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微積分学 (Calculus)

この文書は Gilbert Strang 氏の著書 Calculus を日本語化する目的で書いたものです。
しかし、個人の趣味程度にやっていることなので、内容に誤りがある可能性が高いです。
参照元は MIT OpenCourseWare です。

Strang Calculus Textbook Cover Art
Cover of Calculus, by Professor Gilbert Strang. (Image courtesy of Gilbert Strang.)

目次

  1. 微積分学の導入
    1. 速度と距離
    2. 極限を使わない微積分学
    3. 瞬間速度
    4. 円運動
    5. 三角比の概論
    6. 千の光点
    7. Computing in Calculus
  2. Derivatives
    1. The Derivative of a Function
    2. Powers and Polynomials
    3. The Slope and the Tangent Line
    4. Derivative of the Sine and Cosine
    5. The Product and Quotient and Power Rules
    6. Limits
    7. Continuous Functions
  3. Applications of the Derivative
    1. Linear Approximation
    2. Maximum and Minimum Problems
    3. Second Derivatives: Minimum vs. Maximum
    4. Graphs
    5. Ellipses, Parabolas, and Hyperbolas
    6. Iterations xn+1 = F(xn)
    7. Newton's Method and Chaos
    8. The Mean Value Theorem and l'Hôpital's Rule
  4. The Chain Rule
    1. Derivatives by the Chain Rule
    2. Implicit Differentiation and Related Rates
    3. Inverse Functions and Their Derivatives
    4. Inverses of Trigonometric Functions
  5. Integrals
    1. The Idea of the Integral
    2. Antiderivatives
    3. Summation vs. Integration
    4. Indefinite Integrals and Substitutions
    5. The Definite Integral
    6. Properties of the Integral and the Average Value
    7. The Fundamental Theorem and Its Consequences
    8. Numerical Integration
  6. Exponentials and Logarithms
    1. An Overview
    2. The Exponential ex
    3. Growth and Decay in Science and Economics
    4. Logarithms
    5. Separable Equations Including the Logistic Equation
    6. Powers Instead of Exponentials
    7. Hyperbolic Functions
  7. Techniques of Integration
    1. Integration by Parts
    2. Trigonometric Integrals
    3. Trigonometric Substitutions
    4. Partial Fractions
    5. Improper Integrals
  8. Applications of the Integral
    1. Areas and Volumes by Slices
    2. Length of a Plane Curve
    3. Area of a Surface of Revolution
    4. Probability and Calculus
    5. Masses and Moments
    6. Force, Work, and Energy
  9. Polar Coordinates and Complex Numbers
    1. Polar Coordinates
    2. Polar Equations and Graphs
    3. Slope, Length, and Area for Polar Curves
    4. Complex Numbers
  10. Infinite Series
    1. The Geometric Series
    2. Convergence Tests: Positive Series
    3. Convergence Tests: All Series
    4. The Taylor Series for ex, sin x, and cos x
    5. Power Series
  11. Vectors and Matrices
    1. Vectors and Dot Products
    2. Planes and Projections
    3. Cross Products and Determinants
    4. Matrices and Linear Equations
    5. Linear Algebra in Three Dimensions
  12. Motion along a Curve
    1. The Position Vector
    2. Plane Motion: Projectiles and Cycloids
    3. Tangent Vector and Normal Vector
    4. Polar Coordinates and Planetary Motion
  13. Partial Derivatives
    1. Surfaces and Level Curves
    2. Partial Derivatives
    3. Tangent Planes and Linear Approximations
    4. Directional Derivatives and Gradients
    5. The Chain Rule
    6. Maxima, Minima, and Saddle Points
    7. Constraints and Lagrange Multipliers
  14. Multiple Integrals
    1. Double Integrals
    2. Changing to Better Coordinates
    3. Triple Integrals
    4. Cylindrical and Spherical Coordinates
  15. Vector Calculus
    1. Vector Fields
    2. Line Integrals
    3. Green's Theorem
    4. Surface Integrals
    5. The Divergence Theorem
    6. Stokes' Theorem and the Curl of F
  16. Mathematics after Calculus
    1. Linear Algebra
    2. Differential Equations
    3. Discrete Mathematics