1. Vectors and Matrices
Course Outline
Unit 1: Vectors and Matrices
Part A: Vectors, Determinants, and Planes
» Session 1: Vectors
» Session 2: Dot Products
» Session 3: Uses of the Dot Product: Lengths and Angles
» Session 4: Vector Components
» Session 5: Area and Determinants in 2D
» Session 6: Volumes and Determinants in Space
» Session 7: Cross Products
» Session 8: Equations of Planes
» Problem Set 1
Part B: Matrices and Systems of Equations
» Session 9: Matrix Multiplication
» Session 10: Meaning of Matrix Multiplication
» Session 11: Matrix Inverses
» Session 12: Equations of Planes II
» Session 13: Linear Systems and Planes
» Session 14: Solutions to Square Systems
» Problem Set 2
Part C: Parametric Equations for Curves
» Session 15: Equations of Lines
» Session 16: Intersection of a Line and a Plane
» Session 17: General Parametric Equations; the Cycloid
» Session 18: Point (Cusp) on Cycloid
» Session 19: Velocity and Acceleration
» Session 20: Velocity and Arc Length
» Session 21: Kepler's Second Law
» Problem Set 3
» Practice Exam
» Session 22: Review of Topics
» Session 23: Review of Problems
» Exam
Unit 2: Partial Derivatives
Part A: Functions of Two Variables, Tangent Approximation and Optimization
» Session 24: Functions of Two Variables: Graphs
» Session 25: Level Curves and Contour Plots
» Session 26: Partial Derivatives
» Session 27: Approximation Formula
» Session 28: Optimization Problems
» Session 29: Least Squares
» Session 30: Second Derivative Test
» Session 31: Example
» Problem Set 4
Part B: Chain Rule, Gradient and Directional Derivatives
» Session 32: Total Differentials and the Chain Rule
» Session 33: Examples
» Session 34: The Chain Rule with More Variables
» Session 35: Gradient: Definition, Perpendicular to Level Curves
» Session 36: Proof
» Session 37: Example
» Session 38: Directional Derivatives
» Problem Set 5
Part C: Lagrange Multipliers and Constrained Differentials
» Session 39: Statement of Lagrange Multipliers and Example
» Session 40: Proof of Lagrange Multipliers
» Session 41: Advanced Example
» Session 42: Constrained Differentials
» Session 43: Clearer Notation
» Session 44: Example
» Problem Set 6
» Practice Exam
» Session 45: Review of Topics
» Session 46: Review of Problems
» Exam Materials
Unit 3: Double Integrals and Line Integrals in the Plane
» Session 47: Definition of Double Integration
» Session 48: Examples of Double Integration
» Session 49: Exchanging the Order of Integration
» Session 50: Double Integrals in Polar Coordinates
» Session 51: Applications: Mass and Average Value
» Session 52: Applications: Moment of Inertia
» Session 53: Change of Variables
» Session 54: Example: Polar Coordinates
» Session 55: Example
» Problem Set 7
Part B: Vector Fields and Line Integrals
» Session 56: Vector Fields
» Session 57: Work and Line Integrals
» Session 58: Geometric Approach
» Session 59: Example: Line Integrals for Work
» Session 60: Fundamental Theorem for Line Integrals
» Session 61: Conservative Fields, Path Independence, Exact Differentials
» Session 62: Gradient Fields
» Session 63: Potential Functions
» Session 64: Curl
» Problem Set 8
» Session 65: Green's Theorem
» Session 66: Curl(F) = 0 Implies Conservative
» Session 67: Proof of Green's Theorem
» Session 68: Planimeter: Green's Theorem and Area
» Session 69: Flux in 2D
» Session 70: Normal Form of Green's Theorem
» Session 71: Extended Green's Theorem: Boundaries with Multiple Pieces
» Session 72: Simply Connected Regions and Conservative
» Problem Set 9
Unit 4: Triple Integrals and Surface Integrals in 3-Space
» Session 74: Triple Integrals: Rectangular and Cylindrical Coordinates
» Session 75: Applications and Examples
» Session 76: Spherical Coordinates
» Session 77: Triple Integrals in Spherical Coordinates
» Session 78: Applications: Gravitational Attraction
» Problem Set 10
Part B: Flux and the Divergence Theorem
» Session 79: Vector Fields in Space
» Session 80: Flux Through a Surface
» Session 81: Calculating Flux; Finding ndS
» Session 82: ndS for a Surface z = f(x, y)
» Session 83: Other Ways to Find ndS
» Session 84: Divergence Theorem
» Session 85: Physical Meaning of Flux; Del Notation
» Session 86: Proof of the Divergence Theorem
» Session 87: Diffusion Equation
» Problem Set 11
Part C: Line Integrals and Stokes' Theorem
» Session 88: Line Integrals in Space
» Session 89: Gradient Fields and Potential Functions
» Session 90: Curl in 3D
» Session 91: Stokes' Theorem
» Session 92: Proof of Stokes' Theorem
» Session 93: Example
» Session 94: Simply Connected Regions; Topology
» Session 95: Stokes' Theorem and Surface Independence
» Session 96: Summary of Multiple Integration
» Problem Set 12
» Practice Exam
» Exam
» Session 97: Curl and Physics
» Session 98: Maxwell's Equations
Final Exam
» Session 99: Unit 1 Review
» Session 100: Unit 2 Review
» Session 101: Unit 3 Review
» Session 102: Unit 4 Review
