Mathematics, Coding and Science Education

Calculus III (Multivariable Calculus)

Course Information

I generally teach Honors Calculus I and Calculus II to high schoolers. In cases where the Calculus II students complete the course work before the end of the year, I offer choices as to what to do with the rest of the time. One of these is Calculus III. Generally, we are able to cover

  • Course Overview – This is what I would handout on the first day of class. It contains a quick summary of the course, requirements.
  • Course Plan – This is how I would break down my units in Calculus III and the number of days I reserve for specific topics for a semester of College.
  • Supplement Plan – This is how I use Calculus III as a supplement to a Calculus II course with limited time. Based on four years of experience.
  • Formula Sheets

Required Knowledge: Students engaging with Multivariable Calculus should be familar with the following concepts covered in the previous courses of Calculus I and II:

  • Limits
  • Derivatives
  • Summation
  • Integrals
  • Series
  • Differential Equations

To practice these concepts, see previous courses of Calculus I and Calculus II.

Table of Contents

Chapter 11: Parametric and Polar Equations – An Introduction to Alternate Coordinate Systems

  • Parametric Equations
  • Derivatives, Arclengths and Calculus with Parametric Equations
  • Polar Equations
  • Graphing Polar Equations
  • Area and Arclength of Polar Equations

Chapter 12: Vectors – An Introductions to Vectors and Operations on Vectors

  • Two Dimensional Vectors
  • Three Dimensional Vectors
  • The Dot Product
  • The Cross Product
  • Equations of Lines and PLanes
  • Cylinderical and Spherical Coordinates
  • Curvature

Chapter 13: Partial Derivatives – Using Derivatives in Multivariable Equations

  • Functions of Several Variables
  • Limits and Continuity of Several Variables
  • Partial Derivatives
  • Tangent Planes
  • The Chain Rule
  • The Gradient and Directional Derivatives
  • Optimization of Several Variables
  • Lagrange Multipliers

Chapter 14: Multiple Integrals – Using Integrals in More than Two Dimensions and with Alternate Coordinate Systems

  • Integration of Several Variables
  • Double Integrals in General Regions
  • Double Integrals in Polar Coordinates
  • Applications of Doulbe Integrals
  • Triple Integrals
  • Triple Integrals in Cylindrical Coordinates
  • Triple Integrals in Spherical Coordinates
  • Change of Variables

Chapter 15: Vector Calculus – Using the Techniques of Multivariable Calculus in Conjuction with Vectors

  • Vector Fields
  • Line Integrals
  • The Fundemantal Theorem of Line Integrals
  • Green’s Theorem
  • Curl and Divergence
  • Parametric Surfaces
  • Surface Integrals
  • Stokes’ Theorem
  • Divergence Theorem