Videos
Video lectures for each lesson, also available on YouTube. Each lesson's notes are also available as slideshows or PDFs.
Lesson 1: Parametric Plotting
Parametrically plotting curves, circles, and ellipses; parametric equations for curves, surfaces, and solids; calculus with parametrically-defined curves
Lesson 2: Vectors
Vector addition, subtraction, scalar multiplication; dot products and projection; magnitude and unit vectors; parametric equations for lines
Lesson 3: Perpendicularity
XYZ and parametric equations for a plane; parallel/perpendicular planes; line of intersection; cross product and its magnitude
Lesson 4: The Gradient
Gradient vectors; linearization and tangent planes; Lagrange multipliers and constrained optimization
Lesson 5: Double Integrals and the Gauss-Green Formula
Double integrals over rectangular and non-rectangular regions; Gauss-Green formula; computing area and volume; changing the order of integration
Lesson 6: Vector Fields
Definition and examples of vector fields; gradient fields, slope fields; trajectories; net flow along and across a curve
Lesson 7: Path Integrals
Path integrals; net flow along/across a curve; flux; path independence for gradient fields; Fundamental Theorem of Line Integrals
Lesson 8: Sources, Sinks & Singularities
Gauss-Green formula (Green's Theorem); rotation and divergence; sources and sinks; singularities; del operator; Laplacian
Lesson 9: 2D Change of Variables
Change of variables; non-area-preserving maps; Jacobian determinant; double integrals in polar coordinates
Lesson 10: 3D Change of Variables
Change of variables for triple integrals; 3D Jacobian; volume and mass calculations
Lesson 11: Spherical and Cylindrical Coordinates
Parameterizing spheres and cylinders; Jacobian for spherical/cylindrical coordinates; triple integrals in both systems
Lesson 12: The Divergence Theorem
Flux across a surface; Divergence Theorem (Gauss' Theorem); substitute surfaces; surface area