Math 267 Lecture Videos

The Three-Dimensional Coordinate System

A Model of the 8 Octants

The Sphere

Geometric Representation of Vectors; Vector Sum, Scalar Multiple, and Difference

Algebraic Representation of Vectors

Algebraic Representation of Vectors II

The Dot Product

Examples and Applications of 2D Vectors

Examples of 3D Vectors

The Dot Product Again


The Vector Product

Properties of the Vector Product

Volume of an Oblique Box

Distance from a Point to a Line

Parametric Equations of Lines

Planes in Space

Symmetric Equations of a Line

Angle between Two Lines

Parallel and Orthogonal Planes

The Intersection of Two Planes

Parallel Planes and Skew Lines

Review 4.1-4.5


Graphing 3D Surfaces

Graphing Cylinders

Graphing Quadric Surfaces

Hyperbolic Paraboloid and Pringles

Paraboloid; Surfaces of Revolution


Vector-Valued Functions

The Twisted Cubic

The Helix

Length of a Curve

Limits and Derivatives

Differentiation Formulas and Finding Parametric Equations of the Tangent Line

Integrating Vector-valued Functions

Necessary Condition for the Orthogonality of the Position and Derivative Vectors


Circular Motion

Projectile Motion

Projectile Motion: Parabolic Path

Projectile Motion: Initial and Final Speeds

Example of Projectile Motion

Unit Tangent and Unit Normal Vectors

Another Example for T and N

Arclength Parametrization

Arclength Parametrization Example

TNB Frame for Motion

Curvature of a Plane Curve

Formula for Curvature When the Cartesian Equation of a Plane Curve is Given

Formula for Curvature When Parametric Equations of a Plane Curve are Given

Curvature for a Space Curve

Tangential and Normal Components of Acceleration

Formulas for the Acceleration Components and Curvature Involving Only the Derivatives of r(t) ††

An Example for the Above


Review of Vector-Valued Functions and Motion

Finding the Center of Curvature

Review Problem#3 and #7

Review Problem #8

Review Problem #14

Review Problem #15



Keplerís Laws of Planetary Motion

Proof that the Orbit is a Plane Curve

Proof of the First Law

Proof of the Second Law

Proof of the Third Law


Functions of Several Variables

Level Curves and Level Surfaces

Limit of a Function of Several Variables

Nick Explains the Definition of a Limit

Evaluating Limits of Multivariable Functions

Example of the Two-Path Rule

Finding the Limit Using the Equations of the Lines of Approach

Jump Discontinuity in 3D

Finding Limits for Functions of Three Variables Using Parametric Equations of the Lines of Approach

Finding Limits Using Polar Coordinates

Continuity of Functions of Several Variables

Partial Derivatives

Increment Definition

Another Form of the Increment

Proof of the Epsilon Form of the Increment

Increment Epsilon Form: Another Example


Applications of Differentials


Chain Rule

Proof of the Chain Rule

Implicit Differentiation

Related Rates

The Directional Derivative

The Gradient Theorem

Tangent Planes and Normal Lines

The Direction of the Gradient

Example for the Direction of the Gradient

Local Extrema and Critical Points

Tests for Local Extrema and Saddle Points

Examples of Finding Extrema and Saddle Points

Example of Critical Point Obtained from Undefined Partial Derivative

Extrema over a Closed and Bounded Region

Example: Minimizing the Distance between Parallel Planes

Lagrange Multiplier Technique

Lagrange Multipliers Examples

Proof of Lagrangeís Theorem


Another Limit Example

More on Lagrange Multipliers

An Optimization Problem: Constructing a Gutter



Double Integration

Rx and Ry Regions

Example of Rx and Ry Regions

Setting Up Iterated Double Integrals

Areas and Volumes Using Double Integrals

Rx and Ry Regions Explained Again

More Examples of Finding Volumes

Double Integration in Polar Coordinates

Cartesian to Polar Integration

Volumes Using Polar Coordinates

Surface Area

Triple Integration

Volumes Using Triple Integrals

Mass of Laminas and Solids

Moments and Center of Mass

Moment of Inertia

Cylindrical Coordinates

Volumes and Centroids Using Cylindrical Coordinates

Spherical Coordinates

Multiple Integration Using Spherical Coordinates

The Jacobian††††††††


Some Review Problems

Warning about the Use of Symmetry


Introduction to Vector Fields

Conservative Vector Fields

Divergence and Curl

Line Integrals

Line Integral along Different Paths between Two Points

Mass and Center of Mass of a Wire

Application of the Line Integral to Force Fields

Necessary and Sufficient Condition for Independence of Path

Finding the Potential Function

Vector Fields Recap

Another Example of Finding the Potential Function

Another Necessary and Sufficient Condition for Independence of Path

The Law of Conservation of Energy

Greenís Theorem

Greenís Theorem Examples

Using Greenís Theorem to Find Areas

Vector Form of Greenís Theorem

Surface Integrals

Surface Integrals Part 2

Surface Integrals Part 3

Flux Integrals

The Divergence Theorem

The Divergence Theorem Part 2

Stokesí Theorem

Stokesí Theorem Part 2

The Last Lecture