Calculus 2

The following pages from Paul's Online Notes cover the same topics as PWr's "Analysis 2" class, but some details will be different between these notes and our lectures/quizzes/exams.

Functions of two variables

• Graphs and level curves
• Path integrals of scalar functions
• Partial derivatives
  • Partial derivatives - notation and calculation
  • Higher order partial derivatives
  • Partial derivatives - interpretation
  • Gradient and directional derivatives
  • Linear approximation
  • Tangent planes and normal lines
  • Relative extrema
  • Absolute extrema on a solid rectangle
• Multiply integrals
  • Double integrals - interpretation
  • ∬ over rectangular regions
  • ∬ over advanced regions
  • ∬ in polar coordinates
  • Triple integrals - interpretation
  • ∭ in cylindrical coordinates
  • ∭ in spherical coordinates

Differential equations

• Basic definitions
• Slope fields
• First-order ODEs
  • Autonomous ODEs
  • Separable ODEs
  • Substitutions
  • Linear ODEs
  • Laplace transforms
• Second-order linear ODEs
  • Superposition
  • Distinct real roots
  • Distinct complex roots
  • Repeated roots
  • Non-homogenous case
• Systems of linear ODEs
  • Review of eigenvalues
  • Vocabulary
  • Basic systems
  • Phase plane
  • Real eigenvalues
  • Complex eigenvalues
  • Repeated eigenvalues
  • Non-homogenous systems