130005ｨｨ04ｨｬｨ 14 0709010905080507030707 040309090201 00030809000905080201 14.1 03ｨｦｨｮｨ 00ｨｴ00ｨｦｨｺ06ｨｰ 06ｨｪｨｪ07ｨｦ02ｨｰ 070007 ｨｬ05ｨｨ04ｨｬｨ ｨ 0100ｨｹ ｨｨｨ 0008ｨｪ02ｨｦ ｨｬｨｦｨ 0002ｨｪ08ｨｺ0201ｨｮ04 0004ｨｰ 070104 00ｨｪｨｴｨｮ0007ｨｰ ｨｮ0007ｨｪ ｨ ｨｪｨ 00ｨｪ06ｨｮ0004 06ｨｪｨｪ07ｨｦｨ ｨｰ 0004ｨｰ 0809ｨ 00ｨｬｨ 00ｨｦｨｺ07ｨｰ ｨｮ01ｨｪ05090004ｨｮ04ｨｰ ｨｬｨｦｨ ｨｰ 0809ｨ 00ｨｬｨ 00ｨｦｨｺ07ｨｰ ｨｬ0200ｨ 0905040007ｨｰ ｨｮ02 010507, ｨ ｨｪ0008ｨｮ0007ｨｦ ﾂｨ 000902ｨｦｨｰ ｨｬ0200ｨ 0905040006ｨｰ. 1 14.1.1 0309ｨｦｨｮｨｬ0708 0309ｨｦｨｮｨｬｨｹｨｰ 14.1.1-1 (ｨｮ01ｨｪ05090004ｨｮ04ｨｰ 0807050506ｨｪ ｨｬ0200ｨ 0905040006ｨｪ) 00ｨｮ00ｨｴ D 6ﾛ7 R 2, ｨ ｨｪ0008ｨｮ0007ｨｦ ﾂｨ D 6ﾛ7 R 3 ｨｺｨ ｨｦ T 6ﾛ7 R 010507 0001 ﾂｨｹｨｪ00ｨ ｨｬ04 ｨｺ02ｨｪ05 ｨｮ05ｨｪ0705ｨ. 08ｨｹ0002 ｨｬ08ｨ ｨｮ01ｨｪ05090004ｨｮ04 010507, ｨ ｨｪ0008ｨｮ0007ｨｦ ﾂｨ 0009ｨｦ06ｨｪ ｨｬ0200ｨ 0905040006ｨｪ ｨｬ02 0802010807 0709ｨｦｨｮｨｬ0705 0007 D ｨｺｨ ｨｦ 0802010807 00ｨｦｨｬ06ｨｪ 0007 T 0208ｨｪｨ ｨｦ ｨｬ08ｨ ｨｬ07ｨｪ07ｨｮ07ｨｬｨ ｨｪ0004 ｨ 0802ｨｦｨｺｨｹｨｪｨｦｨｮ04, 06ｨｮ00ｨｴ f, 000701 ｨｮ01ｨｪｨｹ050701 10005060802 020808ｨｮ04ｨｰ 09ｨｦ0905ｨｦ070009ｨ 0208ｨ [3, 4]. 601
13602 0701ｨｪｨ 090007ｨｮ02ｨｦｨｰ 0807050506ｨｪ ｨｬ0200ｨ 0905040006ｨｪ 08ｨ ｨｨ. 09. 0008090500ｨｮ07ｨｰ D ｨｮ0007 T, 00060007ｨｦｨ 06ｨｮ0002: D 6ﾒ6 (x; y) 6ﾓ1 f (x; y) = w ﾊ T; ｨ ｨｪ0008ｨｮ0007ｨｦ ﾂｨ (14.1.1-1) D 6ﾒ6 (x; y; z) 6ﾓ1 f (x; y; z) = w ﾊ T: 08ｨ x; y, ｨ ｨｪ0008ｨｮ0007ｨｦ ﾂｨ x; y; z 0208ｨｪｨ ｨｦ ｨｮ0004ｨｪ 080209080800ｨｴｨｮ04 ｨ 010007 07ｨｦ ｨ ｨｪ0206050900040002ｨｰ ｨｬ0200ｨ 0905040006ｨｰ 07 ｨ 080505 00ｨｦｨ 0201ｨｺ070508ｨ ｨｮ0007 020607ｨｰ ｨｬ0200ｨ 0905040006ｨｰ 07 020808ｨｮ04ｨｰ ｨｹ08ｨｴｨｰ 020808ｨｮ04ｨｰ 0506000200ｨ ｨｦ 00ｨ ｨｮ0007ｨｦ ﾂ0208ｨ (arguments) 0004ｨｰ f, 02ｨｪ06 04 w 0208ｨｪｨ ｨｦ 04 0206ｨ 090004ｨｬ06ｨｪ04 ｨｬ0200ｨ 0905040007. 04ｨｬ07ｨｦｨ, ｨｹ08ｨｴｨｰ ｨｺｨ ｨｦ ｨｮ0004ｨｪ 080209080800ｨｴｨｮ04 0004ｨｰ ｨｬｨｦｨ ｨｰ ｨｬ0200ｨ 0905040007ｨｰ, 04 f 0709080302ｨｦ 0007ｨｪ 00050807 0004ｨｰ ｨｮ01ｨｪ05090004ｨｮ04ｨｰ, 010405ｨ 0107 080209ｨｦ0009050202ｨｦ 0007ｨｪ 0009ｨｹ0807 ｨｬ02 0007ｨｪ 0708070807 0008ｨｪ0200ｨ ｨｦ 04 08ｨ 09ｨ 0805ｨｪｨｴ ｨ 0802ｨｦｨｺｨｹｨｪｨｦｨｮ04. 03 080907ｨｮ01ｨｦ0709ｨｦｨｮｨｬｨｹｨｰ 000701 080201080701 0709ｨｦｨｮｨｬ0705 D 0008ｨｪ0200ｨ ｨｦ ｨｹ08ｨｴｨｰ ｨｺｨ ｨｦ ｨｮ0004ｨｪ 080209080800ｨｴｨｮ04 0004ｨｰ ｨｮ01ｨｪ05090004ｨｮ04ｨｰ ｨｬ02 ｨｬ08ｨ ｨｬ0200ｨ 0905040007, ｨｬ02 0004 01ｨｦｨ 02070905 ｨｹ00ｨｦ 080907ｨｮ01ｨｦ0709080307ｨｪ00ｨ ｨｦ 07ｨｦ 00ｨｦｨｬ06ｨｰ 00ｨｦｨ 00ｨｦｨｰ 0708070802ｨｰ 070908030200ｨ ｨｦ 04 f 00ｨｦｨ ｨｺ05ｨｨ02 ｨｬ0200ｨ 0905040007 x; y, ｨ ｨｪ0008ｨｮ0007ｨｦ ﾂｨ x; y; z ﾂｨｴ09ｨｦｨｮ0005 ｨｺｨ ｨｦ ｨｮ0004 ｨｮ01ｨｪ06 ﾂ02ｨｦｨ 0007 D ｨｴｨｰ 04 06ｨｪｨｴｨｮ04 00ｨｴｨｪ 0208ｨｦｨｬ06090701ｨｰ 08020108ｨｴｨｪ 0709ｨｦｨｮｨｬ0705. 00ｨｦｨ ｨｮ01ｨｪ05090004ｨｮ04 f ｨｬ02 0802010807 0709ｨｦｨｮｨｬ0705 D ｨｨｨ ｨｮ01ｨｬ09070508030200ｨ ｨｦ ｨｮ0007 020607ｨｰ ｨｬ02 f D 07 ｨ ｨｪｨ 050100ｨｦｨｺ05 f(x; y) D, ｨ ｨｪ0008ｨｮ0007ｨｦ ﾂｨ f(x; y; z) D. 08ｨ 08020108ｨ 0709ｨｦｨｮｨｬ0705 ｨｺｨ ｨｦ 00ｨｦｨｬ06ｨｪ 0208ｨｪｨ ｨｦ ｨｬｨｦｨ ｨｺｨ ｨｬ08050504 0208ｨｦ0205ｨｪ02ｨｦｨ 07 0002ｨｪｨｦｨｺｨｹ000209ｨ ｨｬｨｦｨ 0009ｨｦｨｮ01ｨｦ05ｨｮ00ｨ 0004 080209ｨｦ07 ﾂ07 000701 ﾂ06090701. 00ｨｮ00ｨｴ w = f(x; y) D, ｨ ｨｪ0008ｨｮ0007ｨｦ ﾂｨ w = f(x; y; z) D. 08ｨｹ0002 04 0009ｨ 02ｨｦｨｺ07 08ｨ 0905ｨｮ00ｨ ｨｮ04 0004ｨｰ f ｨｨｨ 0208ｨｪｨ ｨｦ 0007 ｨｮ05ｨｪ070507 00ｨｴｨｪ ｨｮ04ｨｬ0208ｨｴｨｪ {((x; y); w) ﾊ D ﾁ T; ｨ ｨｪ0008ｨｮ0007ｨｦ ﾂｨ ((x; y; z); w) ﾊ D ﾁ T:} 04ｨ 09050102ｨｦ00ｨｬｨ 14.1.1-1 01ｨ 010807050700ｨｦｨｮ000208 0007 0802010807 0709ｨｦｨｮｨｬ0705 00ｨｴｨｪ ｨｮ01ｨｪｨ 090007ｨｮ02ｨｴｨｪ f 1 (x; y) = ﾌ x + y; f 2 (x; y) = ﾌ x+ ﾌ y ｨｺｨ ｨｦ f 3 (x; y) = ln ( 4 6ﾓ1 x 2 6ﾓ1 4y 2) : 0905ｨｮ04. 030802ｨｦ0107 ｨ 08ｨｹ 0007ｨｪ 00050807 0004ｨｰ f 1 0809060802ｨｦ ｨｪｨ 080907ｨｺ05080002ｨｦ 0809ｨ 00ｨｬｨ 00ｨｦｨｺｨｹｨｰ ｨ 09ｨｦｨｨｨｬｨｹｨｰ, 0007 0802010807 0709ｨｦｨｮｨｬ0705 D 1 ｨｨｨ 0208ｨｪｨ ｨｦ D 1 = {(x; y) ﾊ R 2 : x + y ﾝ 0}:
130309ｨｦｨｮｨｬ0708 603 y 1.0 0.5 1 71.0 1 70.5 0.5 1.0 x 1 70.5 1 71.0 (a) y 1.0 0.5 1 70.5 1 71.0 0.5 1.0 1.5 2.0 2.5 3.0 x (b) 07 ﾂ07ｨｬｨ 14.1.1-1: 04ｨ 09050102ｨｦ00ｨｬｨ 14.1.1-1: (a) 0007 0802010807 0709ｨｦｨｮｨｬ0705 D 1 = {(x; y) ﾊ R 2 : x + y ﾝ 0; } 0004ｨｰ ｨｮ01ｨｪ05090004ｨｮ04ｨｰ f 1 (x; y) = ﾌ x + y. 05 ｨｬ080502 0201ｨｨ0208ｨ 06 ﾂ02ｨｦ 020608ｨｮｨｴｨｮ04 x + y = 0. (b) 0807 0802010807 0709ｨｦｨｮｨｬ0705 D 2 = {(x; y) ﾊ R 2 : x ﾝ 0; y ﾝ 0} 0004ｨｰ f 2 (x; y) = ﾌ x + ﾌ y. 0109ｨ 02ｨｦｨｺ05 0007 D 1 070908030200ｨ ｨｦ ｨ 08ｨｹ 0007 ｨｮ05ｨｪ070507 00ｨｴｨｪ ｨｮ04ｨｬ0208ｨｴｨｪ 000701 0208ｨｦ0806010701 080701 090908ｨｮｨｺ07ｨｪ00ｨ ｨｦ ｨｮ0007 05ｨｪｨｴ ｨｬ060907ｨｰ 0004ｨｰ 0201ｨｨ0208ｨ ｨｰ x + y = 0 (07 ﾂ. 14.1.1-3a). 2 04ｨｬ07ｨｦｨ 0007 0802010807 0709ｨｦｨｮｨｬ0705 D 2 0004ｨｰ f 2 ｨｨｨ 0208ｨｪｨ ｨｦ D 2 = {(x; y) ﾊ R 2 : x ﾝ 0; y ﾝ 0}; 010405ｨ 0107 0007 107 000200ｨ 090004ｨｬｨｹ09ｨｦ07 000701 07 ﾂ. 14.1.1-3b. 08060507ｨｰ, 020802ｨｦ0107 04 050700ｨ 09ｨｦｨｨｨｬｨｦｨｺ07 ｨｮ01ｨｪ05090004ｨｮ04 070908030200ｨ ｨｦ ｨｬｨｹｨｪ07 00ｨｦｨ ｨｨ0200ｨｦｨｺ06ｨｰ 00ｨｦｨｬ06ｨｰ 0004ｨｰ ｨｬ0200ｨ 0905040007ｨｰ 0004ｨｰ, 00ｨｦｨ 0007 0802010807 0709ｨｦｨｮｨｬ0705 D 3 0004ｨｰ f 3 0809060802ｨｦ 4 6ﾓ1 x 2 6ﾓ1 4y 2 > 0 07 1 > x2 4 + y2, 0708ｨｹ0002 D 3 = {(x; y) ﾊ R 2 : x 2 4 + y2 < 1}; 010405ｨ 0107 0007 0802010807 0709ｨｦｨｮｨｬ0705 0208ｨｪｨ ｨｦ 0007 02ｨｮｨｴ000209ｨｦｨｺｨｹ 0004ｨｰ 06050502ｨｦ0304ｨｰ ｨｬ02 020608ｨｮｨｴｨｮ04 x2 4 +y2 = 1 (07 ﾂ. 14.1.1-2a). 070007 07 ﾂ. 14.1.1-2b 0108ｨｪ0200ｨ ｨｦ 04 0009ｨ 02ｨｦｨｺ07 08ｨ 0905ｨｮ00ｨ ｨｮ04 0004ｨｰ f 3. 2090802ｨｪｨｨ01ｨｬ08030200ｨ ｨｦ ｨｹ00ｨｦ 04 ｨ ｨｪｨｦｨｮｨｹ000400ｨ Ax + By + ｦ > 0 0505ｨｪ0200ｨ ｨｦ 0009ｨ 02ｨｦｨｺ05, ｨｹ00ｨ ｨｪ ﾂｨ 09ｨ ﾂｨｨ0208 04 0201ｨｨ0208ｨ 02 : Ax + By + ｦ = 0 ｨｺｨ ｨｦ ｨｨ02ｨｴ0907ｨｮ0701ｨｬ02 0007 ｨｮ05ｨｪ070507 00ｨｴｨｪ ｨｮ04ｨｬ0208ｨｴｨｪ (x; y) ﾊ R 2, 080701 0208ｨｪｨ ｨｦ ｨｮ0007 05ｨｪｨｴ ｨｬ060907ｨｰ 0004ｨｰ 02.
13604 0701ｨｪｨ 090007ｨｮ02ｨｦｨｰ 0807050506ｨｪ ｨｬ0200ｨ 0905040006ｨｪ 08ｨ ｨｨ. 09. 0008090500ｨｮ07ｨｰ y 1.0 0.5 1 72 1 71 1 2 x 1 70.5 1 71.0 (a) (b) 07 ﾂ07ｨｬｨ 14.1.1-2: 04ｨ 09050102ｨｦ00ｨｬｨ 14.1.1-1: (a) 0007 0802010807 0709ｨｦｨｮｨｬ0705 D 3 = {(x; y) ﾊ R 2 : x 2 4 +y2 < 1} 0004ｨｰ ｨｮ01ｨｪ05090004ｨｮ04ｨｰ f 3 (x; y) = ln ( 4 6ﾓ1 x 2 6ﾓ1 4y 2). 05 01ｨｦｨ ｨｺ02ｨｺ07ｨｬｨｬ06ｨｪ04 ｨｺｨｹｨｺｨｺｨｦｨｪ04 ｨｺｨ ｨｬ08050504 0208ｨｪｨ ｨｦ 04 06050502ｨｦ0304 ｨｬ02 020608ｨｮｨｴｨｮ04 x2 4 +y2 = 1. (b) 05 0009ｨ 02ｨｦｨｺ07 08ｨ 0905ｨｮ00ｨ ｨｮ04 0004ｨｰ f 3 (x; y). 05 ｨｺｨｹｨｺｨｺｨｦｨｪ04 ｨｺｨ ｨｬ08050504 0102ｨｪ ｨｮ01ｨｬ080209ｨｦ05ｨ ｨｬ0905ｨｪ0200ｨ ｨｦ ｨｮ0007 01ｨｦ050009ｨ ｨｬｨｬｨ. 04ｨ 09050102ｨｦ00ｨｬｨ 14.1.1-2 01ｨ 010807050700ｨｦｨｮ000208 0007 0802010807 0709ｨｦｨｮｨｬ0705 0004ｨｰ ｨｮ01ｨｪ05090004ｨｮ04ｨｰ f(x; y) = sin 6ﾓ11 x + ﾌ xy: 0905ｨｮ04. 00ｨｮ00ｨｴ f 1 (x; y) = sin 6ﾓ11 x ｨｺｨ ｨｦ f 2 (x; y) = ﾌ xy: 08ｨｹ0002, ｨｹ08ｨｴｨｰ 0208ｨｪｨ ｨｦ 070104 00ｨｪｨｴｨｮ00ｨｹ ｨ 08ｨｹ 0007 0005ｨｨ04ｨｬｨ 0409ｨ 00ｨｬｨ 00ｨｦｨｺ06ｨｰ 0701ｨｪｨ 090007ｨｮ02ｨｦｨｰ, ｨｮ0004 ｨｮ01ｨｪ05090004ｨｮ04 sin x, ｨｹ00ｨ ｨｪ 0007 0802010807 0709ｨｦｨｮｨｬ0705 080209ｨｦ0709ｨｦｨｮ000208 ｨｮ0007 [6ﾓ1=2; =2], 070908030200ｨ ｨｦ 04 ｨ ｨｪ0008ｨｮ0009070204 ｨｮ01ｨｪ05090004ｨｮ04 sin 6ﾓ11 x 07 arcsin x ｨｺｨ ｨｦ 06 ﾂ02ｨｦ 0802010807 0709ｨｦｨｮｨｬ0705 0007 [6ﾓ11; 1], 010405ｨ 0107 0007 0802010807 00ｨｦｨｬ06ｨｪ 0004ｨｰ sin x. 030807ｨｬ06ｨｪｨｴｨｰ 0007 0802010807 0709ｨｦｨｮｨｬ0705 D 1 0004ｨｰ f 1 0208ｨｪｨ ｨｦ D 1 = {(x; y) ﾊ R 2 : 6ﾓ11 ﾜ x ﾜ 1}: 05 ｨｮ01ｨｪ05090004ｨｮ04 f 2 (x; y) = ﾌ xy
130309ｨｦｨｮｨｬ0708 605 070908030200ｨ ｨｦ, ｨｹ00ｨ ｨｪ xy ﾝ 0, 010405ｨ 0107, ｨｹ00ｨ ｨｪ 00ｨ x, y 0208ｨｪｨ ｨｦ 07ｨｬｨｹｨｮ04ｨｬｨ. 0409ｨ 05ｨ ｨｬ0905ｨｪ07ｨｪ00ｨ ｨｰ 0108ｨｹ0304 ｨｺｨ ｨｦ 0007 D 1 0007 0802010807 0709ｨｦｨｮｨｬ0705 0004ｨｰ f ｨｨｨ 0208ｨｪｨ ｨｦ D f = D 2 ﾈ D 3, ｨｹ00ｨ ｨｪ D 2 = {(x; y) ﾊ R 2 : 6ﾓ11 ﾜ x ﾜ 0; y ﾜ 0} ｨｺｨ ｨｦ D 3 = {(x; y) ﾊ R 2 : 0 ﾜ x ﾜ 1; y ﾝ 0}: 04ｨ 09050102ｨｦ00ｨｬｨ 14.1.1-3 04ｨｬ07ｨｦｨ 0007 0802010807 0709ｨｦｨｮｨｬ0705 00ｨｴｨｪ ｨｮ01ｨｪｨ 090007ｨｮ02ｨｴｨｪ f(x; y) = ﾌ x + y ｨｺｨ ｨｦ g(x; y) = ﾌ x + ﾌ y: 0905ｨｮ04. 00ｨｮ00ｨｴ D f 0007 0802010807 0709ｨｦｨｮｨｬ0705 0004ｨｰ f, ｨ ｨｪ0008ｨｮ0007ｨｦ ﾂｨ D g 0004ｨｰ g. 08ｨｹ0002 08090702ｨ ｨｪ06ｨｰ 0208ｨｪｨ ｨｦ ｨ ｨｪ0008ｨｮ0007ｨｦ ﾂｨ D f = {(x; y) ﾊ R 2 : x + y ﾝ 0 (07 ﾂ. 14:1:1 6ﾓ1 3a) }; D g = {(x; y) ﾊ R 2 : x ﾝ 0 ｨｺｨ ｨｦ y ﾝ 0 (07 ﾂ. 14:1:1 6ﾓ1 3b) }: y 1.0 0.5 1 71.0 1 70.5 0.5 1.0 x 1 70.5 1 71.0 (a) y 1.0 0.5 1 70.5 1 71.0 0.5 1.0 1.5 2.0 2.5 3.0 x (b) 07 ﾂ07ｨｬｨ 14.1.1-3: 04ｨ 09050102ｨｦ00ｨｬｨ 14.1.1-4: (a) 0007 0802010807 0709ｨｦｨｮｨｬ0705 D f = {(x; y) ﾊ R 2 : x + y ﾝ 0} 0004ｨｰ ｨｮ01ｨｪ05090004ｨｮ04ｨｰ f(x; y) = ﾌ x + y. 05 ｨｬ080502 0201ｨｨ0208ｨ 06 ﾂ02ｨｦ 020608ｨｮｨｴｨｮ04 x + y = 0 ｨｺｨ ｨｦ (b) 0007 0802010807 0709ｨｦｨｮｨｬ0705 D g = {(x; y) ﾊ R 2 : x ﾝ 0; y ﾝ 0} 0004ｨｰ g(x; y) = ﾌ x + ﾌ y.
13606 0701ｨｪｨ 090007ｨｮ02ｨｦｨｰ 0807050506ｨｪ ｨｬ0200ｨ 0905040006ｨｪ 08ｨ ｨｨ. 09. 0008090500ｨｮ07ｨｰ 04ｨ 09050102ｨｦ00ｨｬｨ 14.1.1-4 04ｨｬ07ｨｦｨ 00ｨｴｨｪ ｨｮ01ｨｪｨ 090007ｨｮ02ｨｴｨｪ f(x; y; z) = ln(x 6ﾓ1 y + 4z) ｨｺｨ ｨｦ g(x; y; z) = 1 ﾌ x 2 + y 2 + z 2 6ﾓ1 9 : 0905ｨｮ04. 030802ｨｦ0107 04 050700ｨ 09ｨｦｨｨｨｬｨｦｨｺ07 ｨｮ01ｨｪ05090004ｨｮ04 070908030200ｨ ｨｦ ｨｬｨｹｨｪ07 00ｨｦｨ ｨｨ0200ｨｦｨｺ06ｨｰ 00ｨｦｨｬ06ｨｰ 0004ｨｰ ｨｬ0200ｨ 0905040007ｨｰ 0004ｨｰ, 0007 0802010807 0709ｨｦｨｮｨｬ0705 D f 0004ｨｰ f ｨｨｨ 0208ｨｪｨ ｨｦ D f = {(x; y; z) ﾊ R 3 : x 6ﾓ1 y + 4z > 0}; 010405ｨ 0107 0809ｨｹｨｺ02ｨｦ00ｨ ｨｦ 00ｨｦｨ 0007 05ｨｪｨｴ ｨｬ060907ｨｰ 000701 0208ｨｦ0806010701 08 ｨｬ02 020608ｨｮｨｴｨｮ04 08 : x 6ﾓ1 y + 4z = 0: 090802ｨｪｨｨ01ｨｬ08030200ｨ ｨｦ ｨｮ0007 ｨｮ04ｨｬ020807 ｨ 0100ｨｹ ｨ 08ｨｹ 0007 0005ｨｨ04ｨｬｨ 09ｨｪｨ 050100ｨｦｨｺ07 0102ｨｴｨｬ02000908ｨ ｨｹ00ｨｦ 04 0002ｨｪｨｦｨｺ07 ｨｬ07090207 0004ｨｰ 020608ｨｮｨｴｨｮ04ｨｰ 000701 0208ｨｦ0806010701 0208ｨｪｨ ｨｦ 080701, ｨｹ00ｨ ｨｪ 0501ｨｨ0208 ｨｴｨｰ 080907ｨｰ z, ｨｦｨｮ070105ｨｪｨ ｨｬｨ 000905020200ｨ ｨｦ ｨｺｨ ｨｦ ax + by + cz = d; (14.1.1-2) z = f(x; y) = Ax + By + D: (14.1.1-3) 05 0009ｨ 02ｨｦｨｺ07 08ｨ 0905ｨｮ00ｨ ｨｮ04 02ｨｪｨｹｨｰ 0208ｨｦ0806010701 0002ｨｪｨｦｨｺ05 0008ｨｪ0200ｨ ｨｦ ｨｬ02 0007ｨｪ 080907ｨｮ01ｨｦ0709ｨｦｨｮｨｬｨｹ 00ｨｴｨｪ ｨｮ04ｨｬ0208ｨｴｨｪ 0007ｨｬ07ｨｰ 000701 0208ｨｦ0806010701 ｨｬ02 000701ｨｰ 050607ｨｪ02ｨｰ ｨｮ01ｨｪ000200ｨ 00ｨｬ06ｨｪｨｴｨｪ. 08ｨｹ0002 02ｨｪ06ｨｪ07ｨｪ00ｨ ｨｰ 00ｨ 000908ｨ 08ｨ 09ｨ 0805ｨｪｨｴ ｨｮ04ｨｬ0208ｨ 0007ｨｬ07ｨｰ 0007 0104ｨｬｨｦ070109000705ｨｬ02ｨｪ07 00090800ｨｴｨｪ07 010208 ﾂｨｪ02ｨｦ ｨｺｨ ｨｦ 0004 ｨｬ07090207 000701 0208ｨｦ0806010701. 01ｨｦｨ 08ｨ 09050102ｨｦ00ｨｬｨ, 06ｨｮ00ｨｴ ｨｹ00ｨｦ 030400020800ｨ ｨｦ 04 0009ｨ 02ｨｦｨｺ07 08ｨ 0905ｨｮ00ｨ ｨｮ04 000701 0208ｨｦ0806010701 3x+4y +z = 12, 080701 0208ｨｪｨ ｨｦ 0004ｨｰ ｨｬ07090207ｨｰ (14:1:16ﾓ12) ｨｺｨ ｨｦ ｨｮ05ｨｬ02ｨｴｨｪｨ ｨｬ02 0004ｨｪ (14:1:1 6ﾓ1 3) ｨｦｨｮ070105ｨｪｨ ｨｬｨ 000905020200ｨ ｨｦ z = 12 6ﾓ1 3x 6ﾓ1 4y; 010405ｨ 0107 f(x; y) = 12 6ﾓ1 3x 6ﾓ1 4y: (14.1.1-4) 08ｨｹ0002 ｨｨ060007ｨｪ00ｨ ｨｰ ｨｮ0004ｨｪ (14:1:16ﾓ14) x = y = 0 080907ｨｮ01ｨｦ070908030200ｨ ｨｦ ｨｹ00ｨｦ 0007 ｨｮ04ｨｬ020807 0007ｨｬ07ｨｰ 000701 0208ｨｦ0806010701 ｨｬ02 0007ｨｪ z-050607ｨｪｨ 0208ｨｪｨ ｨｦ 0007 (0; 0; 12). 04ｨｬ07ｨｦｨ 0007 ｨｮ04ｨｬ020807 0007ｨｬ07ｨｰ ｨｬ02 0007ｨｪ x-050607ｨｪｨ 0208ｨｪｨ ｨｦ 0007 (4; 0; 0) ｨｺｨ ｨｦ ｨｬ02 0007ｨｪ y-050607ｨｪｨ 0007 (0; 3; 0). 05 ｨ ｨｪｨｦｨｮｨｹ000400ｨ ax+by+cz > 0 0505ｨｪ0200ｨ ｨｦ 0009ｨ 02ｨｦｨｺ05, ｨｹ00ｨ ｨｪ ｨ 09 ﾂｨｦｨｺ05 0008ｨｪ02ｨｦ 04 0009ｨ 02ｨｦｨｺ07 08ｨ 0905ｨｮ00ｨ ｨｮ04 000701 0208ｨｦ0806010701 08 : ax + by + cz = 0
130309ｨｦｨｮｨｬ0708 607 ｨｺｨ ｨｦ ｨｮ0004 ｨｮ01ｨｪ06 ﾂ02ｨｦｨ ｨｨ02ｨｴ0904ｨｨ0208 0007 ｨｮ05ｨｪ070507 00ｨｴｨｪ ｨｮ04ｨｬ0208ｨｴｨｪ (x; y; z) ﾊ R 3, 080701 0208ｨｪｨ ｨｦ ｨｮ0007 05ｨｪｨｴ ｨｬ060907ｨｰ 000701 08. 0807 0802010807 0709ｨｦｨｮｨｬ0705 D g 0004ｨｰ g, 05ｨｹ00ｨｴ 0004ｨｰ 00020009ｨ 00ｨｴｨｪｨｦｨｺ07ｨｰ 090803ｨ ｨｰ ｨｺｨ ｨｦ 000701 08ｨ 0907ｨｪ07ｨｬｨ ｨｮ0007, ｨｨｨ 0208ｨｪｨ ｨｦ D g = {(x; y; z) ﾊ R 3 : x 2 + y 2 + z 2 < 9}; 010405ｨ 0107 0007 02ｨｮｨｴ000209ｨｦｨｺｨｹ 0004ｨｰ ｨｮ02ｨ 0809ｨ ｨｰ ｨｬ02 ｨｺ06ｨｪ000907 0007 ｨｮ04ｨｬ020807 (0; 0; 0) ｨｺｨ ｨｦ ｨ ｨｺ0008ｨｪｨ R = 3. 0908ｨｹ 0007 04ｨ 09050102ｨｦ00ｨｬｨ 14.1.1-4 080907ｨｺ05080002ｨｦ ｨｹ00ｨｦ ｨｮ00ｨｦｨｰ 080209ｨｦ080006ｨｮ02ｨｦｨｰ ｨｮ01ｨｪｨ 090007ｨｮ02ｨｴｨｪ 0009ｨｦ06ｨｪ ｨｬ0200ｨ 0905040006ｨｪ 0007 0802010807 0709ｨｦｨｮｨｬ0705 0208ｨｪｨ ｨｦ 07 ｨｬｨｦｨ 0208ｨｦ0205ｨｪ02ｨｦｨ - 080209080800ｨｴｨｮ04 080201080701 0709ｨｦｨｮｨｬ0705 D f - 07 06ｨｪｨ ｨｰ ｨｹ00ｨｺ07ｨｰ - 0802010807 0709ｨｦｨｮｨｬ0705 D g. 05 0009ｨ 02ｨｦｨｺ07 08ｨ 0905ｨｮ00ｨ ｨｮ04 ｨｬｨｦｨ ｨｰ ｨｮ01ｨｪ05090004ｨｮ04ｨｰ, 06ｨｮ00ｨｴ f, ｨｮ0004ｨｪ 080209080800ｨｴｨｮ04 ｨ 010007 0208ｨｪｨ ｨｦ 0101ｨｪｨ 00ｨｹｨｪ ｨｪｨ 0008ｨｪ02ｨｦ ｨ 08ｨｹ 0007 01ｨｦ050009ｨ ｨｬｨｬｨ 000701 080201080701 00ｨｦｨｬ06ｨｪ T 00ｨｴｨｪ ｨｮ04ｨｬ0208ｨｴｨｪ, 010405ｨ 0107 000701 ｨｮ01ｨｪｨｹ050701 T = {f(x; y; z) ｨｬ02 (x; y; z) ﾊ D}, ｨｹ00ｨ ｨｪ D 0007 0802010807 0709ｨｦｨｮｨｬ0705 0004ｨｰ f ｨｺｨ ｨｦ 0208ｨｪｨ ｨｦ 0002ｨｪｨｦｨｺ05 ｨｬｨｦｨ 0208ｨｦ0205ｨｪ02ｨｦｨ 07 ｨｺｨ ｨｦ 06ｨｪｨ ｨｰ ｨｹ00ｨｺ07ｨｰ 000701 ﾂ06090701 00ｨｴｨｪ 0009ｨｦ06ｨｪ 01ｨｦｨ ｨｮ0005ｨｮ02ｨｴｨｪ. 04ｨｮｨｺ04ｨｮ04 08ｨｴｨｪ 08ｨ 09ｨ ｨｺ0500ｨｴ ｨｮ01ｨｪｨ 090007ｨｮ02ｨｴｨｪ ｨｪｨ 080907ｨｮ01ｨｦ0709ｨｦｨｮ000208 0007 0802010807 0709ｨｦｨｮｨｬ0705 ｨｺｨ ｨｦ ｨｪｨ 0008ｨｪ02ｨｦ 04 0009ｨ 02ｨｦｨｺ07 08ｨ 0905ｨｮ00ｨ ｨｮ04: i) ( 4 6ﾓ1 x 2 6ﾓ1 y 2) 1=2 v) 1= ln (x + y + z), ii) ln(x 6ﾓ1 y) vi) tan 6ﾓ11 y + ﾌ xy, iii) iv) ( 9 6ﾓ1 x 2 ) 1=2 + ( 4 6ﾓ1 y 2 ) 1=2 sin 6ﾓ11 ( y x ) vii) ln(xyz), viii) ln ( x 2 + y 2 6ﾓ1 z 2). 0908ｨ ｨｪ0007ｨｮ02ｨｦｨｰ (i) x 2 + y 2 ﾝ 0,, (ii) x 6ﾓ1 y > 0, (iii) 6ﾓ13 ﾜ x ﾜ 3 ｨｺｨ ｨｦ 6ﾓ12 ﾜ y ﾜ 2, (iv) y ﾜ x ｨｺｨ ｨｦ x ﾙ 0, (v) x + y + z > 0 ｨｺｨ ｨｦ x + y + z ﾙ 1, (vi) xy ﾝ 0, (vii) xyz > 0, (viii) x 2 + y 2 > z 2.
13608 0701ｨｪｨ 090007ｨｮ02ｨｦｨｰ 0807050506ｨｪ ｨｬ0200ｨ 0905040006ｨｪ 08ｨ ｨｨ. 09. 0008090500ｨｮ07ｨｰ 14.1.2 070500ｨｺ05ｨｦｨｮ04 ｨｮ01ｨｪｨ 090007ｨｮ02ｨｴｨｪ 010507 ｨｺｨ ｨｦ 0009ｨｦ06ｨｪ ｨｬ0200ｨ 0905040006ｨｪ 0309ｨｦｨｮｨｬｨｹｨｰ 14.1.2-1 (010507 ｨｬ0200ｨ 0905040006ｨｪ). 0802010807 0709ｨｦｨｮｨｬ0705 D 6ﾛ7 R 2. 08ｨｹ0002 ｨｨｨ 0208ｨｪｨ ｨｦ 00ｨｮ00ｨｴ 04 ｨｮ01ｨｪ05090004ｨｮ04 f(x; y) ｨｬ02 lim f(x; y) = l; (14.1.2-1) (x;y) (x 0 ;y 0 ) 00ｨｹ0002 ｨｺｨ ｨｦ ｨｬｨｹｨｪ07ｨｪ ｨｹ00ｨ ｨｪ 00ｨｦｨ ｨｺ05ｨｨ02 " > 0 01080509 ﾂ02ｨｦ = (") > 0, 0600ｨｮｨｦ 06ｨｮ0002 f(x; y) 6ﾓ1 l < " 00ｨｦｨ ｨｺ05ｨｨ02 (x; y) ﾊ D; ｨｺｨ ｨｦ ﾌ (x 6ﾓ1 x 0 ) 2 + (y 6ﾓ1 y 0 ) 2 < : 0309ｨｦｨｮｨｬｨｹｨｰ 14.1.2-2 (0009ｨｦ06ｨｪ ｨｬ0200ｨ 0905040006ｨｪ). 00ｨｮ00ｨｴ 04 ｨｮ01ｨｪ05090004ｨｮ04 f(x; y; z) ｨｬ02 0802010807 0709ｨｦｨｮｨｬ0705 D 6ﾛ7 R 3. 08ｨｹ0002 ｨｨｨ 0208ｨｪｨ ｨｦ lim f(x; y; z) = l; (14.1.2-2) (x;y;z) (x 0 ;y 0 ;z 0 ) 00ｨｹ0002 ｨｺｨ ｨｦ ｨｬｨｹｨｪ07ｨｪ ｨｹ00ｨ ｨｪ 00ｨｦｨ ｨｺ05ｨｨ02 " > 0 01080509 ﾂ02ｨｦ = (") > 0, 0600ｨｮｨｦ 06ｨｮ0002 f(x; y; z) 6ﾓ1 l < " 00ｨｦｨ ｨｺ05ｨｨ02 (x; y; z) ﾊ D; ｨｺｨ ｨｦ ﾌ (x 6ﾓ1 x 0 ) 2 + (y 6ﾓ1 y 0 ) 2 + (z 6ﾓ1 z 0 ) 2 < : 07 ﾂ0200ｨｦｨｺ05 ｨｬ02 0004 01ｨｦｨ 01ｨｦｨｺｨ ｨｮ08ｨ 010807050700ｨｦｨｮｨｬ0705 00ｨｴｨｪ 0208ｨｦｨｬ06090701ｨｰ 0709ｨｦｨ ｨｺ06ｨｪ 00ｨｦｨｬ06ｨｪ ｨｮ0004ｨｪ 080209080800ｨｴｨｮ04 000701 0309ｨｦｨｮｨｬ0705 14.1.2-1 ｨｦｨｮ ﾂ0502ｨｦ 04 08ｨ 09ｨ ｨｺ0500ｨｴ 0809ｨｹ00ｨ ｨｮ04: 3 0409ｨｹ00ｨ ｨｮ04 14.1.2-1. 00ｨｮ00ｨｴ 04 ｨｮ01ｨｪ05090004ｨｮ04 f(x; y) ｨｬ02 (x; y) ﾊ D 6ﾛ7 R 2 ｨ ｨｪ07ｨｦｨｺ00ｨｹ ｨｮ05ｨｪ070507 ｨｺｨ ｨｦ ｨｮ04ｨｬ020807 (x 0 ; y 0 ) ﾊ D. 09ｨｪ lim f(x; y) = l (x;y) (x 0 ;y 0 ) 309ｨｪ0505070004 0809ｨｹ00ｨ ｨｮ04 ｨｦｨｮ ﾂ0502ｨｦ ｨｺｨ ｨｦ 00ｨｦｨ 0004ｨｪ 080209080800ｨｴｨｮ04 000701 0309ｨｦｨｮｨｬ0705 14.1.2-2 (0905060802 09ｨｦ0905ｨｦ070009ｨ 0208ｨ ).
13070500ｨｺ05ｨｦｨｮ04 ｨｮ01ｨｪｨ 090007ｨｮ02ｨｴｨｪ 010507 ｨｺｨ ｨｦ 0009ｨｦ06ｨｪ ｨｬ0200ｨ 0905040006ｨｪ 609 ｨｺｨ ｨｦ 01080509 ﾂ0701ｨｪ ｨｮ0007 R 07ｨｦ 0709ｨｦｨ ｨｺ06ｨｰ 00ｨｦｨｬ06ｨｰ lim f(x; y) ｨｺｨ ｨｦ lim f(x; y); x x 0 y y 0 00ｨｹ0002 [ ] lim f(x; y) = lim lim f(x; y) (x;y) (x 0 ;y 0 ) x x 0 y y 0 [ ] = lim lim f(x; y) y y 0 x x 0 = l: (14.1.2-3) 0807 ｨ ｨｪ0008ｨｮ0009070207 0102ｨｪ ｨｦｨｮ ﾂ0502ｨｦ 0805ｨｪ00070002, ｨｹ08ｨｴｨｰ ｨ 0100ｨｹ 080907ｨｺ05080002ｨｦ ｨ 08ｨｹ 0007 08ｨ 09ｨ ｨｺ0500ｨｴ 08ｨ 09050102ｨｦ00ｨｬｨ : 04ｨ 09050102ｨｦ00ｨｬｨ 14.1.2-1 00ｨｮ00ｨｴ 04 ｨｮ01ｨｪ05090004ｨｮ04 f(x; y) = x 6ﾓ1 y x + y ｨｬ02 0802010807 0709ｨｦｨｮｨｬｨｹ05 D = {(x; y) ﾊ R2 ｨｬ02 (x; y) ﾙ (0; 0)}: 08ｨｹ0002 02ｨｪ06 x 6ﾓ1 y lim f(x; y) = lim x 0 x 0 x + y = x 6ﾓ1 y lim f(x; y) = lim y 0 y 0 x + y = 61 6462 6463 61 6462 6463 0 6ﾓ1 y 0 + y = 6ﾓ11 ｨ ｨｪ y ﾙ 0 x 6ﾓ1 0 lim x 0 x + 0 = lim x x 0 x = 1 ｨ ｨｪ y = 0; x 6ﾓ1 0 x + 0 = 1 ｨ ｨｪ x ﾙ 0 0 6ﾓ1 y lim y 0 0 + y = lim 6ﾓ1y y 0 y = 6ﾓ11 ｨ ｨｪ x = 0; 0409ｨ [ ] lim lim f(x; y) = 1; ｨ ｨｪ0008ｨｮ0007ｨｦ ﾂｨ lim x 0 y 0 y 0 [ ] lim f(x; y) = 6ﾓ11; x 0 0708ｨｹ0002 ｨｮ05ｨｬ02ｨｴｨｪｨ ｨｬ02 0004ｨｪ 0409ｨｹ00ｨ ｨｮ04 14.1.2-1 0007 lim (x;y) (0;0) f(x; y) 0102ｨｪ 01080509 ﾂ02ｨｦ.
13610 0701ｨｪｨ 090007ｨｮ02ｨｦｨｰ 0807050506ｨｪ ｨｬ0200ｨ 0905040006ｨｪ 08ｨ ｨｨ. 09. 0008090500ｨｮ07ｨｰ 0704ｨｬ02ｨｦ06ｨｮ02ｨｦｨｰ 14.1.2-1 09ｨｪ05050700ｨ ｨｬ02 00ｨｦｨｰ ｨｦ01ｨｦｨｹ00040002ｨｰ 00ｨｴｨｪ 070908ｨｴｨｪ 00ｨｴｨｪ ｨｮ01ｨｪｨ 090007ｨｮ02ｨｴｨｪ ｨｬｨｦｨ ｨｰ ｨｬ0200ｨ 0905040007ｨｰ ｨｦｨｮ ﾂ0502ｨｦ ｨｹ00ｨｦ: 6ｦ1 0007 ｨｹ09ｨｦ07 0202ｨｹｨｮ07ｨｪ 01080509 ﾂ02ｨｦ, 0208ｨｪｨ ｨｦ ｨｬ07ｨｪｨ 01ｨｦｨｺｨｹ, 6ｦ1 0007 ｨｹ09ｨｦ07 000701 ｨ ｨｨ090708ｨｮｨｬｨ 0007ｨｰ, 0004ｨｰ 01ｨｦｨ 02070905ｨｰ ｨｺｨ ｨｦ 000701 00ｨｦｨｪ07ｨｬ06ｨｪ0701 ｨｦｨｮ070500ｨ ｨｦ ｨｬ02 0007 05ｨｨ0907ｨｦｨｮｨｬｨ 00ｨｴｨｪ 070908ｨｴｨｪ, 0004ｨｰ 01ｨｦｨ 02070905ｨｰ ｨｺｨ ｨｦ 000701 00ｨｦｨｪ07ｨｬ06ｨｪ0701. 04ｨｬ07ｨｦｨ 000701 08040508ｨｺ0701, ｨｹ00ｨ ｨｪ 0007 ｨｹ09ｨｦ07 000701 08ｨ 0907ｨｪ07ｨｬｨ ｨｮ0007 0208ｨｪｨ ｨｦ 01ｨｦ0502070907 000701 ｨｬ040102ｨｪｨｹｨｰ, ｨｦｨｮ070500ｨ ｨｦ ｨｬ02 0007 08040508ｨｺ07 00ｨｴｨｪ 070908ｨｴｨｪ. 04ｨｮｨｺ04ｨｮ04 01ｨ 010807050700ｨｦｨｮ000705ｨｪ 07ｨｦ 0709ｨｦｨ ｨｺ06ｨｰ 00ｨｦｨｬ06ｨｰ 00ｨｴｨｪ 08ｨ 09ｨ ｨｺ0500ｨｴ ｨｮ01ｨｪｨ 090007ｨｮ02ｨｴｨｪ ｨｮ0007 ｨｮ04ｨｬ020807 (0; 0) i) x 6ﾓ1 y 2 x + y 2 iv) x 6ﾓ1 2y x + y ii) xy xy v) x 3 6ﾓ1 xy 2 x 2 + y 2 iii) y x 2 + y 2 vi) (1 + y) sin2 x. x 0908ｨ ｨｪ0007ｨｮ02ｨｦｨｰ (i) lim x 0 f(x; y) = 6ﾓ11, lim y 0 f(x; y) = 1, (ii) lim x 0 f(x; y) = lim x 0 f(x; y) = 1, ｨｹ00ｨ ｨｪ 00ｨ x; y 07ｨｬｨｹｨｮ04ｨｬｨ ｨｺｨ ｨｦ 6ﾓ11, ｨｹ00ｨ ｨｪ 02000209ｨｹｨｮ04ｨｬｨ, (iii) lim x 0 f(x; y) = 1 y, lim y 0 f(x; y) = 0, (iv) lim x 0 f(x; y) = 6ﾓ12, lim y 0 f(x; y) = 1, (v) lim x 0 f(x; y) = 0, lim y 0 f(x; y) = x, (vi) lim x 0 f(x; y) = 0, lim y 0 f(x; y) = sin2 x x. 14.1.3 0701ｨｪ06 ﾂ02ｨｦｨ ｨｮ01ｨｪｨ 090007ｨｮ02ｨｴｨｪ 010507 ｨｺｨ ｨｦ 0009ｨｦ06ｨｪ ｨｬ0200ｨ 0905040006ｨｪ 09ｨｪ05050700ｨ ｨｬ02 0004ｨｪ 04ｨ 09050009ｨ 0207 14.1.2 0108ｨｪ0200ｨ ｨｦ ｨｺｨ ｨｦ ｨｮ0004ｨｪ 080209080800ｨｴｨｮ04 ｨ 010007 07 0709ｨｦｨｮｨｬｨｹｨｰ 0004ｨｰ ｨｮ01ｨｪ06 ﾂ02ｨｦｨ ｨｰ ｨｬｨｦｨ ｨｰ ｨｮ01ｨｪ05090004ｨｮ04ｨｰ 010507, ｨ ｨｪ0008ｨｮ0007ｨｦ ﾂｨ 0009ｨｦ06ｨｪ ｨｬ0200ｨ 0905040006ｨｪ.
130701ｨｪ06 ﾂ02ｨｦｨ ｨｮ01ｨｪｨ 090007ｨｮ02ｨｴｨｪ 010507 ｨｺｨ ｨｦ 0009ｨｦ06ｨｪ ｨｬ0200ｨ 0905040006ｨｪ 611 0309ｨｦｨｮｨｬｨｹｨｰ 14.1.3-1 (ｨｮ01ｨｪ06 ﾂ02ｨｦｨ ｨｰ). 0008ｨ ｨｮ01ｨｪ05090004ｨｮ04 f(x; y), ｨ ｨｪ0008ｨｮ0007ｨｦ ﾂｨ f(x; y; z) ｨｬ02 0802010807 0709ｨｦｨｮｨｬ0705, 06ｨｮ00ｨｴ D 6ﾛ7 R 2, ｨ ｨｪ0008ｨｮ0007ｨｦ ﾂｨ D 6ﾛ7 R 3, ｨｨｨ 0208ｨｪｨ ｨｦ ｨｮ01ｨｪ02 ﾂ07ｨｰ ｨｮ0007 ｨｮ04ｨｬ020807 (x 0 ; y 0 ) ﾊ D, ｨ ｨｪ0008ｨｮ0007ｨｦ ﾂｨ (x 0 ; y 0 ; z 0 ) ﾊ D 00ｨｹ0002 ｨｺｨ ｨｦ ｨｬｨｹｨｪ07ｨｪ, ｨｹ00ｨ ｨｪ lim f(x; y) = f (x 0; y 0 ) ; (x;y) (x 0 ;y 0 ) ｨ ｨｪ0008ｨｮ0007ｨｦ ﾂｨ lim f(x; y; z) = f (x 0; y 0 ; z 0 ) : (x;y;z) (x 0 ;y 0 ;z 0 ) 03ｨｦ 08ｨ 09ｨ 0805ｨｪｨｴ 0709ｨｦｨ ｨｺ06ｨｰ 00ｨｦｨｬ06ｨｰ 010807050700080307ｨｪ00ｨ ｨｦ ｨｮ05ｨｬ02ｨｴｨｪｨ ｨｬ02 000701ｨｰ 0309ｨｦｨｮｨｬ0705ｨｰ 14.1.2-1, ｨ ｨｪ0008ｨｮ0007ｨｦ ﾂｨ 14.1.2-2. 04ｨ 09050102ｨｦ00ｨｬｨ 14.1.3-1 05 ｨｮ01ｨｪ05090004ｨｮ04 61 6462 f(x; y) = 6463 x 2 y x 2 + y 2 ｨ ｨｪ (x; y) ﾙ (0; 0) 0 ｨ ｨｪ (x; y) = (0; 0) 0208ｨｪｨ ｨｦ ｨｮ01ｨｪ02 ﾂ07ｨｰ ｨｮ0007 (0; 0), 020802ｨｦ0107 ｨｬ02 ｨ ｨｪ050507000701ｨｰ 010807050700ｨｦｨｮｨｬ0705ｨｰ ｨｬ02 02ｨｺ0208ｨｪ0701ｨｰ 000701 04ｨ 09ｨ 01020800ｨｬｨ 0007ｨｰ 14.1.2-1 080907ｨｺ05080002ｨｦ ｨｹ00ｨｦ [ ] [ ] lim lim f(x; y) = 0; ｨ ｨｪ0008ｨｮ0007ｨｦ ﾂｨ lim lim f(x; y) = 0; x 0 y 0 y 0 x 0 0708ｨｹ0002 ｨｮ05ｨｬ02ｨｴｨｪｨ ｨｬ02 0004ｨｪ 0409ｨｹ00ｨ ｨｮ04 14.1.2-1 0208ｨｪｨ ｨｦ lim f(x; y) = 0; (x;y) (0;0) 010405ｨ 0107 01080509 ﾂ02ｨｦ 04 0709ｨｦｨ ｨｺ07 00ｨｦｨｬ07 ｨｺｨ ｨｦ ｨｦｨｮ070500ｨ ｨｦ ｨｬ02 0004ｨｪ 00ｨｦｨｬ07 0004ｨｰ ｨｮ01ｨｪ05090004ｨｮ04ｨｰ ｨｮ0007 ｨｮ04ｨｬ020807 ｨ 0100ｨｹ.
13612 0701ｨｪｨ 090007ｨｮ02ｨｦｨｰ 0807050506ｨｪ ｨｬ0200ｨ 0905040006ｨｪ 08ｨ ｨｨ. 09. 0008090500ｨｮ07ｨｰ 04ｨ 09050102ｨｦ00ｨｬｨ 14.1.3-2 05 ｨｮ01ｨｪ05090004ｨｮ04 61 6462 f(x; y) = 6463 x 2 x 2 + y 2 ｨ ｨｪ (x; y) ﾙ (0; 0) 0 ｨ ｨｪ (x; y) = (0; 0) 0102ｨｪ 0208ｨｪｨ ｨｦ ｨｮ01ｨｪ02 ﾂ07ｨｰ ｨｮ0007 (0; 0). 05 0505ｨｮ04, 080701 080907ｨｺ05080002ｨｦ ｨｬ02 010807050700ｨｦｨｮｨｬ0705ｨｰ ｨ ｨｪ050507000701ｨｰ 00ｨｴｨｪ 04ｨ 09ｨ 0102ｨｦ00ｨｬ0500ｨｴｨｪ 14.1.2-1 ｨｺｨ ｨｦ 14.1.3-1, ｨ 0207ｨｪ0200ｨ ｨｦ ｨｴｨｰ 05ｨｮｨｺ04ｨｮ04. 0701ｨｦｨｹ00040002ｨｰ ｨｮ01ｨｪ02 ﾂ06ｨｪ ｨｮ01ｨｪｨ 090007ｨｮ02ｨｴｨｪ 03ｨｦ 08ｨ 09ｨ ｨｺ0500ｨｴ 0809070005ｨｮ02ｨｦｨｰ 080701 ｨ ｨｪｨ 02060907ｨｪ00ｨ ｨｦ ｨｮ00ｨｦｨｰ ｨｦ01ｨｦｨｹ00040002ｨｰ 00ｨｴｨｪ ｨｮ01ｨｪ02 ﾂ06ｨｪ ｨｮ01ｨｪｨ 090007ｨｮ02ｨｴｨｪ 010507 ｨｬ0200ｨ 0905040006ｨｪ ｨ 08070002050705ｨｪ ｨｬｨｦｨ 0002ｨｪ08ｨｺ0201ｨｮ04 00ｨｴｨｪ ｨ ｨｪ0008ｨｮ0007ｨｦ ﾂｨｴｨｪ 000701 00ｨ ｨｨ07ｨｬｨ 0007ｨｰ 0701ｨｪ06 ﾂ02ｨｦｨ 0701ｨｪ05090004ｨｮ04ｨｰ, 080701 ｨ ｨｪｨ 0206090200ｨ ｨｦ ｨｮ02 ｨｮ01ｨｪｨ 090007ｨｮ02ｨｦｨｰ ｨｬｨｦｨ ｨｰ ｨｬ0200ｨ 0905040007ｨｰ. 0409ｨｹ00ｨ ｨｮ04 14.1.3-1. 09ｨｪ f; g D ｨｮ01ｨｪ02 ﾂ0208ｨｰ ｨｮ01ｨｪｨ 090007ｨｮ02ｨｦｨｰ ｨｮ0007 ｨｮ04ｨｬ020807 (x 0 ; y 0 ) ﾊ D, 00ｨｹ0002 ｨｺｨ ｨｦ 07ｨｦ ｨｮ01ｨｪｨ 090007ｨｮ02ｨｦｨｰ f ﾀg ｨｺｨ ｨｦ fg 0208ｨｪｨ ｨｦ ｨｮ01ｨｪ02 ﾂ0208ｨｰ ｨｮ0007 ｨｮ04ｨｬ020807 (x 0 ; y 0 ) ﾊ D. 0409ｨｹ00ｨ ｨｮ04 14.1.3-2. 09ｨｪ f; g D ｨｮ01ｨｪ02 ﾂ0208ｨｰ ｨｮ01ｨｪｨ 090007ｨｮ02ｨｦｨｰ ｨｮ0007 ｨｮ04ｨｬ020807 (x 0 ; y 0 ) ﾊ D ｨｺｨ ｨｦ f (x 0 ; y 0 ) ﾙ (0; 0), 00ｨｹ0002 01080509 ﾂ02ｨｦ 080209ｨｦ07 ﾂ07 $ (x 0 ; y 0 ), 00060007ｨｦｨ 06ｨｮ0002 f (x 0 ; y 0 ) ﾙ (0; 0) 00ｨｦｨ ｨｺ05ｨｨ02 x ﾊ $ (x 0 ; y 0 ), 0708ｨｹ0002 04 ｨｮ01ｨｪ05090004ｨｮ04 1=f 06 ﾂ02ｨｦ 06ｨｪｨｪ07ｨｦｨ 00ｨｦｨ ｨｺ05ｨｨ02 x ﾊ D ﾉ $ (x 0 ; y 0 ) ｨｺｨ ｨｦ 0208ｨｪｨ ｨｦ ｨｮ01ｨｪ02 ﾂ07ｨｰ ｨｮ0007 ｨｮ04ｨｬ020807 (x 0 ; y 0 ) ﾊ D. 0704ｨｬ02ｨｦ06ｨｮ02ｨｦｨｰ 14.1.3-1 6ｦ1 09ｨｪ0505070002ｨｰ 0809070005ｨｮ02ｨｦｨｰ ｨｦｨｮ ﾂ050701ｨｪ ｨｺｨ ｨｦ ｨｮ0004ｨｪ 080209080800ｨｴｨｮ04 ｨｮ01ｨｪｨ 090007ｨｮ02ｨｴｨｪ 0009ｨｦ06ｨｪ ｨｬ0200ｨ 0905040006ｨｪ. 6ｦ1 03ｨｦ 08070501ｨｴｨｪ01ｨｬｨｦｨｺ06ｨｰ ｨｺｨ ｨｦ 07ｨｦ 09040006ｨｰ ｨｮ01ｨｪｨ 090007ｨｮ02ｨｦｨｰ 0208ｨｪｨ ｨｦ ｨｮ01ｨｪ02 ﾂ0208ｨｰ ｨｮ01ｨｪｨ 090007ｨｮ02ｨｦｨｰ ｨｮ00ｨ 08020108ｨ 0709ｨｦｨｮｨｬ0705 00ｨｴｨｪ. 04ｨｬ07ｨｦｨ 07ｨｦ 02ｨｺｨｨ0200ｨｦｨｺ06ｨｰ, 0009ｨｦ00ｨｴｨｪ07ｨｬ020009ｨｦｨｺ06ｨｰ, 01080209090705ｨｦｨｺ06ｨｰ ｨｺｨ ｨｦ 07ｨｦ ｨ ｨｪ0008ｨｮ0009070202ｨｰ ｨ 010006ｨｪ ｨｮ01ｨｪｨ 090007ｨｮ02ｨｦｨｰ.
130309ｨｦｨｮｨｬｨｹｨｰ ｨｺｨ ｨｦ ｨｦ01ｨｦｨｹ00040002ｨｰ 613 04ｨｮｨｺ04ｨｮ04 01ｨ 02060200ｨ ｨｮ000705ｨｪ ｨｴｨｰ 080907ｨｰ 0004 ｨｮ01ｨｪ06 ﾂ02ｨｦｨ 07ｨｦ 08ｨ 09ｨ ｨｺ0500ｨｴ ｨｮ01ｨｪｨ 090007ｨｮ02ｨｦｨｰ: x i) sin(x + y) iv) x 2 + y 2 ii) ln ( x 2 + y 2 + z 2) v) iii) x + y x 6ﾓ1 y vi) x + y 1 6ﾓ1 cos x 1 x + y : 0908ｨ ｨｪ0007ｨｮ02ｨｦｨｰ (i) ｨｮ01ｨｪ02 ﾂ07ｨｰ ｨｮ0007 R 2, (ii) ｨｹｨｬ07ｨｦｨ, (iii) ｨｮ01ｨｪ02 ﾂ07ｨｰ ｨｮ0007 R 2 ｨｬ02 x ﾙ y, (iv) ｨｮ01ｨｪ02 ﾂ07ｨｰ ｨｮ0007 R 2, (v) ｨｮ01ｨｪ02 ﾂ07ｨｰ ｨｮ0007 R 2 ｨｬ02 x ﾙ k + 2 ; (vi) ｨｮ01ｨｪ02 ﾂ07ｨｰ ｨｮ0007 R 2 ｨｬ02 x ﾙ 6ﾓ1y. 14.2 000209ｨｦｨｺ07 08ｨ 090500ｨｴ0007ｨｰ 14.2.1 0309ｨｦｨｮｨｬ0708 03 00ｨｪｨｴｨｮ00ｨｹｨｰ 0709ｨｦｨｮｨｬｨｹｨｰ 0004ｨｰ 08ｨ 09ｨ 0006000701 ｨｮ01ｨｪ05090004ｨｮ04ｨｰ ｨｬｨｦｨ ｨｰ ｨｬ0200ｨ 0905040007ｨｰ 4 020802ｨｺ000208ｨｪ0200ｨ ｨｦ ｨｺｨ ｨｦ ｨｮ0004ｨｪ 080209080800ｨｴｨｮ04 ｨｬｨｦｨ ｨｰ ｨｮ01ｨｪ05090004ｨｮ04ｨｰ 010507, ｨ ｨｪ0008ｨｮ0007ｨｦ ﾂｨ 0009ｨｦ06ｨｪ ｨｬ0200ｨ 0905040006ｨｪ 00ｨｦｨ ｨｺ05ｨｨ02 ｨｬ0200ｨ 0905040007 ﾂｨｴ09ｨｦｨｮ0005 ｨｨ02ｨｴ0906ｨｪ00ｨ ｨｰ ｨｹ0502ｨｰ 00ｨｦｨｰ 05050502ｨｰ ｨｬ0200ｨ 0905040006ｨｰ ｨｴｨｰ ｨｮ00ｨ ｨｨ020906ｨｰ ｨｺｨ ｨｦ 0506000200ｨ ｨｦ ｨｬ0209ｨｦｨｺ07 08ｨ 090500ｨｴ0007ｨｰ 0004ｨｰ ｨｮ01ｨｪ05090004ｨｮ04ｨｰ ｨｴｨｰ 080907ｨｰ 0004 ｨｨ02ｨｴ090705ｨｬ02ｨｪ04 ｨｬ0200ｨ 0905040007. 070100ｨｺ02ｨｺ09ｨｦｨｬ06ｨｪｨ 06 ﾂ0701ｨｬ02: 4 0309ｨｦｨｮｨｬｨｹｨｰ 08ｨ 09ｨ 0006000701 ｨｮ01ｨｪ05090004ｨｮ04ｨｰ ｨｬｨｦｨ ｨｰ ｨｬ0200ｨ 0905040007ｨｰ: 06ｨｮ00ｨｴ 04 ｨｮ01ｨｪ05090004ｨｮ04 f D, ｨｹ080701 D 6ﾛ7 R ｨ ｨｪ07ｨｦｨｺ00ｨｹ 01ｨｦ05ｨｮ0004ｨｬｨ ｨｺｨ ｨｦ ｨｮ04ｨｬ020807 x 0 ﾊ D. 08ｨｹ0002 00ｨｦｨ ｨｺ05ｨｨ02 x ﾊ D6ﾓ1{x 0 } ｨｬ02 0007ｨｪ 00050807 f(x)6ﾓ1f(x 0) x6ﾓ1x 0 070908030200ｨ ｨｦ ｨｬ08ｨ ｨｮ01ｨｪ05090004ｨｮ04, 080701 0506000200ｨ ｨｦ 08040508ｨｺ07 01ｨｦｨ 02070906ｨｪ 07 ｨｺ0508ｨｮ04 0004ｨｰ f ｨｮ0007 ｨｮ04ｨｬ020807 x 0. 06ｨ 0506000200ｨ ｨｦ ｨｹ00ｨｦ 04 f 08ｨ 09ｨ 00ｨｴ0008030200ｨ ｨｦ ｨｮ0007 ｨｮ04ｨｬ020807 x 0 ﾊ D ｨｺｨ ｨｦ ｨｨｨ ｨｮ01ｨｬ09070508030200ｨ ｨｦ ｨ 0100ｨｹ ｨｬ02 f (x 0 ) 00ｨｹ0002 ｨｺｨ ｨｦ ｨｬｨｹｨｪ07ｨｪ, ｨｹ00ｨ ｨｪ 01080509 ﾂ02ｨｦ 04 0709ｨｦｨ ｨｺ07 00ｨｦｨｬ07: f f (x) 6ﾓ1 f (x 0 ) (x 0 ) = lim x x 0 x 6ﾓ1 x 0 = lim 02x 0 f (x 0 + 6ﾒ2x) 6ﾓ1 f (x 0 ) 6ﾒ2x f (x 0 + h) 6ﾓ1 f (x 0 ) = lim : h 0 h
13614 000209ｨｦｨｺ07 08ｨ 090500ｨｴ0007ｨｰ 08ｨ ｨｨ. 09. 0008090500ｨｮ07ｨｰ 0309ｨｦｨｮｨｬｨｹｨｰ 14.2.1-1 (ｨｬ0209ｨｦｨｺ07 08ｨ 090500ｨｴ0007ｨｰ). 00ｨｮ00ｨｴ ｨｬｨｦｨ ｨｮ01ｨｪ05090004ｨｮ04 f S ｨｹ080701 S ｨ ｨｪ07ｨｦｨｺ00ｨｹ 010807ｨｮ05ｨｪ070507 000701 R 2, ｨ ｨｪ0008ｨｮ0007ｨｦ ﾂｨ 000701 R 3 ｨｺｨ ｨｦ ｨｮ04ｨｬ020807 (x 0 ; y 0 ) ﾊ S, ｨ ｨｪ0008ｨｮ0007ｨｦ ﾂｨ (x 0 ; y 0 ; z 0 ) ﾊ S. 08ｨｹ0002 070908030200ｨ ｨｦ ｨｴｨｰ 104ｨｰ 00050604ｨｰ ｨｬ0209ｨｦｨｺ07 08ｨ 090500ｨｴ0007ｨｰ (partial derivative) 0004ｨｰ f ｨｴｨｰ 080907ｨｰ 0004 ｨｬ0200ｨ 0905040007 x ｨｮ0007 ｨｮ04ｨｬ020807 (x 0 ; y 0 ), ｨ ｨｪ0008ｨｮ0007ｨｦ ﾂｨ (x 0 ; y 0 ; z 0 ), 04 08ｨ 09ｨ ｨｺ0500ｨｴ 0709ｨｦｨ ｨｺ07, 0202ｨｹｨｮ07ｨｪ 01080509 ﾂ02ｨｦ, 00ｨｦｨｬ07: @f (x 0 ; y 0 ) @x = f x (x 0 ; y 0 ) = D x f (x 0 ; y 0 ) (14.2.1-1) f (x 0 + 6ﾒ2x; y 0 ) 6ﾓ1 f (x 0 ; y 0 ) = lim ; 6ﾒ2x 0 6ﾒ2x ｨ ｨｪ0008ｨｮ0007ｨｦ ﾂｨ @f (x 0 ; y 0 ; z 0 ) @x = f x (x 0 ; y 0 ; z 0 ) = D x f (x 0 ; y 0 ; z 0 ) (14.2.1-2) f (x 0 + 6ﾒ2x; y 0 ; z 0 ) 6ﾓ1 f (x 0 ; y 0 ; z 0 ) = lim : 6ﾒ2x 0 6ﾒ2x 04ｨ 09ｨ 00040907ｨｮ02ｨｦｨｰ 14.2.1-1 6ｦ1 05 0709ｨｦｨ ｨｺ07 00ｨｦｨｬ07 (14:2:1 6ﾓ1 1), ｨ ｨｪ0008ｨｮ0007ｨｦ ﾂｨ (14:2:1 6ﾓ1 2) 0208ｨｪｨ ｨｦ, ｨｹ08ｨｴｨｰ ｨｺｨ ｨｦ ｨｮ0004ｨｪ 080209080800ｨｴｨｮ04 0004ｨｰ ｨｬｨｦｨ ｨｰ ｨｬ0200ｨ 0905040007ｨｰ, 0809ｨ 00ｨｬｨ 00ｨｦｨｺｨｹｨｰ ｨ 09ｨｦｨｨｨｬｨｹｨｰ. 6ｦ1 0807 ｨｮ05ｨｬ09070507 (00020502ｨｮ0007ｨｰ) @ @x = @ x = D x 01040506ｨｪ02ｨｦ 104ｨｰ 00050604ｨｰ ｨｬ0209ｨｦｨｺ07 (partial) 08ｨ 090500ｨｴ0007 ｨｴｨｰ 080907ｨｰ 0004 ｨｬ0200ｨ 0905040007 07 ｨｮ01ｨｪｨｦｨｮ0006ｨｮｨ x, ｨｮ02 01ｨｦ05ｨｺ09ｨｦｨｮ04 ｨｬ02 0007ｨｪ 00ｨｪｨｴｨｮ00ｨｹ ｨｮ01ｨｬ090705ｨｦｨｮｨｬｨｹ 00ｨｦｨ ｨｬｨｦｨ ｨｬ0200ｨ 0905040007. D = D 1 = d dx 6ｦ1 04ｨｬ07ｨｦｨ 0709080307ｨｪ00ｨ ｨｦ 07ｨｦ ｨｬ0209ｨｦｨｺ06ｨｰ 08ｨ 090500ｨｴ0007ｨｦ ｨｴｨｰ 080907ｨｰ 00ｨｦｨｰ 05050502ｨｰ ｨｬ0200ｨ 0905040006ｨｰ.
130309ｨｦｨｮｨｬ0708 615 0704ｨｬ02ｨｦ06ｨｮ02ｨｦｨｰ 14.2.1-1 i) 09ｨｪ05050700ｨ ｨｬ02 0004ｨｪ 080209080800ｨｴｨｮ04 0004ｨｰ 08ｨ 09ｨ 0006000701 ｨｬｨｦｨ ｨｰ ｨｬ0200ｨ 0905040007ｨｰ 04 ｨｬ0209ｨｦｨｺ07 08ｨ 090500ｨｴ0007ｨｰ ｨｬｨｦｨ ｨｰ ｨｮ01ｨｪ05090004ｨｮ04ｨｰ, 06ｨｮ00ｨｴ f, ｨｴｨｰ 080907ｨｰ ｨｬｨｦｨ ｨｬ0200ｨ 0905040007 0004ｨｰ x ｨｮ02 06ｨｪｨ ｨｮ04ｨｬ020807 x 0, ｨｨｨ 0709080302ｨｦ 0007ｨｪ ｨｮ01ｨｪ00020502ｨｮ0007 ｨｬ0200ｨ 09070507ｨｰ 0004ｨｰ f ｨｮ0007 ｨｮ04ｨｬ020807 ｨ 0100ｨｹ ｨｺｨ 0005 0007ｨｪ x-050607ｨｪｨ ｨｺｨ ｨｦ 0002ｨｴｨｬ020009ｨｦｨｺ05 ｨｨｨ ｨｦｨｮ070500ｨ ｨｦ ｨｬ02 0004ｨｪ 0202ｨ 080007ｨｬ06ｨｪ04 0004ｨｰ 00ｨｴｨｪ08ｨ ｨｰ 07 01ｨｦｨ 0207090200ｨｦｨｺ05 ｨｬ02 0007ｨｪ ｨｮ01ｨｪ00020502ｨｮ0007 01ｨｦ0205ｨｨ01ｨｪｨｮ04ｨｰ 0004ｨｰ 0202ｨ 0800ｨｹｨｬ02ｨｪ04ｨｰ 0201ｨｨ0208ｨ ｨｰ 000701 01ｨｦｨ 000905ｨｬｨｬｨ 0007ｨｰ 0004ｨｰ ｨｮ01ｨｪ05090004ｨｮ04ｨｰ ｨｮ0007 ｨｮ04ｨｬ020807 (x 0 ; f (x 0 )). 04ｨｬ07ｨｦｨ 00ｨｦｨ 00ｨｦｨｰ 05050502ｨｰ ｨｬ0200ｨ 0905040006ｨｰ. ii) 03ｨｦ ｨｮ01ｨｪ00020502ｨｮ0006ｨｰ ｨｬ0200ｨ 09070507ｨｰ 00ｨｴｨｪ ｨｬ0200ｨ 0905040006ｨｪ ｨｮ0004ｨｪ 080209080800ｨｴｨｮ04 (i) 0208ｨｪｨ ｨｦ 0101ｨｪｨ 00ｨｹｨｪ ｨｪｨ 0208ｨｪｨ ｨｦ 01ｨｦｨ 0207090200ｨｦｨｺ0708 ｨｬ0200ｨ 0605 000701ｨｰ, 010405ｨ 0107 ｨｪｨ 06 ﾂ0701ｨｬ02 00ｨ ﾂ0500020904 ｨｬ0200ｨ 09070507 ｨｴｨｰ 080907ｨｰ x ｨｮ02 ｨｮ0500ｨｺ09ｨｦｨｮ04 ｨｬ02 0004 ｨｬ0200ｨ 09070507 ｨｴｨｰ 080907ｨｰ y, ｨｺ.0508. iii) 0408ｨｴｨｰ ｨｺｨ ｨｦ ｨｮ0004ｨｪ 080209080800ｨｴｨｮ04 0004ｨｰ 08ｨ 09ｨ 0006000701 ｨｮ01ｨｪ05090004ｨｮ04ｨｰ ｨｬｨｦｨ ｨｰ ｨｬ0200ｨ 0905040007ｨｰ ｨ ｨｪ 00ｨｦｨ 0004 ｨｬ0209ｨｦｨｺ07 08ｨ 090500ｨｴ0007 ｨｬｨｦｨ ｨｰ ｨｮ01ｨｪ05090004ｨｮ04ｨｰ, 06ｨｮ00ｨｴ 0004ｨｪ f x, ｨｦｨｮ ﾂ0502ｨｦ ｨｹ00ｨｦ: 6ｦ1 f x (x 0 ; f (x 0 )) = 0, 00ｨｹ0002 04 0202ｨ 080007ｨｬ06ｨｪ04 0201ｨｨ0208ｨ ｨｮ0007 ｨｮ04ｨｬ020807 (x 0 ; f (x 0 )) 0208ｨｪｨ ｨｦ 08ｨ 09050505040504 ｨｮ0004 01ｨｦ0205ｨｨ01ｨｪｨｮ04 000701 x-050607ｨｪｨ, 02ｨｪ06, ｨ ｨｪ 6ｦ1 f x (x 0 ; f (x 0 )) = + ﾞ, 00ｨｹ0002 04 0202ｨ 080007ｨｬ06ｨｪ04 0201ｨｨ0208ｨ ｨｮ0007 (x 0 ; f (x 0 )) 0208ｨｪｨ ｨｦ ｨｺ05ｨｨ020004 ｨｮ0007ｨｪ x-050607ｨｪｨ. 04ｨ 090500ｨｴ0007ｨｦ ｨ ｨｪ0600020904ｨｰ 00050604ｨｰ 0309ｨｦｨｮｨｬｨｹｨｰ 14.2.1-2 (ｨｬ0209ｨｦｨｺ07 08ｨ 090500ｨｴ0007ｨｰ). 00ｨｮ00ｨｴ ｨｬｨｦｨ ｨｮ01ｨｪ05090004ｨｮ04 f S ｨｹ080701 S ｨ ｨｪ07ｨｦｨｺ00ｨｹ 010807ｨｮ05ｨｪ070507 000701 R 2, ｨ ｨｪ0008ｨｮ0007ｨｦ ﾂｨ 000701 R 3 ｨｺｨ ｨｦ ｨｮ04ｨｬ020807 (x 0 ; y 0 ) ﾊ S, ｨ ｨｪ0008ｨｮ0007ｨｦ ﾂｨ (x 0 ; y 0 ; z 0 ) ﾊ S. 08ｨｹ0002, ｨ ｨｪ 04 104ｨｰ 00050604ｨｰ ｨｬ0209ｨｦｨｺ07 08ｨ 090500ｨｴ0007ｨｰ 0004ｨｰ f ｨｴｨｰ 080907ｨｰ 0004 ｨｬ0200ｨ 0905040007, 06ｨｮ00ｨｴ x, 01080509 ﾂ02ｨｦ 00ｨｦｨ ｨｺ05ｨｨ02 (x 0 ; y 0 ) ﾊ S, ｨ ｨｪ0008ｨｮ0007ｨｦ ﾂｨ (x 0 ; y 0 ; z 0 ) ﾊ S, 00ｨｹ0002 070908030200ｨ ｨｦ 04 ｨｬ0209ｨｦｨｺ07 08ｨ 090500ｨｴ0007ｨｰ f x ｨｮ0007 S. 09ｨｪ0505070007ｨｰ 0709ｨｦｨｮｨｬｨｹｨｰ ｨｦｨｮ ﾂ0502ｨｦ ｨｺｨ ｨｦ 00ｨｦｨ 00ｨｦｨｰ ｨｬ0200ｨ 0905040006ｨｰ y ｨｺｨ ｨｦ z. 00ｨｮ00ｨｴ 04 ｨｮ01ｨｪ05090004ｨｮ04 f S. 09ｨｪ 01080509 ﾂ02ｨｦ 04 104ｨｰ 00050604ｨｰ ｨｬ0209ｨｦｨｺ07 08ｨ 090500ｨｴ0007ｨｰ 0004ｨｰ f, 06ｨｮ00ｨｴ ｨｴｨｰ 080907ｨｰ x, 00ｨｹ0002 070908030200ｨ ｨｦ 04 204ｨｰ 00050604ｨｰ ｨｬ0209ｨｦｨｺ07 08ｨ 090500ｨｴ0007ｨｰ 0004ｨｰ f ｨｮ0007 x ｨｴｨｰ 020607ｨｰ: f x x = f 2x = @ 2 f @x 2 = @ @x ( ) @f ; (14.2.1-3) @x
13616 000209ｨｦｨｺ07 08ｨ 090500ｨｴ0007ｨｰ 08ｨ ｨｨ. 09. 0008090500ｨｮ07ｨｰ ｨｹ080701 ｨｹｨｬ07ｨｦｨ 0007 ｨｮ05ｨｬ09070507 @ 2 @x 2 = @ xx = @ 2x = D xx 01040506ｨｪ02ｨｦ 204ｨｰ 00050604ｨｰ ｨｬ0209ｨｦｨｺ07 08ｨ 090500ｨｴ0007 ｨｴｨｰ x. 04ｨｬ07ｨｦｨ 0709080307ｨｪ00ｨ ｨｦ 07ｨｦ 304ｨｰ, 404ｨｰ ｨｺｨ ｨｦ 0002ｨｪｨｦｨｺ05 04-00050604ｨｰ ｨｬ0209ｨｦｨｺ07 08ｨ 090500ｨｴ0007ｨｰ 0004ｨｰ f ｨｮ0007 x ｨｴｨｰ 020607ｨｰ: f x x x = f 3x = @ 3 f @x 3 = @ @x f x x x x = f 4x = @ 4 f @x 4 = @ @x f x = @ f @x = @ @x ( @ 6ﾓ11 f @x 6ﾓ11 ( @ 2 f @x 2 ( @ 3 f @x 3 ) ; ) ; ｨｺｨ ｨｦ 0002ｨｪｨｦｨｺ05 030808ｨｮ04ｨｰ 0709080307ｨｪ00ｨ ｨｦ 07ｨｦ 08ｨ 090500ｨｴ0007ｨｦ 00ｨｴｨｪ 08ｨ 09ｨ ｨｺ0500ｨｴ ｨｬ07090206ｨｪ f x y = @ 2 f @x @y = @ ( ) @f ; @x @y ) : (14.2.1-4) f x x y = @ 3 f @x 2 @y = @ @x 2 ( ) @f ; @y f x y y = @ 3 f @x @y 2 = @ @x ( @ 2 ) f @y 2 ; ｨｺ.0508. (14.2.1-5) 03ｨｦ 08ｨ 090500ｨｴ0007ｨｦ ｨ 010006ｨｰ 05060007ｨｪ00ｨ ｨｦ 0807050506ｨｰ 02070906ｨｰ ｨ ｨｪ05ｨｬ02ｨｦｨｺ0002ｨｰ 07 ｨｺｨ ｨｦ 0208050505040502ｨｰ. 04ｨ 09ｨ 00070904ｨｮ04 14.2.1-1 03ｨｦ 08ｨ 090500ｨｴ0007ｨｦ f x ; f xx ; : : : ; f x 0208ｨｪｨ ｨｦ ｨｮ01ｨｪｨ 090007ｨｮ02ｨｦｨｰ, 02ｨｪ06 07ｨｦ ｨ ｨｪ0008ｨｮ0007ｨｦ ﾂ02ｨｰ 08ｨ 090500ｨｴ000708 00ｨｴｨｪ ｨｮ0007 ｨｮ04ｨｬ020807 (x 0 ; y 0 ), ｨ ｨｪ0008ｨｮ0007ｨｦ ﾂｨ (x 0 ; y 0 ; z 0 ) 0208ｨｪｨ ｨｦ 0809ｨ 00ｨｬｨ 00ｨｦｨｺ0708 ｨ 09ｨｦｨｨｨｬ0708. 09ｨｪ0505070004 08ｨ 09ｨ 00070904ｨｮ04 ｨｦｨｮ ﾂ0502ｨｦ ｨｺｨ ｨｦ 00ｨｦｨ 00ｨｦｨｰ ｨｬ0200ｨ 0905040006ｨｰ y ｨｺｨ ｨｦ z. 09ｨｪ0505070004 08ｨ 09ｨ 00070904ｨｮ04 ｨｦｨｮ ﾂ0502ｨｦ 00ｨｦｨ 00ｨｦｨｰ 0208050505040502ｨｰ 08ｨ 09ｨ 0006000701ｨｰ.
13090807050700ｨｦｨｮｨｬｨｹｨｰ 08ｨ 09ｨ 000600ｨｴｨｪ 617 0704ｨｬ0208ｨｴｨｮ04 14.2.1-1 070004ｨｪ 080209080800ｨｴｨｮ04 00ｨｴｨｪ 02080505050405ｨｴｨｪ 08ｨ 09ｨ 000600ｨｴｨｪ 04 08ｨ 09ｨ 000600ｨｦｨｮ04 ｨ 09 ﾂ080302ｨｦ ｨ 08ｨｹ 0007ｨｪ 010206ｨｦｨｹ 010208ｨｺ0004, 010405ｨ 0107 ｨ ｨｪ 00ｨｦｨ 08ｨ 09050102ｨｦ00ｨｬｨ 030400020800ｨ ｨｦ 04 ｨｬ0209ｨｦｨｺ07 08ｨ 090500ｨｴ0007ｨｰ f xy, 00ｨｹ0002 04 ｨｮ02ｨｦ0905 08ｨ 09ｨ 000600ｨｦｨｮ04ｨｰ 0208ｨｪｨ ｨｦ: f y ｨｺｨ ｨｦ ｨｮ0004 ｨｮ01ｨｪ06 ﾂ02ｨｦｨ 04 08ｨ 090500ｨｴ00ｨｹｨｰ 0004ｨｰ f y ｨｴｨｰ 080907ｨｰ x, 010405ｨ 0107 f xy = (f y ) x : 14.2.2 090807050700ｨｦｨｮｨｬｨｹｨｰ 08ｨ 09ｨ 000600ｨｴｨｪ 08ｨ ｨｪｨｹｨｪ02ｨｰ 08ｨ 09ｨ 000600ｨｦｨｮ04ｨｰ 5 03ｨｦ 00ｨｪｨｴｨｮ000708 ｨｺｨ ｨｪｨｹｨｪ02ｨｰ 08ｨ 09ｨ 000600ｨｦｨｮ04ｨｰ 00ｨｴｨｪ ｨｮ01ｨｪｨ 090007ｨｮ02ｨｴｨｪ ｨｬｨｦｨ ｨｰ ｨｬ0200ｨ 0905040007ｨｰ ｨｦｨｮ ﾂ050701ｨｪ ｨｺｨ ｨｦ ｨｮ0004ｨｪ 080209080800ｨｴｨｮ04 0004ｨｰ ｨｬ0209ｨｦｨｺ07ｨｰ 08ｨ 09ｨ 0006000701. 080908ｨｪ0200ｨ ｨｦ ｨｮｨｺｨｹ08ｨｦｨｬ07 ｨｮ0007 ｨｮ04ｨｬ020807 ｨ 0100ｨｹ ｨｪｨ 0008ｨｪ02ｨｦ ｨｬｨｦｨ 010802ｨｪｨｨ05ｨｬｨｦｨｮ04 ｨｬ02 0004 ｨｬ07090207 0809070005ｨｮ02ｨｴｨｪ 00ｨｴｨｪ 08ｨ 09ｨ ｨｺ0500ｨｴ ｨｺｨ ｨｪｨｹｨｪｨｴｨｪ 08ｨ 09ｨ 000600ｨｦｨｮ04ｨｰ 00ｨｴｨｪ ｨｮ01ｨｪｨ 090007ｨｮ02ｨｴｨｪ ｨｬｨｦｨ ｨｰ ｨｬ0200ｨ 0905040007ｨｰ: 0409ｨｹ00ｨ ｨｮ04 (08ｨ 090500ｨｴ0007ｨｰ ｨｮ00ｨ ｨｨ020905ｨｰ ｨｮ01ｨｪ05090004ｨｮ04ｨｰ). 00ｨｮ00ｨｴ 04 ｨｮ01ｨｪ05090004ｨｮ04 f R ｨｹ080701 f(x) = c ｨｮ00ｨ ｨｨ020905 00ｨｦｨ ｨｺ05ｨｨ02 x ﾊ R. 08ｨｹ0002 f (x) = 0 00ｨｦｨ ｨｺ05ｨｨ02 x ﾊ R: 0409ｨｹ00ｨ ｨｮ04 (08ｨ 090500ｨｴ0007ｨｰ ｨ ｨｨ090708ｨｮｨｬｨ 0007ｨｰ). 00ｨｮ00ｨｴ ｨｹ00ｨｦ 07ｨｦ ｨｮ01ｨｪｨ 090007ｨｮ02ｨｦｨｰ f, g D 0208ｨｪｨ ｨｦ 08ｨ 09ｨ 00ｨｴ0008ｨｮｨｦｨｬ02ｨｰ ｨｮ0007 D. 08ｨｹ0002 ｨｦｨｮ ﾂ0502ｨｦ (f(x) + g(x)) = f (x) + g (x) 00ｨｦｨ ｨｺ05ｨｨ02 x ﾊ D: 05 ｨｦ01ｨｦｨｹ000400ｨ 0002ｨｪｨｦｨｺ02050200ｨ ｨｦ. 0409ｨｹ00ｨ ｨｮ04 (08ｨ 090500ｨｴ0007ｨｰ 00ｨｦｨｪ07ｨｬ06ｨｪ0701). 00ｨｮ00ｨｴ ｨｹ00ｨｦ 07ｨｦ ｨｮ01ｨｪｨ 090007ｨｮ02ｨｦｨｰ f; g D 0208ｨｪｨ ｨｦ 08ｨ 09ｨ 00ｨｴ0008ｨｮｨｦｨｬ02ｨｰ ｨｮ0007 D. 08ｨｹ0002 ｨｦｨｮ ﾂ0502ｨｦ (f(x)g(x)) = f (x)g(x) + f(x)g (x) 00ｨｦｨ ｨｺ05ｨｨ02 x ﾊ D: 50005060802 09. 0008090500ｨｮ07ｨｰ [1] 080202. 6.
13618 000209ｨｦｨｺ07 08ｨ 090500ｨｴ0007ｨｰ 08ｨ ｨｨ. 09. 0008090500ｨｮ07ｨｰ 04ｨｬ07ｨｦｨ 04 ｨｦ01ｨｦｨｹ000400ｨ 0002ｨｪｨｦｨｺ02050200ｨ ｨｦ. 030802ｨｦ0107 08090702ｨ ｨｪ06ｨｰ ｨｦｨｮ ﾂ0502ｨｦ (05f(x)) = 05f (x) ｨｬ02 05 ﾊ R ｨｮ00ｨ ｨｨ020905 ｨ 08ｨｹ 00ｨｦｨｰ 08ｨ 09ｨ 0805ｨｪｨｴ 0809070005ｨｮ02ｨｦｨｰ 080907ｨｺ05080002ｨｦ 000205ｨｦｨｺ05 04 08ｨ 09ｨ ｨｺ0500ｨｴ 0009ｨ ｨｬｨｬｨｦｨｺ07 ｨｦ01ｨｦｨｹ000400ｨ : 00ｨｦｨ ｨｺ05ｨｨ02 x ﾊ D ｨｺｨ ｨｦ k; 05 ﾊ R. (kf(x) + 05g(x)) = kf (x) + 05g (x) 0409ｨｹ00ｨ ｨｮ04. 09ｨｪ 04 ｨｮ01ｨｪ05090004ｨｮ04 f D 08ｨ 09ｨ 00ｨｴ0008030200ｨ ｨｦ ｨｮ0007 D ｨｺｨ ｨｦ 0208ｨｦ08050607ｨｪ 01080509 ﾂ02ｨｦ x 0 ﾊ D, 0600ｨｮｨｦ 06ｨｮ0002 f (x 0 ) ﾙ 0, 00ｨｹ0002 ( ) 1 = 6ﾓ1 f (x 0) f(x) x=x 0 f 2 (x) : 0409ｨｹ00ｨ ｨｮ04 (08ｨ 090500ｨｴ0007ｨｰ 08040508ｨｺ0701). 00ｨｮ00ｨｴ ｨｹ00ｨｦ 07ｨｦ ｨｮ01ｨｪｨ 090007ｨｮ02ｨｦｨｰ f; g D 0208ｨｪｨ ｨｦ 08ｨ 09ｨ 00ｨｴ0008ｨｮｨｦｨｬ02ｨｰ ｨｮ0007 D ｨｺｨ ｨｦ 0208ｨｦ08050607ｨｪ g (x) ﾙ 0 00ｨｦｨ ｨｺ05ｨｨ02 x ﾊ D. 08ｨｹ0002 ｨｦｨｮ ﾂ0502ｨｦ [ ] f(x) = f (x)g(x) 6ﾓ1 f(x)g (x) 00ｨｦｨ ｨｺ05ｨｨ02 x ﾊ D: g(x) g 2 (x) 04ｨ 090500ｨｴ0007ｨｰ ｨｮ05ｨｪｨｨ020004ｨｰ ｨｮ01ｨｪ05090004ｨｮ04ｨｰ 00ｨｮ00ｨｴ ｨｬｨｦｨ ｨｮ01ｨｪ05090004ｨｮ04 f 010507, ｨ ｨｪ0008ｨｮ0007ｨｦ ﾂｨ 0009ｨｦ06ｨｪ ｨｬ0200ｨ 0905040006ｨｪ. 09ｨｪ 04 f ｨｨ02ｨｴ0904ｨｨ0208 ｨｴｨｰ ｨｮ01ｨｪ05090004ｨｮ04 ｨｬｨｹｨｪ07ｨｪ 0004ｨｰ ｨｬ0200ｨ 0905040007ｨｰ x, 02ｨｪ06 07ｨｦ 05050502ｨｰ ｨｬ0200ｨ 0905040006ｨｰ ｨｴｨｰ ｨｮ00ｨ ｨｨ020906ｨｰ, 00ｨｹ0002 080907ｨｺ05080002ｨｦ 07 08ｨ 09ｨ ｨｺ0500ｨｴ ｨｺｨ ｨｪｨｹｨｪｨ ｨｰ 08ｨ 09ｨ 000600ｨｦｨｮ04ｨｰ ｨｮ05ｨｪｨｨ020004ｨｰ ｨｮ01ｨｪ05090004ｨｮ04ｨｰ: 0602060904ｨｬｨ 14.2.2-1 (08ｨ 090500ｨｴ0007ｨｰ ｨｮ05ｨｪｨｨ020004ｨｰ ｨｮ01ｨｪ05090004ｨｮ04ｨｰ). 00ｨｮ00ｨｴ 07ｨｦ ｨｮ01ｨｪｨ 090007ｨｮ02ｨｦｨｰ y = f(w) D 1 ｨｺｨ ｨｦ w = g(x) D 2 ｨｹ080701 g (D 2 ) 6ﾛ7 D 1 ｨｺｨ ｨｦ D 1, D 2 ｨ ｨｪ07ｨｦｨｺ0005 01ｨｦｨ ｨｮ0007ｨｬｨ 00ｨ ｨｺｨ ｨｦ 04 080907ｨｺ0508000701ｨｮｨ ｨｮ05ｨｪｨｨ020004 ｨｮ01ｨｪ05090004ｨｮ04 h(x) = (f 71 g) (x) = f(g(x)) 00ｨｦｨ ｨｺ05ｨｨ02 x ﾊ D 2 : 00ｨｮ00ｨｴ 020808ｨｮ04ｨｰ ｨｹ00ｨｦ 00ｨｦｨ 06ｨｪｨ ｨｮ04ｨｬ020807 x 0 ﾊ D 2 01080509 ﾂ02ｨｦ 04 08ｨ 090500ｨｴ0007ｨｰ g (x 0 ) = w 0 ｨｺｨ ｨｦ 04 ｨ ｨｪ0008ｨｮ0007ｨｦ ﾂ04 y 0 = f (w 0 ) ｨｮ0007 ｨｮ04ｨｬ020807 w 0 = g (x 0 ) ｨｬ02 w 0 ﾊ D 1. 08ｨｹ0002 01080509 ﾂ02ｨｦ ｨｺｨ ｨｦ 04 08ｨ 090500ｨｴ0007ｨｰ 0004ｨｰ ｨｮ05ｨｪｨｨ020004ｨｰ ｨｮ01ｨｪ05090004ｨｮ04ｨｰ h(x) D 2 ｨｮ0007 ｨｮ04ｨｬ020807 x 0 ﾊ D 2 ｨｺｨ ｨｦ ｨｦｨｮ ﾂ0502ｨｦ dh(x) dx ｨO = df(w) ｨO x = x0 dw ｨO w = w0 dg(x) dx ｨO = y 0 w 0: x = x0
13090807050700ｨｦｨｮｨｬｨｹｨｰ 08ｨ 09ｨ 000600ｨｴｨｪ 619 0807 ｨｨ02060904ｨｬｨ ｨ 0100ｨｹ, 080701 0208ｨｪｨ ｨｦ 00ｨｪｨｴｨｮ00ｨｹ ｨｴｨｰ ｨｺｨ ｨｪｨｹｨｪｨ ｨｰ ｨ 0501ｨｮｨｦ01ｨｴ0007ｨｰ 08ｨ 09ｨ 000600ｨｦｨｮ04ｨｰ (chain rule) 00ｨｦｨ ｨｮ01ｨｪｨ 090007ｨｮ02ｨｦｨｰ ｨｬｨｦｨ ｨｰ ｨｬ0200ｨ 0905040007ｨｰ, ｨｹｨｬ07ｨｦｨ 0202ｨ 09ｨｬｨｹ030200ｨ ｨｦ ｨｺｨ ｨｦ 00ｨｦｨ 00ｨｦｨｰ 05050502ｨｰ ｨｬ0200ｨ 0905040006ｨｰ. 08ｨｹ0002 ｨｮ05ｨｬ02ｨｴｨｪｨ ｨｬ02 0007 0602060904ｨｬｨ 14.2.2-1, ｨ ｨｪ 00ｨｦｨ ｨｺ05ｨｨ02 x ﾊ D 2 01080509 ﾂ02ｨｦ 04 08ｨ 090500ｨｴ0007ｨｰ g (x) ｨｺｨ ｨｦ 0208ｨｦ08050607ｨｪ ｨｹ00ｨｦ 00ｨｦｨ 0004ｨｪ ｨ ｨｪ0008ｨｮ0007ｨｦ ﾂ04 00ｨｦｨｬ07 g(x) = w ﾊ D 1 01080509 ﾂ02ｨｦ 04 f (w) = f (g(x)), ｨｨｨ 01080509 ﾂ02ｨｦ ｨｺｨ ｨｦ 04 08ｨ 090500ｨｴ0007ｨｰ 0004ｨｰ f(g(x)) ｨｴｨｰ 080907ｨｰ x 00ｨｦｨ ｨｺ05ｨｨ02 x ﾊ D 2 ｨｺｨ ｨｦ ｨｨｨ 0108ｨｪ0200ｨ ｨｦ ｨ 08ｨｹ 0004 ｨｮ ﾂ06ｨｮ04 dh(x) dx = df(g(x)) dx = df(g(x)) dg(x) dg(x) dx = f g g x: (14.2.2-1) 0002 0007ｨｪ 00050807 14.2.2-1 010807050700080307ｨｪ00ｨ ｨｦ 07ｨｦ 08ｨ 090500ｨｴ0007ｨｦ 00ｨｴｨｪ ｨｮ05ｨｪｨｨ0200ｨｴｨｪ ｨｮ01ｨｪｨ 090007ｨｮ02ｨｴｨｪ ｨｬｨｦｨ ｨｰ ｨｬ0200ｨ 0905040007ｨｰ, 06ｨｮ00ｨｴ x, 07ｨｦ ｨｺ0109ｨｦｨｹ00020902ｨｰ 00ｨｴｨｪ 07080708ｨｴｨｪ 0108ｨｪ07ｨｪ00ｨ ｨｦ ｨｮ0007ｨｪ 0408ｨｪｨ ｨｺｨ 14.2.2-1. 04ｨ 09050102ｨｦ00ｨｬｨ 14.2.2-1 01ｨ 010807050700ｨｦｨｮ000705ｨｪ 07ｨｦ 104ｨｰ 00050604ｨｰ ｨｬ0209ｨｦｨｺ06ｨｰ 08ｨ 090500ｨｴ0007ｨｦ 0004ｨｰ ｨｮ01ｨｪ05090004ｨｮ04ｨｰ 0905ｨｮ04. 02ｨｦｨ 0107 ﾂｨｦｨｺ05 06 ﾂ0701ｨｬ02 f(x; y) = x 4 + 4 ﾌ y 6ﾓ1 5 f x = ( ) x 4 + 4 y 1=2 6ﾓ1 5 = ( { x 4) ( }} ){ x x + 4 y 1=2 6ﾓ1 5 0 x = 4x 3 ; f y = 1 2 y 1 2 6ﾓ11 ( ) {}}{ 0 ( x 4 + 4 y 1=2 6ﾓ1 5 = 4 y 1=2) { + ( }}{ x 4 6ﾓ1 5 ) y y y = 2 y 6ﾓ11=2 :
13620 000209ｨｦｨｺ07 08ｨ 090500ｨｴ0007ｨｰ 08ｨ ｨｨ. 09. 0008090500ｨｮ07ｨｰ 0408ｨｪｨ ｨｺｨ ｨｰ 14.2.2-1: 08ｨ 09ｨ 000600ｨｴｨｪ 00ｨｴｨｪ ｨｺ0109ｨｦｨｹ000209ｨｴｨｪ ｨｮ05ｨｪｨｨ0200ｨｴｨｪ ｨｮ01ｨｪｨ 090007ｨｮ02ｨｴｨｪ ｨｬ02 ｨｬ0200ｨ 0905040007 x. ｨ / ｨ 0701ｨｪ05090004ｨｮ04 04ｨ 090500ｨｴ0007ｨｰ 1 f a (x) af (x)f a6ﾓ11 (x) 2 e f(x) f (x)e f(x) 3 ln f(x) f (x) f(x) 4 sin f(x) f (x) cos f(x) 5 cos f(x) 6ﾓ1f (x) sin f(x) f 6 (x) tan f(x) cos 2 f(x) 7 cot f(x) 6ﾓ1 f (x) sin 2 f(x) 8 tan 6ﾓ11 f (x) f(x) 1 + f 2 (x) 9 sin 6ﾓ11 f(x) f (x) ﾌ 1 6ﾓ1 f 2 (x) 10 cos 6ﾓ11 f (x) f(x) 6ﾓ1 ﾌ 1 6ﾓ1 f 2 (x) 11 sinh f(x) f (x) cosh f(x) 12 cosh f(x) f (x) sinh f(x) 13 tanh f(x) f (x) cosh 2 f(x) = f (x) [ 1 6ﾓ1 tanh 2 f(x) ] 14 coth f(x) 6ﾓ1 f (x) sinh 2 f(x) = f (x) [ 1 6ﾓ1 coth 2 f(x) ]
1304ｨ 09050102ｨｦ00ｨｬｨ 14.2.2-2 04ｨｬ07ｨｦｨ 0004ｨｰ ｨｮ01ｨｪ05090004ｨｮ04ｨｰ 090807050700ｨｦｨｮｨｬｨｹｨｰ 08ｨ 09ｨ 000600ｨｴｨｪ 621 h(s; t) = t 2 ln ( s 2 + 1 ) + 9 t 3 6ﾓ1 3 ﾌ s 4 : 0905ｨｮ04. 00 ﾂ0701ｨｬ02 h s = [ t 2 ln ( s 2 + 1 ) + 9 t 6ﾓ13 6ﾓ1 s 4=3] s = [ t 2 ln ( s 2 + 1 )] s + 9 0 {}}{ ( t 6ﾓ13 ) s 6ﾓ1 ( s 4=3) s 00050807ｨｰ 3 000701 0408ｨｪｨ ｨｺｨ 14.2.2-1 {}}{ = t 2 [ ( ln s 2 + 1 )] 6ﾓ1 4 s 3 s 4 3 6ﾓ11 2 s = t 2 1 {( }}{ s 2 s 2 + 1 ) + 1 s 6ﾓ14 3 s1=3 = 2 s t2 s 2 + 1 6ﾓ1 4 3 s1=3 ; h t = [ t 2 ln ( s 2 + 1 ) + 9 t 6ﾓ13 6ﾓ1 s 4=3] t = ln(s 2 +1) (t 2 ) t {[ }}{ t 2 ln ( s 2 + 1 )] 6ﾓ13 t 6ﾓ14 0 {}}{ t +9 ( t 6ﾓ13 ) {( }}{ s 6ﾓ1 s 4=3) s = 2 t ln ( s 2 + 1 ) 6ﾓ1 27 t 6ﾓ14 : 04ｨ 09050102ｨｦ00ｨｬｨ 14.2.2-3 04ｨｬ07ｨｦｨ 0004ｨｰ g(x; y; z) = x 2 y 6ﾓ1 y 2 z 3 + sin(xy):
13622 000209ｨｦｨｺ07 08ｨ 090500ｨｴ0007ｨｰ 08ｨ ｨｨ. 09. 0008090500ｨｮ07ｨｰ 0905ｨｮ04. 00 ﾂ0701ｨｬ02 g x = [ x 2 y 6ﾓ1 y 2 z 3 + sin(xy) ] x = y (x 2 ) x = y 2x {( }}{ x 2 y ) 0 (xy) {}}{ x cos(xy) ( 6ﾓ1 x y 2 z 3) {}}{ x + [sin(xy)] x = 2 xy + y(x) x = y {}}{ (xy) x cos(xy) = 2xy + y cos(xy); g y = [ x 2 y 6ﾓ1 y 2 z 3 + sin(xy) ] y = x 2 (y ) y = x 2 z 3 (y 2 ) y = z 3 (2y) ({}}{ x 2 y ) {( }}{ 6ﾓ1 y y 2 z 3) + [sin(xy)] y y = x 2 6ﾓ1 2 y z 3 + (xy) y cos(xy) = x 2 6ﾓ1 2 y z 3 + x cos(xy); g z = [ x 2 y 6ﾓ1 y 2 z 3 + sin(xy) ] z = 0 ({}}{ y2 (z3 ) x 2 y ) z = y2 (3z2 ) 0 {}}{ y 6ﾓ1 ( y 2 z 3) {}}{ + [sin(xy)] y z = 6ﾓ13y 2 z 2 : 04ｨ 09050102ｨｦ00ｨｬｨ 14.2.2-4 01ｨ 010807050700ｨｦｨｮ000705ｨｪ 07ｨｦ 104ｨｰ ｨｺｨ ｨｦ 07ｨｦ 204ｨｰ 00050604ｨｰ ｨｬ0209ｨｦｨｺ06ｨｰ 08ｨ 090500ｨｴ0007ｨｦ 0004ｨｰ ｨｮ01ｨｪ05090004ｨｮ04ｨｰ f(x; y; z) = x y e6ﾓ1x + z 2 :
130905ｨｮ04. 04ｨｬ07ｨｦｨ 01ｨｦｨ 0107 ﾂｨｦｨｺ05 06 ﾂ0701ｨｬ02 090807050700ｨｦｨｮｨｬｨｹｨｰ 08ｨ 09ｨ 000600ｨｴｨｪ 623 f x = ( ) x y e6ﾓ1x + z 2 x = ( ) x y e6ﾓ1x x 0 ({}}{ + z 2 ) x = 1 y ( x e 6ﾓ1x ) x = 1 y 65 1 66 {}}{ 67(x) x e 6ﾓ1x + x (6ﾓ1x) x e 6ﾓ1x = 6ﾓ1 e 6ﾓ1x ({}}{ e 6ﾓ1x ) x 68 69 60 = (1 6ﾓ1 x) e6ﾓ1x y ; [ ] (1 6ﾓ1 x) e 6ﾓ1x f xx = [f x (x; y; z)] x = y = 1 [ (1 6ﾓ1 x) e 6ﾓ1x ] y x = 1 y x 65 68 6ﾓ11 6ﾓ1e 66 {}}{ 6ﾓ1x 67 (1 6ﾓ1 x) x e 6ﾓ1x ({}}{ + (1 6ﾓ1 x) e 6ﾓ1x ) 69 x 60 = (x 6ﾓ1 2) e6ﾓ1x y ; f y = ( ) x y e6ﾓ1x + z 2 y = ( ) x y e6ﾓ1x y 0 {( }}{ + z 2 ) y = ( 6ﾓ1y6ﾓ116ﾓ11 = 6ﾓ1y6ﾓ12 x e 6ﾓ1x) {}}{ = 6ﾓ1x y 6ﾓ12 e 6ﾓ1x ; ( y 6ﾓ11 ) y
13624 000209ｨｦｨｺ07 08ｨ 090500ｨｴ0007ｨｰ 08ｨ ｨｨ. 09. 0008090500ｨｮ07ｨｰ f yy = ( 6ﾓ12 y 6ﾓ1x e 6ﾓ1x y 6ﾓ12) { 6ﾓ13 ( }}{ = 6ﾓ1x e6ﾓ1x y y 6ﾓ12 ) y = 2 x y 6ﾓ13 e 6ﾓ1x ; f z = ( ) x y e6ﾓ1x + z 2 z 0 {( }} ){ x = y e6ﾓ1x + ( z 2) z = 2z; z f zz = (2z) z = 2; f xy = (f y ) x 04 f y 06 ﾂ02ｨｦ 070104 010807050700ｨｦｨｮ000208 08ｨ 09ｨ 0805ｨｪｨｴ = ( 6ﾓ1x e 6ﾓ1x y 6ﾓ12) x = 6ﾓ1y 6ﾓ12 ( x e 6ﾓ1x) x = 6ﾓ1 1 y 2 65 68 1 6ﾓ1e 66 {}}{{ 6ﾓ1x 67 (x) x e 6ﾓ1x ( }}{ + x e 6ﾓ1x ) 69 x60 = (x 6ﾓ1 1) e6ﾓ1x y 2 ; f yx = (f x ) y 04 f x 06 ﾂ02ｨｦ 070104 ｨｹｨｬ07ｨｦｨ 010807050700ｨｦｨｮ000208 = [ ] (1 6ﾓ1 x) e 6ﾓ1x y y = (1 6ﾓ1 x) e 6ﾓ1x ( 1 y ( = (1 6ﾓ1 x) e 6ﾓ1x 6ﾓ1 1 ) (x 6ﾓ1 1) e6ﾓ1x y 2 = y 2 ; ｨｺｨ ｨｦ ｨｹｨｬ07ｨｦｨ f yz = f zy = 0; ) y f xz = f zx = 0:
13090807050700ｨｦｨｮｨｬｨｹｨｰ 08ｨ 09ｨ 000600ｨｴｨｪ 625 0704ｨｬ0208ｨｴｨｮ04 14.2.2-1 0908ｨｹ 0007 04ｨ 09050102ｨｦ00ｨｬｨ 14.2.2-4 080907ｨｺ05080002ｨｦ ｨｹ00ｨｦ f xy = f yx ; f yz = f zy ｨｺｨ ｨｦ f xz = f zx ; 010405ｨ 0107 07ｨｦ ｨ ｨｪ05ｨｬ02ｨｦｨｺ0002ｨｰ 08ｨ 090500ｨｴ0007ｨｦ 204ｨｰ 00050604ｨｰ 00ｨｴｨｪ 0801ｨｦｨｴｨｪ ｨ ｨｪ05 010507 ｨｬ0200ｨ 0905040006ｨｪ 0208ｨｪｨ ｨｦ 08ｨｮ02ｨｰ. 07 ﾂ0200ｨｦｨｺ05 ｨｦｨｮ ﾂ0502ｨｦ 0007 08ｨ 09ｨ ｨｺ0500ｨｴ ｨｨ02060904ｨｬｨ : 0602060904ｨｬｨ 14.2.2-2 (Schwarz). 00ｨｮ00ｨｴ 04 ｨｮ01ｨｪ05090004ｨｮ04 f(x; y) 6ﾛ7 R 2, ｨｹ080701 S ｨ ｨｪ07ｨｦｨｺ00ｨｹ ｨｮ05ｨｪ070507, 0004ｨｰ 07080708ｨ ｨｰ 01080509 ﾂ0701ｨｪ 07ｨｦ 204ｨｰ 00050604ｨｰ ｨｬ0209ｨｦｨｺ06ｨｰ 08ｨ 090500ｨｴ0007ｨｦ ｨｺｨ ｨｦ 0208ｨｪｨ ｨｦ ｨｮ01ｨｪ02 ﾂ0208ｨｰ ｨｮ0007 S. 08ｨｹ0002 f xy = f yx 00ｨｦｨ ｨｺ05ｨｨ02 (x; y) ﾊ S: (14.2.2-2) 0704ｨｬ0208ｨｴｨｮ04 14.2.2-2 0807 08ｨ 09ｨ 0805ｨｪｨｴ ｨｨ02060904ｨｬｨ, 080701 0208ｨｪｨ ｨｦ 020808ｨｮ04ｨｰ 00ｨｪｨｴｨｮ00ｨｹ ｨｺｨ ｨｦ ｨｴｨｰ ｨｨ02060904ｨｬｨ 00ｨｴｨｪ Schwarz- Clairaut, 0002ｨｪｨｦｨｺ02050200ｨ ｨｦ 00ｨｦｨ 000902ｨｦｨｰ ｨｺｨ ｨｦ 080209ｨｦｨｮｨｮｨｹ00020902ｨｰ ｨｬ0200ｨ 0905040006ｨｰ. 04ｨ 09050102ｨｦ00ｨｬｨ 14.2.2-5 00ｨｮ00ｨｴ 04 ｨｮ01ｨｪ05090004ｨｮ04 f(x; y) = y x + y : 01ｨ 010807050700ｨｦｨｮ000208 04 00ｨｦｨｬ07 f xyy (1;0). 0905ｨｮ04. 0909 ﾂｨｦｨｺ05 ｨ 08ｨｹ 0004 0704ｨｬ0208ｨｴｨｮ04 14.2.1-1, ｨｮ05ｨｬ02ｨｴｨｪｨ ｨｬ02 0004ｨｪ 07080708ｨ ｨｮ0004ｨｪ 080209080800ｨｴｨｮ04 00ｨｴｨｪ 02080505050405ｨｴｨｪ 08ｨ 09ｨ 000600ｨｴｨｪ 04 08ｨ 09ｨ 000600ｨｦｨｮ04 ｨ 09 ﾂ080302ｨｦ ｨ 08ｨｹ 0007ｨｪ 010206ｨｦｨｹ 010208ｨｺ0004, 0208ｨｪｨ ｨｦ f xyy = @3 f(x; y; z) @x @y 2 = @ @x 090807050700ｨｦｨｮｨｬｨｹｨｰ 0004ｨｰ f yy : 01ｨｦｨ 0107 ﾂｨｦｨｺ05 06 ﾂ0701ｨｬ02 ( ) @f(x; y; z) @y 2 = (f yy ) x : (1) f y = ( ) y = x + y y 1 {}}{ (y) y (x + y) 6ﾓ1 y 0+1 {}}{ (x + y) y (x + y) 2 = x (x + y) 2 ;
13626 000209ｨｦｨｺ07 08ｨ 090500ｨｴ0007ｨｰ 08ｨ ｨｨ. 09. 0008090500ｨｮ07ｨｰ f yy = [ x ] (x + y) 2 y = x [ (x + y) 6ﾓ12] y 65 68 1 {}}{ = x 676ﾓ12 (x + y) y (x + y) 6ﾓ126ﾓ11 60 0708ｨｹ0002 ｨ ｨｪ00ｨｦｨｺｨ ｨｨｨｦｨｮ0006ｨｪ00ｨ ｨｰ ｨｮ0004ｨｪ (1) 06 ﾂ0701ｨｬ02 = x [ 6ﾓ12(x + y) 6ﾓ13] = 6ﾓ1 2 x (x + y) 3 ; f xyy = [ 6ﾓ1 2 x ] [ (x + y) 3 = 6ﾓ12 x x (x + y) 3 ] x = 6ﾓ12 1 3(x+y) x (x+y) 36ﾓ11 {}}{ (x) x (x + y) 3 [{}}{ 6ﾓ1 x (x + y) 3 ] x (x + y) 6 = 6ﾓ12 (x + y)3 6ﾓ1 x [ 3(x + y) 2] 2(2x 6ﾓ1 y) (x + y) 6 = (x + y) 4 : 0409ｨ f xyy (1;0) = 2(2x 6ﾓ1 y) (x + y) 4 ｨO = (1;0) 2(2 1 6ﾓ1 0) (1 + 0) 4 = 4: 04ｨ 09050102ｨｦ00ｨｬｨ 14.2.2-6 00ｨｮ00ｨｴ 04 ｨｮ01ｨｪ05090004ｨｮ04 f(x; y; z) = ( x 2 + y 2 + z 2) 6ﾓ11=2. 020208060002 ｨｹ00ｨｦ 6 f xx + f yy + f zz = 0: (14.2.2-3) 605 020608ｨｮｨｴｨｮ04 (14:2:2 6ﾓ1 3), 080701 0208ｨｪｨ ｨｦ 00ｨｪｨｴｨｮ0007 ｨｴｨｰ 04 020608ｨｮｨｴｨｮ04 000701 Laplace (Laplace equation), 06 ﾂ02ｨｦ ｨｮ04ｨｬｨ ｨｪ00ｨｦｨｺ06ｨｰ 0202ｨ 09ｨｬ070006ｨｰ ｨｮ00ｨ 0302ｨ 09ｨｬ07ｨｮｨｬ06ｨｪｨ 00ｨ ｨｨ04ｨｬｨ 00ｨｦｨｺ05 (0905060802 09ｨｦ0905ｨｦ070009ｨ 0208ｨ ｨｺｨ ｨｦ 09. 0008090500ｨｮ07ｨｰ [2] 080202. 4-0306ｨｦｨｮ06ｨｮ02ｨｦｨｰ Maxwell). 05 ｨｮ01ｨｪ05090004ｨｮ04 f, 080701 0208ｨ 0504ｨｨ020502ｨｦ 0004ｨｪ (14:2:2 6ﾓ1 3), 0506000200ｨ ｨｦ 00ｨｹ0002 ｨｺｨ ｨｦ ｨ 09ｨｬ07ｨｪｨｦｨｺ07 ｨｮ01ｨｪ05090004ｨｮ04.
13090807050700ｨｦｨｮｨｬｨｹｨｰ 08ｨ 09ｨ 000600ｨｴｨｪ 627 0905ｨｮ04. 00 ﾂ0701ｨｬ02 f x = [ (x 2 + y 2 + z 2) 6ﾓ11=2 ] x (1) = 6ﾓ1 1 2 ( x 2 + y 2 + z 2) 6ﾓ1 1 2 6ﾓ11 2x {}}{ ( x 2 + y 2 + z 2) x = 6ﾓ1x ( x 2 + y 2 + z 2) 6ﾓ13=2 ; 65 68 1 {}}{ ( f xx = 6ﾓ1 67(x) x x 2 + y 2 + z 2) [ 6ﾓ13=2 (x 60 6ﾓ1 x 2 + y 2 + z 2) ] 6ﾓ13=2 x = 6ﾓ1 ( x 2 + y 2 + z 2) 6ﾓ13=2 65 68 2x 66 6ﾓ1x 676ﾓ1 3 ( x 2 + y 2 + z 2) {}}{ 6ﾓ1 3 2 6ﾓ11 ( x 2 + y 2 + z 2) 69 2 x60 = 6ﾓ1 ( x 2 + y 2 + z 2) 6ﾓ13=2 + 3 2 x2 ( x 2 + y 2 + z 2) 6ﾓ15=2 : (1) 09ｨｹ00ｨｴ 0004ｨｰ ｨｮ01ｨｬｨｬ02000908ｨ ｨｰ 0004ｨｰ f ｨｹｨｬ07ｨｦｨ 06 ﾂ0701ｨｬ02 f xx = 6ﾓ1 ( x 2 + y 2 + z 2) 6ﾓ13=2 + 3 2 y2 ( x 2 + y 2 + z 2) 6ﾓ15=2 ; (2) f xx = 6ﾓ1 ( x 2 + y 2 + z 2) 6ﾓ13=2 + 3 2 z2 ( x 2 + y 2 + z 2) 6ﾓ15=2 : (3) 040907ｨｮｨｨ060007ｨｪ00ｨ ｨｰ ｨｺｨ 0005 ｨｬ060504 00ｨｦｨｰ (1), (2) ｨｺｨ ｨｦ (3) 080907ｨｺ05080002ｨｦ 000205ｨｦｨｺ05 04 (14:2:2 6ﾓ1 3). 09ｨｮｨｺ07ｨｮ02ｨｦｨｰ 1. 01ｨ 010807050700ｨｦｨｮ000705ｨｪ ｨｹ0502ｨｰ 07ｨｦ 104ｨｰ ｨｺｨ ｨｦ 204ｨｰ 00050604ｨｰ ｨｬ0209ｨｦｨｺ06ｨｰ 08ｨ 090500ｨｴ0007ｨｦ 00ｨｴｨｪ 08ｨ 09ｨ ｨｺ0500ｨｴ ｨｮ01ｨｪｨ 090007ｨｮ02ｨｴｨｪ:
13628 000209ｨｦｨｺ07 08ｨ 090500ｨｴ0007ｨｰ 08ｨ ｨｨ. 09. 0008090500ｨｮ07ｨｰ i) ﾌ x 2 + y 2 v) x x 2 + y 2 ii) e 6ﾓ1x2 6ﾓ1y 2 vi) ( ) x ln y iii) sin 2 (x 6ﾓ1 y) vii) ln ( x 2 6ﾓ1 y 2) iv) x x + y viii) x y + z 2. 09ｨｪ f(x; y; z) = ln(xy + z), ｨｪｨ 010807050700ｨｦｨｮ000705ｨｪ 07ｨｦ 08ｨ 090500ｨｴ0007ｨｦ f x ; f y ｨｺｨ ｨｦ f z ｨｮ0007 ｨｮ04ｨｬ020807 P (1; 2; 0). 3. 020208060002 ｨｹ00ｨｦ 04 ｨｮ01ｨｪ05090004ｨｮ04 f(x; y) = e x sin y 0208ｨｪｨ ｨｦ ｨ 09ｨｬ07ｨｪｨｦｨｺ07. 4. 020208060002 ｨｹ00ｨｦ, ｨ ｨｪ f(x; y) = ln ( x 2 + xy + y 2), 00ｨｹ0002 x f x + y f y = 2: 5. 04ｨｬ07ｨｦｨ, ｨ ｨｪ f(x; y; z) = x + x 6ﾓ1 y y 6ﾓ1 z ; 00ｨｹ0002 f x + f y + f z = 1: 6. 09ｨｪ x = r cos ｨｺｨ ｨｦ y = r sin, ｨｪｨ 010807050700ｨｦｨｮ000208 04 00ｨｦｨｬ07 0004ｨｰ 070908030701ｨｮｨ ｨｰ A = ｨO x r y r 7. 0807 02ｨｬ09ｨ 01ｨｹｨｪ E 000701 0009ｨ 080203080701 ｨｬ02 0905ｨｮ02ｨｦｨｰ a, b ｨｺｨ ｨｦ 050307ｨｰ h 0108ｨｪ0200ｨ ｨｦ ｨ 08ｨｹ 0007ｨｪ 00050807 x y ｨO : E = 1 (a + b)h: 2 01ｨ 010807050700ｨｦｨｮ000705ｨｪ 07ｨｦ 08ｨ 090500ｨｴ0007ｨｦ E a, E b ｨｺｨ ｨｦ E h ｨｺｨ ｨｦ ｨｮ0004 ｨｮ01ｨｪ06 ﾂ02ｨｦｨ ｨｪｨ 0107ｨｨ0208 04 0002ｨｴｨｬ020009ｨｦｨｺ07 0209ｨｬ04ｨｪ0208ｨ 000701ｨｰ.
130302ｨ 0800ｨｹｨｬ02ｨｪ07 02080808020107 629 0908ｨ ｨｪ0007ｨｮ02ｨｦｨｰ 1. (i) f x = xy ﾌ x, f x 2 +y 2 y = y ﾌ x 2 +y 2, f xx = y 2 (x 2 +y 2 ) 3=2, f yy = x 2 (x 2 +y 2 ) 3=2, f xy = 6ﾓ1. (x 2 +y 2 ) 3=2 (ii) ｨｮ01ｨｬｨｬ020009ｨｦｨｺ07 f x = 6ﾓ12xe 6ﾓ1x2 6ﾓ1y 2, f xx = 2 ( 2x 2 6ﾓ1 1 ) e 6ﾓ1x2 6ﾓ1y 2, f xy = 4xye 6ﾓ1x2 6ﾓ1y 2 ｨｺ.0508. (iii) f x = sin 2(x 6ﾓ1 y), f y = 6ﾓ1 sin 2(x 6ﾓ1 y), f xx = f yy = 2 cos 2(x 6ﾓ1 y), f xy = 6ﾓ12 cos 2(x 6ﾓ1 y). (iv) f x = y (x+y) 2 ; f y = 6ﾓ1 x (x+y) 2 ; f xx = 6ﾓ1 2y (x+y) 3 ; f yy = 2x (x+y) 3 ; f xy = x6ﾓ1y (x+y) 3 : (v) f x = y2 6ﾓ1x 2 ; f (x 2 +y 2 ) 2 y = 6ﾓ1 2xy ; (x 2 +y 2 ) 2 f xx = 2x(x2 6ﾓ13y2 ) ; (x 2 +y 2 ) 3 f yy = 6ﾓ1 2x(x2 6ﾓ13y2 ) ; (x 2 +y 2 ) 3 f xy = 2y(3x2 6ﾓ1y 2 ) : (x 2 +y 2 ) 3 (vi) f x = 1 x ; fy = 6ﾓ1 1 x ; fxx = 6ﾓ1 1 x 2 ; f yy = 1 y 2 ; f xy = 0. (vii) f x = 2x x 2 6ﾓ1y ; f 2 y = 6ﾓ1 2y x 2 6ﾓ1y ; f 2 xx = f yy = 6ﾓ1 2(x2 +y 2 ) ; f (x 2 6ﾓ1y 2 ) 2 xy = 4xy : (x 2 6ﾓ1y 2 ) 2 (viii) f x = 1 x y+z ; fy = fz = 6ﾓ1 ; f (y+z) 2 xx = 0, f yy = f zz = 2x ;, f (y+z) 3 xy = f xz = 6ﾓ1 1 ; f (y+z) 2 yz = 2x : (y+z) 3 2. f x (P ) = 1, f y (P ) = f z (P ) = 1 2 : 3-5. 04090702ｨ ｨｪ0208ｨｰ. 6. A = r. 7. E a = 1 2 bh ｨｺ.0508. 14.2.3 0302ｨ 0800ｨｹｨｬ02ｨｪ07 02080808020107 0308ｨｪｨ ｨｦ 070104 00ｨｪｨｴｨｮ00ｨｹ ｨｮ0007ｨｪ ｨ ｨｪｨ 00ｨｪ06ｨｮ0004 ｨｹ00ｨｦ 0002ｨｴｨｬ020009ｨｦｨｺ05 04 08ｨ 090500ｨｴ0007ｨｰ ｨｬｨｦｨ ｨｰ ｨｮ01ｨｪ05090004ｨｮ04ｨｰ ｨｬｨｦｨ ｨｰ ｨｬ0200ｨ 0905040007ｨｰ, 7 06ｨｮ00ｨｴ f, ｨｮ02 06ｨｪｨ ｨｮ04ｨｬ020807 x 0 000701 080201080701 0709ｨｦｨｮｨｬ0705 0004ｨｰ ｨｦｨｮ070500ｨ ｨｦ ｨｬ02 0004ｨｪ 0202ｨ 080007ｨｬ06ｨｪ04 0004ｨｰ 00ｨｴｨｪ08ｨ ｨｰ 07 01ｨｦｨ 0207090200ｨｦｨｺ05 ｨｬ02 0007ｨｪ ｨｮ01ｨｪ00020502ｨｮ0007 01ｨｦ0205ｨｨ01ｨｪｨｮ04ｨｰ 0004ｨｰ 0202ｨ 0800ｨｹｨｬ02ｨｪ04ｨｰ 0201ｨｨ0208ｨ ｨｰ 000701 01ｨｦｨ 000905ｨｬｨｬｨ 0007ｨｰ 0004ｨｰ ｨｮ01ｨｪ05090004ｨｮ04ｨｰ ｨｮ0007 ｨｮ04ｨｬ020807 (x 0 ; f (x 0 )). 08ｨｹ0002 04 020608ｨｮｨｴｨｮ04 0004ｨｰ 0202ｨ 0800ｨｹｨｬ02ｨｪ04ｨｰ 0201ｨｨ0208ｨ ｨｰ ｨｮ0007 ｨｮ04ｨｬ020807 ｨ 0100ｨｹ 0108ｨｪ0200ｨ ｨｦ ｨ 08ｨｹ 0007ｨｪ 00050807 y 6ﾓ1 f (x 0 ) = f (x 0 ) (x 6ﾓ1 x 0 ) : 030802ｨｺ000208ｨｪ07ｨｪ00ｨ ｨｰ 0004ｨｪ 08ｨ 09ｨ 0805ｨｪｨｴ 0002ｨｴｨｬ020009ｨｦｨｺ07 0209ｨｬ04ｨｪ0208ｨ ｨｨ02ｨｴ090705ｨｬ02 ｨｬｨｦｨ ｨｮ01ｨｪ05090004ｨｮ04 010507 ｨｬ0200ｨ 0905040006ｨｪ, 06ｨｮ00ｨｴ z = f(x; y), ｨｬ02 0802010807 0709ｨｦｨｮｨｬ0705 0007 D 6ﾛ7 R 2 ｨｺｨ ｨｦ ｨｮ04ｨｬ020807 z 0 = (x 0 ; y 0 ) ﾊ D ｨｮ0007 0708070807 ｨｪｨ 01080509 ﾂ0701ｨｪ 07ｨｦ ｨｬ0209ｨｦｨｺ06ｨｰ 08ｨ 090500ｨｴ0007ｨｦ f x (x 0 ; y 0 ) ｨｺｨ ｨｦ f y (x 0 ; y 0 ). 08ｨｹ0002 ｨｨ02ｨｴ0906ｨｪ00ｨ ｨｰ 0007 y ｨｮ00ｨ ｨｨ0209ｨｹ, 04 f x ｨｨｨ 0208ｨｪｨ ｨｦ ｨｬｨｦｨ ｨｮ01ｨｪ05090004ｨｮ04 70005060802 0005ｨｨ04ｨｬｨ 04ｨ 090500ｨｴ0007ｨｰ 0701ｨｪ05090004ｨｮ04ｨｰ - 0102ｨｴｨｬ020009ｨｦｨｺ07 ｨｮ04ｨｬｨ ｨｮ08ｨ 08ｨ 09ｨ 0006000701.
13630 000209ｨｦｨｺ07 08ｨ 090500ｨｴ0007ｨｰ 08ｨ ｨｨ. 09. 0008090500ｨｮ07ｨｰ 0004ｨｰ ｨｬ0200ｨ 0905040007ｨｰ x ｨｬ02 01ｨｦ050009ｨ ｨｬｨｬｨ C x, ｨｹ080701 ｨｮ05ｨｬ02ｨｴｨｪｨ ｨｬ02 00ｨｦｨｰ 0704ｨｬ02ｨｦ06ｨｮ02ｨｦｨｰ 14.2.1-1 (i) 04 f x (x 0 ; y 0 ) ｨｨｨ 0709080302ｨｦ 0007ｨｪ ｨｮ01ｨｪ00020502ｨｮ0007 01ｨｦ0205ｨｨ01ｨｪｨｮ04ｨｰ 0004ｨｰ 0202ｨ 0800ｨｹｨｬ02ｨｪ04ｨｰ 0201ｨｨ0208ｨ ｨｰ 02 x 000701 01ｨｦｨ 000905ｨｬｨｬｨ 0007ｨｰ C x ｨｮ0007 ｨｮ04ｨｬ020807 f (x 0 ; y 0 ). 04ｨｬ07ｨｦｨ ｨｨ02ｨｴ0906ｨｪ00ｨ ｨｰ 0007 x ｨｮ00ｨ ｨｨ0209ｨｹ ｨｮ0004ｨｪ f y, 080701 0208ｨｪｨ ｨｦ ｨｬｨｦｨ ｨｮ01ｨｪ05090004ｨｮ04 000701 y ｨｬ02 01ｨｦ050009ｨ ｨｬｨｬｨ C y, 04 08ｨ 090500ｨｴ0007ｨｰ f y (x 0 ; y 0 ) ｨｨｨ 0709080302ｨｦ 0007ｨｪ ｨｮ01ｨｪ00020502ｨｮ0007 01ｨｦ0205ｨｨ01ｨｪｨｮ04ｨｰ 0004ｨｰ 0202ｨ 0800ｨｹｨｬ02ｨｪ04ｨｰ 0201ｨｨ0208ｨ ｨｰ 02 y 000701 01ｨｦｨ 000905ｨｬｨｬｨ 0007ｨｰ C y ｨｮ0007 f (x 0 ; y 0 ). 030807ｨｬ06ｨｪｨｴｨｰ, ｨｹ00ｨ ｨｪ 07ｨｦ ｨｬ0200ｨ 0905040006ｨｰ (x; y) ｨｬ0200ｨ 0905050507ｨｪ00ｨ ｨｦ ｨｮ0007 D 07ｨｦ ｨｬ0209ｨｦｨｺ06ｨｰ 08ｨ 090500ｨｴ0007ｨｦ f x ｨｺｨ ｨｦ f y ｨｨｨ ｨｬ0200ｨ 0905050507ｨｪ00ｨ ｨｦ ｨｮ0007ｨｪ 0002ｨｴｨｬ020009ｨｦｨｺｨｹ 00ｨｹ0807, 080701 070908030200ｨ ｨｦ ｨ 08ｨｹ 0004ｨｪ 0007ｨｬ07 00ｨｴｨｪ 0201ｨｨ02ｨｦ06ｨｪ 02 x ｨｺｨ ｨｦ 02 y, 010405ｨ 0107 ｨｮ0007 02080808020107, 06ｨｮ00ｨｴ, 080701 07ｨｦ 0201ｨｨ020802ｨｰ ｨ 010006ｨｰ 070908030701ｨｪ. 0807 0709080302ｨｦ ｨｮ0004ｨｪ 080209080800ｨｴｨｮ04 ｨ 010007 0007 0202ｨ 0800ｨｹｨｬ02ｨｪ07 02080808020107 (tangent plane) 0004ｨｰ 0208ｨｦ0205ｨｪ02ｨｦｨ ｨｰ z = f(x; y) ｨｺｨ ｨｦ 04 020608ｨｮｨｴｨｮ07 000701 ｨ 08070102ｨｦｨｺｨｪ050200ｨ ｨｦ ｨｹ00ｨｦ 0108ｨｪ0200ｨ ｨｦ ｨ 08ｨｹ 0007ｨｪ 00050807 8 z = f (x 0 ; y 0 ) + f x (x 0 ; y 0 ) (x 6ﾓ1 x 0 ) + f y (x 0 ; y 0 ) (y 6ﾓ1 y 0 ) : (14.2.3-1) 08ｨｹ0002 04 020608ｨｮｨｴｨｮ04 000701 ｨ ｨｪ0008ｨｮ0007ｨｦ ﾂ0701 ｨｺ05ｨｨ02000701 0208ｨｦ0806010701 (normal plane) 0108ｨｪ0200ｨ ｨｦ ｨ 08ｨｹ 0007ｨｪ 00050807 04ｨ 09ｨ 00070904ｨｮ04 14.2.3-1 x 6ﾓ1 x 0 f x (x 0 ; y 0 ) = y 6ﾓ1 y 0 f y (x 0 ; y 0 ) = z 6ﾓ1 z 0 : (14.2.3-2) 6ﾓ11 0705ｨｬ02ｨｴｨｪｨ ｨｺｨ ｨｦ ｨｬ02 00ｨｦｨｰ 0704ｨｬ02ｨｦ06ｨｮ02ｨｦｨｰ 14.2.1-1 080209080800ｨｴｨｮ04 (iii) ｨ ｨｪ ｨｮ000701ｨｰ 08ｨ 09ｨ 0805ｨｪｨｴ 010807050700ｨｦｨｮｨｬ0705ｨｰ 080907ｨｺ050302ｨｦ ｨｹ00ｨｦ 00ｨｦｨ ｨｬｨｦｨ ｨｬ0209ｨｦｨｺ07 08ｨ 090500ｨｴ0007ｨｰ 06ｨｮ00ｨｴ 0004ｨｪ f x, 0208ｨｪｨ ｨｦ f x (x 0 ; y 0 ) = 0, 00ｨｹ0002 0007 0202ｨ 0800ｨｹｨｬ02ｨｪ07 02080808020107 0208ｨｪｨ ｨｦ 08ｨ 09050505040507 ｨｮ0007ｨｪ x-050607ｨｪｨ, 02ｨｪ06 0007 ｨｺ05ｨｨ020007 02080808020107 0006ｨｬｨｪ02ｨｦ 0007ｨｪ x-050607ｨｪｨ ｨｺ05ｨｨ0200ｨ ｨｮ0007 ｨｮ04ｨｬ020807 x = x 0. 04ｨ 09050102ｨｦ00ｨｬｨ 14.2.3-1 01ｨ 010807050700ｨｦｨｮ000208 04 020608ｨｮｨｴｨｮ04 000701 0202ｨ 0800ｨｹｨｬ02ｨｪ0701 0208ｨｦ0806010701 0004ｨｰ 0208ｨｦ0205ｨｪ02ｨｦｨ ｨｰ z = f(x; y) = 3 + x2 16 + y2 ｨｮ0007 ｨｮ04ｨｬ020807 (x 0 ; y 0 ) = (6ﾓ14; 3): 9 80705ｨｬ02ｨｴｨｪｨ ｨｺｨ ｨｦ ｨｬ02 0004ｨｪ 010807ｨｮ04ｨｬ0208ｨｴｨｮ04 000701 04ｨ 09ｨ 01020800ｨｬｨ 0007ｨｰ 14.1.1-4 04 0002ｨｪｨｦｨｺ07 ｨｬ07090207 0004ｨｰ 020608ｨｮｨｴｨｮ04ｨｰ 000701 0208ｨｦ0806010701 ax + by + cz = d, ｨｹ00ｨ ｨｪ 0501ｨｨ0208 ｨｴｨｰ 080907ｨｰ z, ｨｦｨｮ070105ｨｪｨ ｨｬｨ 000905020200ｨ ｨｦ z = f(x; y) = Ax + By + D:
130302ｨ 0800ｨｹｨｬ02ｨｪ07 02080808020107 631 0905ｨｮ04. 02ｨｦｨ 0107 ﾂｨｦｨｺ05 06 ﾂ0701ｨｬ02 07 ﾂ07ｨｬｨ 14.2.3-1: 04ｨ 09050102ｨｦ00ｨｬｨ 14.2.3-1. z = f(x; y) = 3 + x2 16 + y2 9 ; z 0 = f(4; 6ﾓ13) = 5; f x (x; y) = x 8 ; f x(4; 6ﾓ13) = 6ﾓ1 1 2 ; f y (x; y) = 2y 9 ; f y(4; 6ﾓ13) = 0409ｨ ｨｮ05ｨｬ02ｨｴｨｪｨ ｨｬ02 0007ｨｪ 00050807 (14:2:3 6ﾓ1 1) 04 020608ｨｮｨｴｨｮ04 000701 0208ｨｦ0806010701 ｨｨｨ 0208ｨｪｨ ｨｦ (07 ﾂ. 14.2.3-1) z = 5 6ﾓ1 1 2 (x + 4) + 2 (y 6ﾓ1 3); 3 02ｨｪ06 000701 ｨｺ05ｨｨ02000701 0208ｨｦ0806010701 ｨｮ05ｨｬ02ｨｴｨｪｨ ｨｬ02 0007ｨｪ 00050807 (14:2:3 6ﾓ1 2) 010405ｨ 0107 x + 4 6ﾓ1 1 2 = y 6ﾓ1 3 2 3 = z 6ﾓ1 5 6ﾓ11 ; 4(x + 4) = 6ﾓ13(y 6ﾓ1 3) = 2(z 6ﾓ1 5): 2 3 : 070004ｨｪ 080209080800ｨｴｨｮ04 080701 04 020608ｨｮｨｴｨｮ04 0004ｨｰ 0208ｨｦ0205ｨｪ02ｨｦｨ ｨｰ 0102ｨｪ 0208ｨｪｨ ｨｦ 0004ｨｰ 08ｨ 09ｨ 0805ｨｪｨｴ ｨ ｨｪｨ 050100ｨｦｨｺ07ｨｰ (explicit) ｨｬ07090207ｨｰ z = f(x; y), ｨ 050505 070908030200ｨ ｨｦ 080208050200ｨｬ06ｨｪｨ (implicit), 010405ｨ 0107 0208ｨｪｨ ｨｦ 0004ｨｰ ｨｬ07090207ｨｰ f(x; y; z) = 0 07 01ｨｦｨ 0207090200ｨｦｨｺ05, ｨｹ00ｨ ｨｪ 0102ｨｪ 0208ｨｪｨ ｨｦ
13632 000209ｨｦｨｺ07 08ｨ 090500ｨｴ0007ｨｰ 08ｨ ｨｨ. 09. 0008090500ｨｮ07ｨｰ 0101ｨｪｨ 00ｨｹｨｪ ｨｪｨ 0501ｨｨ0208 04 020608ｨｮｨｴｨｮ04 f(x; y; z) = 0 ｨｬ07ｨｪ07ｨｮ07ｨｬｨ ｨｪ00ｨ ｨｴｨｰ 080907ｨｰ z, 00ｨｹ0002 07ｨｦ 08ｨ 09ｨ 0805ｨｪｨｴ 0206ｨｦｨｮ06ｨｮ02ｨｦｨｰ 00ｨｦｨ 0007 ｨｮ04ｨｬ020807 ｨ ｨｪ0008ｨｮ0007ｨｦ ﾂｨ 0009050207ｨｪ00ｨ ｨｦ: 0202ｨ 0800ｨｹｨｬ02ｨｪ07 02080808020107 (x 0 ; y 0 ; z 0 ) ｨｬ02 f (x 0 ; y 0 ; z 0 ) = 0; f x (x 0 ; y 0 ; z 0 ) (x 6ﾓ1 x 0 ) + f y (x 0 ; y 0 ; z 0 ) (y 6ﾓ1 y 0 ) ｨｺ05ｨｨ020007 02080808020107 + f z (x 0 ; y 0 ; z 0 ) (z 6ﾓ1 z 0 ) = 0; (14.2.3-3) x 6ﾓ1 x 0 f x (x 0 ; y 0 ; z 0 ) = y 6ﾓ1 y 0 f y (x 0 ; y 0 ; z 0 ) = z 6ﾓ1 z 0 : (14.2.3-4) f y (x 0 ; y 0 ; z 0 ) 04ｨ 09ｨ 00070904ｨｮ04 14.2.3-2 07ｨｮ ﾂ0502ｨｦ ｨｺｨ ｨｦ ｨｮ0004ｨｪ 080209080800ｨｴｨｮ04 ｨ 010007 ｨ ｨｪ0505070004 04ｨ 09ｨ 00070904ｨｮ04 0004ｨｰ 14.2.3-1. 04ｨ 09050102ｨｦ00ｨｬｨ 14.2.3-2 01ｨ 010807050700ｨｦｨｮ000208 04 020608ｨｮｨｴｨｮ04 000701 0202ｨ 0800ｨｹｨｬ02ｨｪ0701 ｨｺｨ ｨｦ 000701 ｨｺ05ｨｨ02000701 0208ｨｦ0806010701 ｨｮ0004ｨｪ 0208ｨｦ0205ｨｪ02ｨｦｨ xy 6ﾓ1 z 3 = 0 ｨｮ0007 ｨｮ04ｨｬ020807 (x; y) = (1; 6ﾓ11): 0905ｨｮ04. 0908ｨｹ 0004ｨｪ 020608ｨｮｨｴｨｮ04 0004ｨｰ 0208ｨｦ0205ｨｪ02ｨｦｨ ｨｰ 080907ｨｺ05080002ｨｦ ｨｹ00ｨｦ 0 = xy 6ﾓ1 z 3 ｨO ｨO x=1; y=6ﾓ11 = 6ﾓ1z 3 6ﾓ1 1; 010405ｨ 0107 z = 6ﾓ11; 0708ｨｹ0002 0007 0304000705ｨｬ02ｨｪ07 ｨｮ04ｨｬ020807 0208ｨｪｨ ｨｦ 0007 P (1; 6ﾓ11; 6ﾓ11). 0409ｨ f x (x; y; z) = y; f x P = 6ﾓ11; f y (x; y; z) = x; f y P = 1; f z (x; y; z) = 6ﾓ13z 2 ; f z P = 6ﾓ13:
130302ｨ 0800ｨｹｨｬ02ｨｪ07 02080808020107 633 030807ｨｬ06ｨｪｨｴｨｰ ｨｮ05ｨｬ02ｨｴｨｪｨ ｨｬ02 0007ｨｪ 00050807 (14:2:3 6ﾓ1 3) 04 020608ｨｮｨｴｨｮ04 000701 0208ｨｦ0806010701 ｨｨｨ 0208ｨｪｨ ｨｦ 6ﾓ1(x 6ﾓ1 1) + 1(y + 1) 6ﾓ1 3(z + 1) = 0; 010405ｨ 0107 x 6ﾓ1 y + 3z + 1 = 0; 02ｨｪ06 000701 ｨｺ05ｨｨ02000701 0208ｨｦ0806010701 ｨｮ05ｨｬ02ｨｴｨｪｨ ｨｬ02 0007ｨｪ 00050807 (14:2:3 6ﾓ1 4) 010405ｨ 0107 x 6ﾓ1 1 6ﾓ11 = y + 1 = z + 1 1 6ﾓ13 ; 1 6ﾓ1 x = y + 1 = 6ﾓ1 1 (z + 1): 3 04ｨ 09050102ｨｦ00ｨｬｨ 14.2.3-3 04ｨｬ07ｨｦｨ 04 020608ｨｮｨｴｨｮ04 000701 0202ｨ 0800ｨｹｨｬ02ｨｪ0701 ｨｺｨ ｨｦ 000701 ｨｺ05ｨｨ02000701 0208ｨｦ0806010701 ｨｮ0004ｨｪ 0208ｨｦ0205ｨｪ02ｨｦｨ 3xy 6ﾓ1 z 3 = a 3 ｨｮ0007 ｨｮ04ｨｬ020807 (x; y) = (0; a): 0905ｨｮ04. 0908ｨｹ 0004ｨｪ 020608ｨｮｨｴｨｮ04 0004ｨｰ 0208ｨｦ0205ｨｪ02ｨｦｨ ｨｰ 080907ｨｺ05080002ｨｦ ｨｹ00ｨｦ a 3 = 3xy 6ﾓ1 z 3 ｨO ｨO x=0; y=a = 6ﾓ1z 3 ; 010405ｨ 0107 z = 6ﾓ1a; 0708ｨｹ0002 0007 0304000705ｨｬ02ｨｪ07 ｨｮ04ｨｬ020807 0208ｨｪｨ ｨｦ 0007 P (0; a; 6ﾓ1a). 0409ｨ f x (x; y; z) = 3yz; f x P = 6ﾓ13a 2 ; f z (x; y; z) = 3xy 6ﾓ1 3z 2 ; f z P = 6ﾓ13a 2 ; f y (x; y; z) = 3xz; f y P = 0; 0708ｨｹ0002, 020802ｨｦ0107 f y P = 0, ｨｮ05ｨｬ02ｨｴｨｪｨ ｨｬ02 0004ｨｪ 04ｨ 09ｨ 00070904ｨｮ04 14.2.3-2 0007 0202ｨ 0800ｨｹｨｬ02ｨｪ07 02080808020107 ｨｨｨ 0208ｨｪｨ ｨｦ 08ｨ 09050505040507 ｨｮ0007ｨｪ y-050607ｨｪｨ, 02ｨｪ06 0007 ｨｺ05ｨｨ020007 ｨｨｨ 0006ｨｬｨｪ02ｨｦ ｨｺ05ｨｨ0200ｨ 0007ｨｪ y-050607ｨｪｨ ｨｮ0007 ｨｮ04ｨｬ020807 y = a. 030807ｨｬ06ｨｪｨｴｨｰ ｨｮ05ｨｬ02ｨｴｨｪｨ ｨｬ02 0007ｨｪ 00050807 (14:2:3 6ﾓ1 3) 04 020608ｨｮｨｴｨｮ04 000701 0208ｨｦ0806010701 ｨｨｨ 0208ｨｪｨ ｨｦ 6ﾓ13a 2 (x 6ﾓ1 0) + 0(y 6ﾓ1 a) 6ﾓ1 3a 2 (z + a) = 0;