Course Description and Outline

Multivariable Calculus is a continuation of Calculus II, but is mostly concerned with generalizing the main concepts from Calculus I, such as differentiation and integration, to higher dimensions. Topics will cover include vector algebra and geometry; parametric curves in 3-dimensional space; differentiation and integration of scalar functions of several variables with applications to maximum-minimum problems and finding areas and volumes; differentiation and integration of vector fields; and Green’s, Stokes’, and Divergence Theorems (see below for a more detailed list of topics). At the end of the course, you should have a good understanding of how calculus of functions of two or more variables works and be able to apply this knowledge to a variety of problems arising in math and physics, such as motion in space, approximation of functions, optimization, density, mass, etc.

The more general objective of this course is to continue to provide you with a deeper understanding and working knowledge of calculus, while in the process strengthening your analytical skills, increasing your ability to communicate mathematics symbolically and orally, making you comfortable with reading and understanding mathematics on your own, and developing an appreciation for calculus as one of the greatest intellectual developments in history.

Here is a more detailed list of topics we will cover:
  • Vector algebra and geometry
    • vector algebra and interpretation in space
    • dot product, cross product, orthogonality, direction angles, projection
  • Quadric surfaces
    • equations and visualizations of cylinders, paraboloids, hyperboloids, etc.
  • Calculus of one-parameter vector functions in space
    • parametric equations of lines and curves
    • vector equations of lines and curves
    • limits and continuity
    • derivatives of vector functions
    • tangent vectors and lines
    • interaction of the derivative with dot and cross products
    • integrals of vector functions
    • arc length and curvature
    • normal and binormal vectors (time permitting)
    • velocity and acceleration
  • Calculus of functions of two or more variables
    • surfaces and level curves
    • limits and continuity
    • differentiation
      • partial derivatives
      • Clairaut’s Theorem
      • tangent planes
      • linear approximation (time permitting)
      • differentials
      • The Chain Rule
      • implicit differentiation (time permitting)
      • directional derivatives, the gradient, and interpretations
      • optimization
      • Lagrange multipliers (time permitting)
    • integration
      • double integrals over rectangles and their properties
      • integral as volume
      • average value
      • iterated integrals and Fubini’s Theorem
      • double integrals over general regions
      • double integrals in polar coordinates
      • density, mass, center of mass
      • surface area
      • triple integrals over boxes and general regions
      • triple integrals in cylindrical and spherical coordinates
      • general change of variables and the Jacobian (time permitting)
  • Vector calculus
    • vector fields in the plane and the three-space
    • gradient fields
    • line integrals
    • line integral of vector fields and work along a curve
    • The Fundamental Theorem for Line Integrals
    • conservative vector fields and independence of path
    • Green’s Theorem
    • curl and divergence of a vector field and interpretations
    • parametric equations of surfaces
    • grid curves
    • surfaces of revolution
    • tangent planes
    • surface area (time permitting)
    • surface integrals
    • surface integrals (flux) of vector fields
    • Stokes’ Theorem
    • Divergence Theorem (time permitting)