I. Introduction

Calculus III (Multivariable Calculus) extends calculus to functions of multiple variables and adds the language of vectors and vector fields. You learn about partial derivatives, multiple integrals, and the core theorems of vector calculus that is used greatly in engineering & physics such as fields and flux in 2D/3D space.

II. Outline

  • Vectors & Geometry in 3D
    • Vectors, dot/cross product, projections
    • Lines and planes in 3D, distances/angles
    • Quadric surfaces (ellipsoids, paraboloids, hyperboloids)
    • Common coordinate systems
  • Functions of Several Variables
    • Graphs of surfaces and level curves/level surfaces
    • Limits and continuity in higher dimensions
    • Visualizing multivariable behavior (slices, contour plots)
  • Partial Derivatives & Local Linear Models
    • Partial derivatives and higher partials
    • Tangent planes and linearization
    • Directional derivatives and the gradient
    • Multivariable chain rule & implicit differentiation
  • Optimization in Multiple Variables
    • Critical points and second-derivative tests
    • Constrained optimization + Lagrange multipliers
    • Applied modeling
  • Multiple Integrals
    • Iterated integrals and setting bounds for different regions.
    • Area/volume via double/triple integrals
    • Changing order of integration
    • Coordinate changes: polar, cylindrical, spherical
    • Jacobians
  • Vector Fields & Line Integrals
    • Vector fields, flow
    • Line integrals of scalar fields vs vector fields
    • Conservative fields, potential functions, path independence
    • Fundamental theorem for line integrals
  • Surface Integrals & Vector Calculus Theorems
    • Surface parametrization, surface area integrals
    • Flux integrals across surfaces
    • Green’s Theorem
    • Stokes’ Theorem
    • Divergence Theorem

III. Free Books

IV. Video Series

V. See Also