[목 차]

01 해석학_1

1. 실수체계 ········································· 2

1.1 완비성 공리 ····································· 2

2. 수열 ················································ 6

2.1 수열의 극한 ····································· 6

2.2 극한 정리 ········································ 8

2.3 Bolzano-Weierstrass 정리 ········ 12

2.4 Cauchy 수열 ································ 18

 2.5 상극한, 하극한 ···························· 20

3. 함수 ·············································· 22

3.1 함수의 극한 ·································· 22

3.2 함수의 연속 ·································· 28

3.2 함수의 고른연속 ···························· 32

4. 미분 ·············································· 36

4.1 도함수 ·········································· 36

4.2 평균값정리 ··································· 41

4.3 역함수정리 ··································· 48

5. 적분 ·············································· 51

5.1 Riemann 적분 ····························· 51

5.2 특이 Riemann 적분 ······················ 64

5.3 Riemann-Stieltjes 적분 ············· 67

6. 무한 급수 ······································ 69

6.1 무한 급수 ······································ 69

6.2 적분판정법, 비교판정법,

극한비교판정법 ····························· 72

6.3 절대수렴, 근판정법, 비판정법 ········ 77

6.4 교대급수 ······································· 80

7. 함수열 급수 ··································· 85

7.1 함수열의 고른수렴 ························ 85

7.2 함수열 급수의 고른수렴 ················· 93

7.3 멱급수 ·········································· 97

7.4 해석함수 ······································· 99

02 벡터해석학_109

1. 편도함수 ····································· 110

1.1 극한과 연속 ································ 110

1.2 편도함수 ····································· 112

1.3 극댓값과 극솟값 ·························· 126

2. 다중적분 ····································· 130

2.1 반복적분 ····································· 130

2.2 중적분 ········································ 136

2.3 극좌표에서의 중적분 ··················· 140

2.4 삼중적분 ····································· 143

2.5 원기둥좌표에서의 삼중적분 ········· 148

2.6 구면좌표에서의 삼중적분 ············ 150

2.7 다중 적분에서의 변수변환 ··········· 153

3. 벡터미적분학 ······························ 157

3.1 선적분 ········································ 157

3.2 Green의 정리 ···························· 162

3.3 매개변수 곡면의 면적 ·················· 167

3.4 곡면적분 ····································· 171

03 선형대수학_175

1. 행렬 ············································ 176

1.1 역행렬을 구하는 방법 ·················· 176

1.2 부분공간과 일차독립 ··················· 179

1.3 특수 형태의 행렬 ························· 183

1.4 LU-분해 ···································· 188

2. 행렬식 ········································ 190

2.1 행렬식 및 여인수 전개 ················· 190

2.2 행렬식의 성질 ····························· 194

2.3 Cramer의 규칙,    에 대한

공식과 응용 ································ 200

2.4 고유값과 고유벡터 ······················ 213

3. 선형변환 ····································· 219

3.1 변환으로써의 행렬 ······················ 219

3.2 선형연산자의 기하학 ··················· 223

3.3 핵(kernel)과 치역 ······················· 226

3.4 선형변환의 합성과 가역성 ··········· 230

4. 차원과 구조 ································ 234

4.1 기저와 치원 ································ 234

4.2 행렬의 기본공간 ·························· 238

4.3 차원정리 ····································· 244

4.4 계수(rank)정리 ··························· 248

4.5 사영(projection)정리 ·················· 250

4.6 정규직교 기저와 Gram-Schmidt

과정 ··········································· 257

4.7 기저에 관련된 좌표 ····················· 260

5. 대각화 ········································ 264

5.1 대각화 가능성 ····························· 264

04 미분기하학_271

1. 곡선 ············································ 272

1.1 정칙곡선 ····································· 272

1.2 곡률 ··········································· 276

1.3 비틀림률 ····································· 281

2. 곡면 ············································ 288

2.1 정칙곡면 ····································· 288

2.2 접평면, 법선 ······························· 290

2.3 제 1, 2 기본형식 ························· 292

2.4 법곡률 ········································ 294

2.5 Gaussian 곡률 ··························· 300

3. 내재적 기하학 ····························· 307

3.1 등거리사상 ································· 307

3.2 측지선 ········································ 309

3.3 Gauss-Bonnet 정리 ················· 313

05 복소해석학_317

1. 복소수 ········································ 318

1.1 지수형식 ····································· 318

1.2 복소수의 제곱근 ·························· 319

1.3 복소평면의 구역 ·························· 321

2. 해석함수 ····································· 323

2.1 복소변수함수 ······························ 323

2.2 극한과 연속 ································ 325

2.3 미분 ··········································· 328

2.4 Cauchy-Riemann 방정식,

해석함수 ····································· 330

3. 적분 ············································ 333

3.1 적분 ··········································· 333

3.2 경로적분 ····································· 334

3.3 Cauchy의 적분공식 ···················· 337

3.4 Liouville의 정리 ························· 346

3.5 최대절댓값원리 ··························· 348

4. 유수 ············································ 349

4.1 유수 ··········································· 349

4.2 극에서의 유수 ····························· 355

5. 유수의 응용 ································ 361

5.1 특이 적분 ···································· 361

5.2 Sine, Cosine을 포함한 정적분 ····· 369

6. 기본함수에 의한 사상 ·················· 371

6.1 선형분수변환 ······························ 371

06 정수론_375

1. 나눗셈법 ····································· 376

1.1 나눗셈법 ····································· 376

1.2 Euclidean 알고리즘 ··················· 379

1.3 Diophantine 방정식 ·················· 383

2. 합동 ············································ 385

2.1 합동의 기본 성질 ························· 385

2.2 선형 합동식 ································ 388

2.3 Fermat의 정리, Wilson의 정리 ···· 392

3. 수론함수 ····································· 395

3.1 약수의 합과 개수 ························· 395

3.2 Euler의 정리 ······························ 397

4. 원시근, 지표 ······························· 399

4.1 계수(Order) ······························· 399

4.2 원시근 ········································ 402

4.3 지표(Index) ································ 403

5. 이차상호법칙 ······························ 407

5.1 Euler의 판정기준 ························ 407

5.2 Legendre 기호 ·························· 410

5.3 이차상호법칙 ······························ 413

07 현대대수학_415

1. 군, 부분군 ··································· 416

1.1 동형 이항 구조 ···························· 416

1.2 군, 부분군 ·································· 419

1.3 순환군 ········································ 424

2. 치환, 잉여류, 직접곱 ··················· 429

2.1 치환군 ········································ 429

2.2 궤도, 순환, 교대군 ······················ 436

2.3 잉여류, Lagrange 정리 ·············· 440

2.4 직접곱 ········································ 443

3. 정규부분군, 잉여군 ····················· 448

3.1 정규부분군 ································· 448

3.2 잉여군 ········································ 453

4. 환, 체 ········································· 457

4.1 환, 체 ········································· 457

4.2 정역 ··········································· 459

4.3 Fermat의 정리, Euler의 정리 ····· 461

4.4 다항식 환 ···································· 465

4.5 체 위에서의 다항식의 인수분해 ···· 468

5. 아이디얼 ····································· 474

5.1 환준동형사상, 잉여환 ·················· 474

5.2 극대아이디얼, 소아이디얼 ··········· 477

6. 확대체 ········································ 482

6.1 확대체 ········································ 482

6.2 대수적 확대 ································ 487

6.3 유한체 ········································ 491

7. 자기동형사상, Galois 이론 ········· 494

7.1 체의 자기동형사상 ······················ 494

7.2 분해체 ········································ 499

7.3 분리확대체 ································· 500

7.4 Galois 이론 ································ 501

08 위상수학_507

1. 집합, 함수 ··································· 508

1.1 집합 ··········································· 508

1.2 함수 ··········································· 511

2. 위상 ············································ 514

2.1 위상 ··········································· 514

2.2 극한점, 폐포, 내부, 외부, 경계 ····· 515

2.3 부분공간, 상대위상 ····················· 524

2.4 기저 ··········································· 525

2.5 연속함수 ····································· 528

3. 거리, 분리공리 ···························· 531

3.1 거리 ··········································· 531

3.2 분리공리 ····································· 535

4. 콤팩트공간, 연결공간 ················· 541

4.1 콤팩트공간 ································· 541

4.2 연결공간 ····································· 546

5. 적공간, 상공간 ···························· 549

5.1 적공간 ········································ 549

5.2 상공간 ········································ 551

09 이산수학_555

1. 그래프 ········································ 556

1.1 그래프의 표현 ····························· 556

1.2 경로, 회로 ·································· 561

1.3 평면그래프 ································· 564

2. 계산기술 ····································· 568

2.1 이항정리 ····································· 568

2.2 조합 ··········································· 570

3. 점화관계, 생성함수 ····················· 572

3.1 점화관계 ····································· 572

3.2 생성함수 ····································· 575

10 확률 및 통계_579

1. 확률, 확률변수 ···························· 580

1.1 확률 ··········································· 580

1.2 조건부확률 ································· 582

1.3 확률변수 ····································· 584

1.4 기댓값 ········································ 590

2. 다변량분포 ································· 593

2.1 두 개의 확률변수 ························· 593

2.2 조건부분포, 기댓값 ····················· 598

2.3 상관계수 ····································· 600

3. 확률분포 ····································· 603

3.1 이항분포 ····································· 603

3.2 정규분포 ····································· 606

4. 추론 ············································ 610

4.1 확률표본 ····································· 610

4.2 추정, 검정 ·································· 613

참고문헌 ········································· 622