Vector Calculus Home

Packet 01

• Introduction
  • Electricity and Magnetism
  • Fluid Dynamics
  • Plasma Physics: Fluids + EM Fields:
  • Mathematics: Differential Geometry
  • Optimization

Packet 02

• Parametric curves in 2D
  • Parametric curve
• Calculus with parametric curves in 2D
  • Traditional derivatives (unnatural)
  • Distance traveled
  • Length function
  • Speed
  • Curvature
• Polar coordinates
  • Area under polar graphs
• Problems

Packet 03

• Vectors
  • Vector space: addition and scaling
  • Components of vectors
  • Geometric vectors
• Products of vectors
  • Dot product
  • Cross product
  • Triple products
• Geometric applications of vectors
  • Lines and planes
  • Distances
• Quaternions
• Problems

Packet 04

• Surfaces: basic examples
  • Spheres
  • Cylinders generalized
  • Ruled surfaces
  • Quadrics: ellipsoids
  • Quadrics: hyperboloids
  • Quadrics: paraboloids
• Problems

Packet 05

• Parametric space curves: differentiation
  • Derivative rules for parametric space curves
  • Tangent and normal vectors
  • Distance traveled, speed
  • Curvature
  • Acceleration decomposition
• Parametric space curves: integration
  • Arc length parametrization
• Problems

Packet 06

• Parametric space curves: vector integration
  • Vector integrals
  • Position, velocity, acceleration
  • Kepler’s Law of Ellipses
• Line integrals I: scalar fields
• Problems

Packet 08

• Iterated integrals
  • Rectangular domains
  • Non-rectangular domains
  • Volume under a surface
  • Order of integration
• Integration over measures of area and volume
  • Area of a region
• Double integration in polar coordinates
• Applications of multiple integrals
  • Density and total quantity
  • Average value
  • Center of mass
  • Moments
• Problems

Packet 09

• Partial Derivatives
• Linear approximation
  • Differentials
  • Differentiability
• Problems

Packet 10

• Chain Rule
  • Chain rule over a curve
  • Chain rule over several parameters
• Directional derivative
• Vector fields
  • Gradient
• Problems

Packet 11

• Line integrals II: vector fields, aligned
• Optimization
  • Review of 1D theory
  • 2D theory
  • nD theory
• Constrained optimization: Lagrange multipliers
• Problems

Packet 13

• Curl
  • Curl as derivative operator
  • Curl as measure of circulation
• Green’s Theorem
  • Understanding Green’s Theorem
• Path dependence
• Problems

Packet 14

• Fundamental Theorem of Line Integrals
• Line integrals III: vector fields, across
• Divergence
  • Divergence as a derivative operator
  • Divergence as measure of expansion
• Divergence Theorem
• Green’s theorem revisited
• Problems

Packet 15

• Parametric surfaces
  • Change of variables
  • Surface integral: scalar
  • Surface integral: vector: flux across
• Stokes’ Theorem in 3D
  • Vector curl
  • Stokes’ Theorem
  • Path independence
  • Surface independence
• Divergence Theorem in 3D
• Problems

Final exam study

Vector Calculus - Study Guide - Final Exam