0. U ]�^ M0(x0, y0, z0) +�N�O I × J ×K, a î 1 ◦ ˆ I × J m ‰ X ' M(x, y), 6�ï X z ∈ K a F (x, y, z) = 0; 2◦ Ô 1◦ ð • ��+ ® ` z = f(x, y) ∈ C(1)(I × J)( Ô,ï X – t ~�† 6 z0 = f(x0, y0)). 9 š�³�´ F (x, y, z) = 0 • � — ® ` z = f(x, y), U 6 ∂z ∂x = − F ′x(x, y, z) F ′z(x, y, z) , ∂z ∂y = − F ′y(x, y, z) F ′z(x, y, z) . §8.5.2 ˜�s  ™  ‘ ‘ � ’�û!ü�™ ^ X ��+ ¥�¦ G Ô m & ƒ … + n+m � ³�´ F1(x1, x2, · · · , xn; y1, y2, · · · , ym) = 0, F2(x1, x2, · · · , xn; y1, y2, · · · , ym) = 0, · · · · · · (1) Fm(x1, x2, · · · , xn; y1, y2, · · · , ym) = 0, t�A ^ š�K • � m & n ��® ` y1 = f1(x1, x2, · · · , xn), y2 = f2(x1, x2, · · · , xn), · · · · · · (2) ym = fm(x1, x2, · · · , xn). ffi (2) ff › (1), œ � m &  <�= 2jž w �ˆ xi(i = 1, 2, · · · , n) p I G l œ � ± – ³�´ € ∂F1 ∂xi + ∂F1 ∂y1 ∂y1 ∂xi + ∂F1 ∂y2 ∂y2 ∂xi + · · ·+ ∂F1 ∂ym ∂ym ∂xi = 0, ∂F2 ∂xi + ∂F2 ∂y1 ∂y1 ∂xi + ∂F2 ∂y2 ∂y2 ∂xi + · · ·+ ∂F2 ∂ym ∂ym ∂xi = 0, · · · · · · (3) ∂Fm ∂xi + ∂Fm ∂y1 ∂y1 ∂xi + ∂Fm ∂y2 ∂y2 ∂xi + · · ·+ ∂Fm ∂ym ∂ym ∂xi = 0. 10 [�\�³�´ € +�z�` |�“ = detJyF = ∣∣∣∣∣∣∣∣∣∣∣∣∣ ∂F1 ∂y1 ∂F1 ∂y2 · · · ∂F1 ∂ym ∂F2 ∂y1 ∂F2 ∂y2 · · · ∂F2 ∂ym · · · · · · · · · · · · ∂Fm ∂y1 ∂Fm ∂y2 · · · ∂Fm ∂ym ∣∣∣∣∣∣∣∣∣∣∣∣∣ 6= 0, U Ô,³�´ (3) l t�A Ÿ�  y1, y2, · · · , ym ˆ xi(i = 1, 2, · · · , n) + H I�`�2 §8.6 ¡��W¢������� � =X §8.6.1 ff�ú�ý�þ�ß�û�ü�� A B�^ 1) i ) ¯,± > r = r(t) = (x(t), y(t), z(t)) (α ≤ t ≤ β) S X &�¬�­ + X � "�$�� ® `!2 w ffi [α, β] $ X ' t, � Q i ) m X ' M(t) = (x(t), y(t), z(t)). ' M(t) +‡p‡q‡l‡? B X ¥�i ) ¯,± L. r = r(t) �¯,± +�" I = ³�´ G w < £!o ¯,± +�²!`!³�´ L : x = x(t), y = y(t), z = z(t) (α ≤ t ≤ β). š L ¤�6Œ‹Ó ' G L�ˆ!‰!Ü α ≤ t1 < t1 < β, r(t1) 6= r(t2), U ? L B�X ¥�¿ À°¯Œ± ( Ž Jordan ¯Œ± ); š x′(t), y′(t), z′(t) ^ [α, β] ¬‡­ G ¹ ;°º k B�»�G l�? L B�X ¥�¼�½*¯,±�} š r(α) = r(β), l�? L B�X ¥ Ÿ ¯,± 2 × z(t) ≡ 0 k G L l�S Oxy "�#�$ + X ¥*¯,± 2 2) X � "�$ � ® `�+ E J > r = r(t) (α ≤ t ≤ β) S X � "�$ � ® `�24[!\!]!^�"�$ a, a lim t→t0 r(t)− r(t0) t− t0 = a, U ? r(t) ^ t0 t E G œ�? a B r(t) ^ t0 + E J G4P�Q r′(t0) Ž dr dt . ¥ ¦ !   r′(t) = (x′(t), y′(t), z′(t)). 11 º�§ G ¡ t�A � ��"�$ � ® `!+ E { dr(t) = r′(t)dt = (x′(t), y′(t), z′(t))dt. ‘�¢ 8.6.1 > r(t) ø r1(t) R r2(t) ‹ t E G4U 1◦ (r1(t) + r2(t)) ′ = r′1(t) + r ′ 2(t); 2◦ [�\ f(t) t E G4U (f(t)r(t))′ = f(t)r′(t) + f ′(t)r(t); 3◦ (r1(t) · r2(t)) ′ = r′1(t) · r2(t) + r1(t) · r ′ 2(t); 4◦ (r1(t)× r2(t)) ′ = r′1(t)× r2(t) + r1(t)× r ′ 2(t); 5◦ [�\ f(u) t E G4U dr(f(u)) du = f ′(u)r′(f(u)). 3) i ) ¯,± + ¨�"�$ > ²‡` ¯Œ± L $‡%‡' M0 R M , w ‡+#" I {�©‡S r(t0) R r(t). o S −−−⇀ M0M= r(t)− r(t0), ª r(t)− r(t0) t− t0 l x −−−⇀ M0M È ± G œ ;�",²�`�+�«�¬!³�" ( � 8.7). o S r′(t0) = lim t→t0 r(t)− r(t0) t− t0 l�S ­�"�$ r(t)− r(t0) t− t0 + � 24[�\ r′(t0) 6= 0, U r ′(t0) l�S L ^ M0 + ¨�"�$ G œ ;�"ò!` «�¬!+�³ ",2 6�¸ ¨�"�$ G l ~ ®Â¯,± + ¨ ± ³�´ B x− x(t0) x′(t0) = y − y(t0) y′(t0) = z − z(t0) z′(t0) A 7 ¯,± + r "�# ³�´ B x′(t0)(x− x(t0)) + y ′(t0)(y − y(t0)) + z ′(t0)(z − z(t0)) = 0. X ��® ` y = f(x) (a ≤ x ≤ b) 0�1 "�#*¯,± 2  �t�A ! Q i ) ¯Ã± G ©‚" I = ³!´ B r = r(x) = (x, f(x), 0). 12 w + ¨�"�$ B r′(x) = (1, f ′(x), 0). §8.6.2 ¯�ú�ý�þ�ß�û�ü�� A B�^ > D ⊂ R2, r = r(u, v) = (x(u, v), y(u, v), z(u, v)), (u, v) ∈ D. U  �t�A � ��" $ � ® ` r(u, v) + H E J B ∂r ∂u = ( ∂x ∂u , ∂y ∂u , ∂z ∂u ) , ∂r ∂v = ( ∂x ∂v , ∂y ∂v , ∂z ∂v ) . X &@YC� "°$C� ® `‡^ X '‡t�E ×@¹�±‡× w +�² & b@c ® ` ‹ ^�� '‡tCE G œ�¹�6 dr = (dx(u, v), dy(u, v), dz(u, v)). (1) dr = r′udu+ r ′ vdv. (2) Ô Y��‡® ` E { :�= + ; y –�G t œ�� Y � "�$ � ® `�+ E { :�= + ; y –�G L g  u R v SŒ‹y $ Ž ³ S m )�y $ G (2) = ´ S Q … +�2 Y � 8 Z "�$�� ® ` r = r(u, v) ((u, v) ∈ D) ž uv "�#�$ + M!r D $ ‰�Ł X ' (u, v) � B i ) m A r(u, v) B " I + ' M , × (u, v) ^ � ��O*J�y�·!k G � ' M +‡p‡q‡^ i ) m : Q�X & � : G Ä D!? w B Ô r(u, v) Á   + i ) ¯Ã# 2 2 � X & v � Gjµ u ^�© ¶ · �*J�y!·!k G " I r(u, v) + ¸ ' l�^ ¯,#!$�¹‚  X ¥°¯Œ± G ? B u ¯Œ± Gºµ v �‡^‡©�¶�·�� »�¼�J�y ½ G Ï � o v �‡+ u ¯Œ± l‡^ ¯Œ#‡$ y�½ G œ ¾  �¿�&*¯Ã#�} º�§ GÁÀ � u � Geµ v y�½ G " I r(u, v) +�¸ ' l‡^ ¯Œ#‡$�Â�  X ¥*¯,± G ? B v ¯Œ± Gõ u y�Ä ð 6‡+�¶ · � G Ï � o u �‡+ v ¯Œ± l�¾  �¿ p ¯,# 2 ¿ � ¯Ã# l�SÂÔ�� Å u ¯Œ± R v ¯Œ± �Æ�: Q +�2ÈÇ�[ É�# r = (R sin θ cosϕ,R sin θ sinϕ,R cos θ) (0 ≤ θ ≤ pi, 0 ≤ ϕ ≤ 2pi) m G θ À ��k ð œ ��+ ϕ ¯,± l�S Ê ± G : ϕ À ��k ð œ�+ θ ¯,± l�S Ë ± 2 / ¯Œ#‡$ ½ ' +�" I ð ë�í +�³�´ r = r(u, v) ((u, v) ∈ D) ̇0‡1 ¯Œ# G ? B ¯,# + ý Í ] s  , w < £�o # +�³�´ € î  x = x(u, v), y = y(u, v), z = z(u, v). (u, v) ∈ D 13 ? o ³!´ € B Î �!��Ï!ü�s� , /‚" I = ³!´ Ž ²Å`!³Å´6ÌÅ0Å1Å+ ¯4# D6Ð ¿ ? B ²�` ¯,# 2 X & ^ T O D J%���‡+ 8 Z "%$ � ® ` r = r(u, v) ((u, v) ∈ D), [‡\ r′u(u, v) R r′v(u, v) ‹ ^ D J ¬�­ œ�¹�^ D J�Ñ Ñ�6 n(u, v) = r′u(u, v)× r ′ v(u, v) 6= 0, U ? ¯,# r = r(u, v) ((u, v) ∈ D) B�X � ¼�½*¯,# 2 > M0(u0, v0) S ¯,#�$ X ' G { M0 ‰ Ò X ¥ Ó p Ô�^ ¯,#�$ + ¼!½Â¯,± G > ©�³�´�S r = (x(u(t), v(t)), y(u(t), v(t)), z(u(t), v(t))), u(t0) = u0, v(t0) = v0. � ¥Â¯Ã±!t!A ! Q S uv "!#ÂT O D m ¯Ã± î u = u(t), v = v(t) Ë�y!Ë î x = x(u, v), y = y(u, v), z = z(u, v) � Q + i ) ¯,± 2 p E { œ Õ ¯,± ^ M0 + X & ¨�"�$ B r′u(u0, v0)du+ r ′ v(u0, v0)dv, w ^ r′u(u0, v0) (u- ¯Œ± +�¨‡³#" ) x r′v(u0, v0) (v- ¯Œ± +�¨‡³#" ) ð ^‡+ "‡# J G 3‡5 n0 = r ′ u(u0, v0)×r ′ v(u0, v0) x‡A‡$ ¨#"%$ Ï�Ö j�2×� § G { M0 ¹�ԇ^ ¯Œ#‡$ +�‰�Ł X ¥�¼�½*¯Ã± ^ M0 Ñ�+ ¨ ± ‹ Ø ^ { M0 ¹ A n0 B r "�$�+ "�# J,2 n%o �� ‡? { M0 ø A n0 B r "%$‡+ "�# B ¯,# ^ M0 ч+�¨ "‡# G :�? n0 B ¯,# ^ M0 Ñ�+ r "�$�2 ð A�¼!½*¯Ã# 6 ¬!­ y!·!+ r "�$�2 Ô n = r′u × r ′ v t œ n = ( ∂(y, z) ∂(u, v) , ∂(z, x) ∂(u, v) , ∂(x, y) ∂(u, v) ) . ¯,# ^ M Ñ�+ ¨ "�# ³�´�S ∂(y, z) ∂(u, v) (X − x(u, v)) + ∂(z, x) ∂(u, v) (Y − y(u, v)) + ∂(x, y) ∂(u, v) (Z − x(u, v)) = 0, Ž ³ fi Q ∣∣∣∣∣∣∣ X − x(u, v) Y − y(u, v) Z − z(u, v) x′u(u, v) y ′ u(u, v) z ′ u(u, v) x′v(u, v) y ′ v(u, v) z ′ v(u, v) ∣∣∣∣∣∣∣ = 0; 14 Ô Y��‡® ` z = f(x, y) ((x, y) ∈ D) ð 0‡1‡+ ¯Œ# S X K Ù ¤ + ¯,# G D ¿ ? � | ¯,# B 3 = ¯,# G w�t!A ! Q S Y�� "�$�� ® `!+ X & … Ç î r = (x, y, f(x, y)), (x, y) ∈ D. š f ′x(x, y) R f ′ y(x, y) ‹ ^ D ¬�­ G4U r′x = (1, 0, f ′ x) r′y = (0, 1, f ′ y), n = r′x×r ′ y = (−f ′ x,−f ′ y, 1), � Ú r′x R r ′ y ‹ ¬�­ ¹ n 6= 0, n�o�w S X � ¼!½Â¯Ã# 2 ¥�¦ fi‚  3 = ¯Ã# ^ ' (x, y, f(x, y)) Ñ�+ ¨ "�# R r ± ³�´ G w !{ ©!S −f ′x(x, y)(X − x)− f ′ y(x, y)(Y − y) + Z − f(x, y) = 0. R X − x −f ′x(x, y) = Y − y −f ′y(x, y) = Z − f(x, y). §8.6.3 ’ ] Î ��� ^�ý�þ 8 Û�’�] Î � Ü�Ý!� Þ!ý!þ > 8 �‡® ` F (x, y, z) ^ (x0, y0, z0) ß�à‡6 X�R ¬!­ +�H E J F ′x ø F ′ y R F ′ z, á�⠗ ® `‡]‡^�� ã ä ³‡´ F (x, y, z) = 0 å M0(x0, y0, z0) ß�à • �‡¸�æ�ç è é ê E ë — ì í ( î æ ï ð z ñ ì í y ò ), n�ó å M0 ß à ô  �õ æ ö‚÷�ø�ùÁú�û�ü ý�þ ß F (x, y, z) = 0 ��� ï ë ÷�ø���� ñ�����÷�ø�ù � Γ ý ÷�ø� �� M0(x0, y0, z0) ë æ��� ���÷�� ä�����í þ�ß ñ Γ :   x = x(t), y = y(t), z = z(t). t ∈ [α, β] (x(t0), y(t0), z(t0)) = (x0, y0, z0). ý�� Γ å�÷�ø� ä������ F (x(t), y(t), z(t)) = 0 (t ∈ [α, β]), � ������ff t fi�fl ä�ffi�� t = t0, �!�" F ′x(M0)x ′(t0) + F ′ y(M0)y ′(t0) + F ′ z(t0)z ′(t0) = 0. 15 � �# ä%$�ò n = (F ′x(M0), F ′ y(M0), F ′ z(M0)) &�÷�ø� �' æ���� M0 ë ÷�� å M0 (*) ù+��ð,$�ò n * #÷%ø�å M0 ë*- $%ò�ù+.���÷�ø� ����‚÷�ø ë�/�0�1 � ù 2*3*4,5 (−f ′x,−f ′ y, 1) z = f(x, y) ë æ�ç - $�ò�ù76�&�8 ø���fi�! ë�9�: æ ; ë ù §8.7 <>=>?>@>A Taylor B>C>D>E>F §8.7.1 G�H�I�J�K Taylor L�M N�O ì í ë Taylor P�� f(x+ h, y + k) = n∑ m=0 1 m! Dmf(x, y) +Rn = n∑ m=0 1 m! ( h ∂ ∂x + k ∂ ∂y )m f(x, y) +Rn, �RQ Rn = 1 (n+ 1)! Dn+1f(x+ θh, y + θk) = 1 (n+ 1)! ( h ∂ ∂x + k ∂ ∂y )n+1 f(x+ θh, y + θk) (0 < θ < 1). å (0, 0) S�T ë Taylor P���URV Maclaurin P�� ù §8.7.2 W�X�Y�I�J�K�Z�[ � f(x, y) åR\�] D Q�� ï�^ ù M0(x0, y0) ∈ D. _ å M0 ë�` ç�a�]RQ�ä�b � f(x, y) ≤ f(x0, y0), c � M0(x0, y0) f(x, y) ë æ ç�d�e�f�g ä f(x0, y0) V f(x, y) ë æ ç�d�e�f ù h�i ä�_ å M0(x0, y0) ë�` ç�a�]RQ�ä�b�� f(x, y) ≥ f(x0, y0), c � M0(x0, y0) f(x, y) ë æ ç�d�j�f�g ä f(x0, y0) V f(x, y) ë æ ç�d�j�f ù k ì�í*l�æ*m�n�fl í�o ñ�p ë g äqV�ì í ë�r g ù ê*s k ì í ê�t ë d�f�g o� r g ù�u���v�w�x�î�æ�ï ù 16 y�z 8.7.1 � M0(x0, y0) f(x, y) ë�r g äjå M0 ë�` ç�]RQ f(x, y) � N m�è�é*n*fl í�ù|{ A = ∂2f ∂x2 ∣∣∣∣ M0 , B = ∂2f ∂x∂y ∣∣∣∣ M0 , C = ∂2f ∂y2 ∣∣∣∣ M0 } ∆ = AC −B