Calculus

복수전공하고 있는 수학과의 학부 졸업시험을 위해 2024년 5월부터 2024년 7월까지 미적분학 1·2를 다시 공부 했습니다. 다시 공부를 하면서, 교재 연습 문제 중에 재밌었던 문제들과 헷갈리는 개념들을 정리해보았습니다.

공부를 마치고 나서는 “미적분학을 다시 공부하며 든 생각들“라는 제목으로 간단한 회고도 작성해보았습니다 🙂

참고자료 및 서적

• James Stewart - Calculus 8th ed. Early Transcendentals
• George B. Thomas, Jr. - Calculus 13th ed. Early Transcendentals
• Gilbert Strang - Calculus (Open TextBook)
• Joel Feldman - CLP Calculus (Open TextBook)
• APEX Calculus (Open TextBook)

Calculus 1

• Limit and Continuity: Problem Solving
  • 입델 논법이 수립된 과정
  • A Fixed point Theorem
  • Assigning a value to zero power of zero $0^0$
• Derivatives: Problem Solving
  • 모든 점에서 미분 불가능한 연속 함수
  • The cissoid of Diocles
  • Newton’s Serpentine
• Application of Derivatives: Problem Solving
  • Geometric Mean
  • Cauchy’s Mean Value Theorem
  • Proof of l’hôpital’s rule
• Techniques of Integrals: Problem Solving
  • Equivalence of the washer and shell methods
• Taylor Series & Macluarin Series
• 급수의 극한을 판정하는 법
• 교대 급수의 극한을 판정하는 법
  • 절대 수렴과 조건부 수렴
  • 리만 재배열 정리
• Sequence and Series: Problem Solving
  • The zipper theorem
  • A recurive definition of $\pi/2$
  • The Cantor Set
  • The Cauchy condensation test

Calculus 2

• Parametric Equations: Problem Solving
  • The witch of Maria Agnesi
  • Hypocycloid
  • Trochoids
  • Limaçon Curve
  • Lissajous Curve
  • Nephroid
  • Strophoid
  • The nephroid of Freeth
• Vectors and Space: Problem Solving
• Partial Derivatives and Differentiability
• Lagrange Multiplier
  • Constrained Maxima/minima
  • Lagrange Method with Two Constraints
• Multiple Integrals
  • Fubini’s Theorem
  • Jacobian
• Vector Fields
  • Gradient Fields $\nabla f$
  • Conservative Vector Field, and Potential Function
  • Arc Length와 Line Integral
    • Fundamental Theorem for Line Integrals
    • Work done by Gravitational Field
    • Independent of Path
    • On a Closed Curve
• Green Theorem
  • Curve Orientation: CCW is positive orientation
  • Curve Boundary
  • Green Theorem on Not simply-connected(= General Region)
• Divergence and Curl
  • Divergence $\nabla \cdot \mathbf{F}$
  • Curl $\nabla \times \mathbf{F}$
  • Laplace Operator $\nabla^2$
  • Vector form of Green Theorem
    • Tangential Form
    • Normal Form
• Parametric Surface, and Surface Integral
• Stokes’ Theorem
• Divergence Theorem