# Multivariable Calculus Complete Semester Course

See other videos here: https://maultsby.wordpress.ncsu.edu/course-videos/

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

Force

The Dot Product

Vector Projection

Work

Vector Cross Product

Quick Right-Hand Rule Demo

Cross Product Areas and Volumes

Torque

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