Multivariable Calculus Complete Semester Course

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This is the entire semester of my Multivariable Calculus course (MA 242 at NC State University). These videos are based off of my own notes. Here are the seven units in this course:

  • Unit 1: Vectors and Geometry in (x,y) and (x,y,z)-space
  • Unit 2: The geometry of curves
  • Unit 3: Differentiable Multivariable Calculus
  • Unit 4: Multiple integrals (in Cartesian coordinates)
  • Unit 5: Multiple integrals (in polar and spherical coordinates)
  • Unit 6: Vector fields, line and surface integrals
  • Unit 7: the Fundamental Theorems (Greens, Stokes, Divergence)

Unit 1: Vectors and Euclidean Geometry in ℝ³

Welcome to (x,y,z) Space R3
Five Examples in R3
Five More Examples in R3
Introduction to Vectors
Vector Magnitude and the Standard Basis Vectors
The Dot Product
Vector Projection
Vector Cross Product
Quick Right-Hand Rule Demo
Cross Product Areas and Volumes
Lines in R2
Vector Equation of a Line
Vector Equation of a Plane and General Form
Examples with Lines and Planes
Distance from a Point to a Plane

Unit 2: The Geometry of Curves in ℝ³

Introduction to Vector-Valued Functions and Curves
Calculus on Vector-Valued Functions (Curves)
Parametrized Curves
Parameterize the Curve of Intersection
Velocity and Acceleration
Newton's Second Law and Projectile Motion
The Unit Tangent Vector T
Arc Length and the Arc Length Function
Reparametrization with Respect to Arc Length
Curvature κ(t) for a Parametrized Curve
N and B with Visuals
Examples with T, N, B, κ and the Osculating Circle
Example Finding the Osculating Plane and TNB
Example Finding the Osculating Circle
Osculating Plane and Circle
The Decomposition of Acceleration

Unit 3: Differential Multivariable Functions

Introduction to Functions of Multiple Variables
Quadric Surfaces
Level Sets and Contour Maps
Parametric Surfaces r(u,v)
Multivariable Limits and Continuity
Partial Derivatives
Directional Derivatives
Clairaut's Theorem and Higher Order Derivatives
Tangent Planes in Multivariable Calculus
Find the Point on the Sphere Closest to a Plane
Differentials and Linearization in Multivariable Calculus
Directional Derivatives and the Gradient
The Chain Rule in Multivariable Calculus
Gradients and Tangent Planes
Tangent Plane and Normal Line using a Gradient
Multivariable Optimization
Optimization over Bounded Regions
Lagrange Multipliers

Unit 4: Multiple Integrals in Cartesian Coordinates

Definition of Double Riemann Integration
Fubini's Theorem
Double Integration over General Regions
Applications of Double Integrals
Introduction to Triple Integrals
Applications of Triple Integrals

Unit 5: Multiple Integrals in Curvilinear Coordinates

Introduction to Polar Coordinates and Integration
Examples of Integrating with Polar Coordinates
Areas Between Circles with Double Integrals
Double Integral in Polar Example
Average Value with Polar Coordinates Example
Triple Integration with Cylindrical (Polar) Coordinates
Introduction to Spherical Coordinates
Spherical Coordinates Integration Examples
Spherical Integration Example
Computing a Volume with Double and Triple Integrals

Unit 6: Line and Surface Integrals

Vector Fields
Conservative Vector Fields
Finding Potential Functions
Introduction to Line Integrals
Scalar Line Integrals
How to Set Up a Scalar Line Integral
Scalar Line Integrals Practice and Properties
Vector Line Integrals
Work Done by a Vector Field Computed with a Line Integral
Circulation Integrals
Fundamental Theorem for Line Integrals (FTLI)
Conservation of Energy
Examples of Scalar and Vector Line Integrals
Surface Area with a Surface Integral
Example Computing Surface Area with a Surface Integral
Scalar Surface Integrals
Surface Integrals: Why 'r' Isn't Needed When Parametrizing with Polar Coordinates
Example of a Scalar Surface Integral
Flux Integrals
Example of a Flux Integral

Unit 7: Vector Analysis

Curl and Divergence of a Vector Field
Green's Theorem for Circulation
Example Using Green's Theorem to Compute a Circulation Integral
Another Example of Green's Theorem
Green's Theorem for Flux
Stokes' Theorem
Divergence Theorem
One Flux Example Two Ways: Using Stokes' and the Divergence Theorem