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18.013A

Calculus with Applications

Undergraduate · Historical course

Spring 2005

Topics

  • The Spreadsheet
  • Philosophy, Numbers and Functions
  • The Exponential Function and Trigonometric Functions
  • Vectors, Dot Products, Matrix Multiplication and Distance
  • Area of a Parallelogram, Determinants, Volume and Hypervolume, the Vector Product
  • Vectors and Geometry in Two and Three Dimensions
  • Differentiable Functions, the Derivative and Differentials
  • Computation of Derivatives from their Definition
  • Calculation of Derivatives by Rule
  • Derivatives of Vector Fields and the Gradient in Polar Coordinates
  • Higher Derivatives, Taylor Series, Quadratic Approximations and Accuracy of Approximations
  • Quadratic Approximations in Several Dimensions
  • Applications of Differentiation: Direct Use of Linear Approximation
  • Solving Equations
  • Extrema
  • Curves
  • Some Important Examples and a Formulation in Physics
  • The Product Rule and Differentiating Vectors
  • Complex Numbers and Functions of Them
  • The Anti-derivative or Indefinite Integral
  • The Area under a Curve and its Many Generalizations
  • The Fundamental Theorem of Calculus in One Dimension
  • The Fundamental Theorem of Calculus in Higher Dimensions; Additive Measures, Stokes Theorem and the Divergence Theorem
  • Reducing a Line Integral to an Ordinary Integral and Related Reductions
  • Reducing a Surface Integral to a Multiple Integral and the Jacobian
  • Numerical Integration
  • Numerical Solution of Differential Equations
  • Doing Integrals
  • Introduction to Electric and Magnetic Fields
  • Magnetic Fields, Magnetic Induction and Electrodynamics
  • Series
  • Doing Area, Surface and Volume Integrals
  • Some Linear Algebra
  • Second Order Differential Equations

Readings

  • A note to the user: The contents in this section are presented in the format of a textbook. This section is divided into chapters, and each chapter contains a number of pages. Navigation on each page allows the user to move to the next or previous page.
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  • To access the textbook, please open the ZIP file and index.html, which will provide you access to the full textbook. Please note that some of the tools are no longer available.
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  • 18.013A Textbook (ZIP) ↗
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Sources