Канал: Matthew Salomone

596.C4.2 Derivative of a Complex Function: The Amplitwist

596.C4.2 Derivative of a Complex Function: The Amplitwist

596C.C4.1 Rethinking the Meaning of Derivative: Real Case

596C.C4.1 Rethinking the Meaning of Derivative: Real Case

596C.C3.A Applying Complex Exponential and Log to M.C. Escher

596C.C3.A Applying Complex Exponential and Log to M.C. Escher

596.C3.9 Complex Exponential and Logarithm - Mapping Properties

596.C3.9 Complex Exponential and Logarithm - Mapping Properties

596C.C3.8 The Complex Exponential Function - Algebra

596C.C3.8 The Complex Exponential Function - Algebra

596C.C3.7 The Truth About Polar Form: Euler's Identity

596C.C3.7 The Truth About Polar Form: Euler's Identity

596C.C3.6 When Complex Numbers Multiply: A Clue About Polar Form

596C.C3.6 When Complex Numbers Multiply: A Clue About Polar Form

596C.C3.5 The Principal Square Root of a Complex Number

596C.C3.5 The Principal Square Root of a Complex Number

596C.C3.4 Recap: There's More Than One "Square Root Function"

596C.C3.4 Recap: There's More Than One "Square Root Function"

596C.C3.3 Left and Right Inverses and the Real Square Root

596C.C3.3 Left and Right Inverses and the Real Square Root

596C.C3.2 One-to-One and Onto Functions

596C.C3.2 One-to-One and Onto Functions

596C.C3.1 Some Square-Root Conundrums

596C.C3.1 Some Square-Root Conundrums

596C.C2.4 Complex Functions as Transformations, Part 2

596C.C2.4 Complex Functions as Transformations, Part 2

596C.C2.3 Complex Functions as Transformations, Part 1

596C.C2.3 Complex Functions as Transformations, Part 1

596C.C2.2 Real and Imaginary Parts of a Complex Function

596C.C2.2 Real and Imaginary Parts of a Complex Function

596C.C2.1 Introducing Functions of a Complex Variable

596C.C2.1 Introducing Functions of a Complex Variable

Using Adobe Scan to Capture Your Work

Using Adobe Scan to Capture Your Work

596C.T2.1 Your Portfolio and My "Ungrading" Philosophy

596C.T2.1 Your Portfolio and My "Ungrading" Philosophy

596C.T1.1 Reading Klein's "Plan A/Plan B" History

596C.T1.1 Reading Klein's "Plan A/Plan B" History

596C.C1.3 Investigations of Complex Number Arithmetic and Geometry

596C.C1.3 Investigations of Complex Number Arithmetic and Geometry

596C.C1.2 Rectangular and Polar Coordinates in the Complex Plane

596C.C1.2 Rectangular and Polar Coordinates in the Complex Plane

596C.C1.1 Introducing Complex Numbers and the Complex Plane

596C.C1.1 Introducing Complex Numbers and the Complex Plane

596C.0 Welcome and Overview

596C.0 Welcome and Overview

202.V4.3 Intersection of Vector Subspaces

202.V4.3 Intersection of Vector Subspaces

202.V5.2 Adding a Vector to a Linearly Independent Set

202.V5.2 Adding a Vector to a Linearly Independent Set

202.G4 Finding the Eigenspace Associated with an Eigenvalue

202.G4 Finding the Eigenspace Associated with an Eigenvalue

202.G3 Characteristic Polynomials and Geometry of Eigenvalues

202.G3 Characteristic Polynomials and Geometry of Eigenvalues

202.G3 Introduction to Eigenvalues

202.G3 Introduction to Eigenvalues

202.G2 Using Both Row Operations and Dimension Reduction Together for Determinants

202.G2 Using Both Row Operations and Dimension Reduction Together for Determinants

202.G2 Dimension Reduction in Finding Determinants

202.G2 Dimension Reduction in Finding Determinants

202.G1 Recap: Using Row Operations to Find Determinant

202.G1 Recap: Using Row Operations to Find Determinant

202.G2-G4 Determinants and Eigenvalues: Agenda for Today

202.G2-G4 Determinants and Eigenvalues: Agenda for Today

202.G1 Computing Determinant via Row Operations

202.G1 Computing Determinant via Row Operations

202.M2 How Row Operations Affect the Determinant

202.M2 How Row Operations Affect the Determinant

202.G1 Invertible Transformations and the Determinant

202.G1 Invertible Transformations and the Determinant

202.M3 Computing the Inverse of a Matrix

202.M3 Computing the Inverse of a Matrix

202.M3 What Kinds of Matrices are Invertible

202.M3 What Kinds of Matrices are Invertible

202.M2 Inverse Matrices Agenda for Today

202.M2 Inverse Matrices Agenda for Today

202.M1 Matrix Multiplication

202.M1 Matrix Multiplication

202.M1 Composition of Linear Transformations

202.M1 Composition of Linear Transformations

202.A4 Surjective and Bijective Linear Transformations

202.A4 Surjective and Bijective Linear Transformations

202.A4 Injective Linear Transformations

202.A4 Injective Linear Transformations

202.A3 Kernel, Image, Row Space, and Rank+Nullity Theorem

202.A3 Kernel, Image, Row Space, and Rank+Nullity Theorem

202.A3 Image of a Linear Transformation

202.A3 Image of a Linear Transformation

202.3.2.6 A2 The Standard Matrix of a Linear Transformation

202.3.2.6 A2 The Standard Matrix of a Linear Transformation

202.A2 Standard Matrix of a Linear Transformation

202.A2 Standard Matrix of a Linear Transformation

202.A3 Image and Kernel of a Linear Transformation

202.A3 Image and Kernel of a Linear Transformation

202.A1 What can Linear Transformations Do?

202.A1 What can Linear Transformations Do?

202.A1 Using the Definition of Linear Transformation

202.A1 Using the Definition of Linear Transformation

202.A1 The Derivative is a Linear Transformation

202.A1 The Derivative is a Linear Transformation

202.A1 Introducing Linear Transformations

202.A1 Introducing Linear Transformations

202.A1-A3 Linear Transformations Agenda for Today

202.A1-A3 Linear Transformations Agenda for Today

402.4Z2 Uniform Limit of Continuous Functions is Continuous

402.4Z2 Uniform Limit of Continuous Functions is Continuous

402.4Z1 Our First Sequence of Functions, fₙ(x) = xⁿ

402.4Z1 Our First Sequence of Functions, fₙ(x) = xⁿ

402.3Z3 Examples of Metric Spaces

402.3Z3 Examples of Metric Spaces

402.3Z2 The Absolute Value Metric & The "Add-Subtract Trick"

402.3Z2 The Absolute Value Metric & The "Add-Subtract Trick"

402.3Z What is a Metric Space?

402.3Z What is a Metric Space?

202.V9 Homogeneous Linear Systems

202.V9 Homogeneous Linear Systems

202.V8 Visualizing "Polynomials as Vectors"

202.V8 Visualizing "Polynomials as Vectors"

202.V8 Doing Linear Algebra with Polynomials

202.V8 Doing Linear Algebra with Polynomials

202.V8 Coordinate Vectors are How We Do Linear Algebra with "Other" Vectors

202.V8 Coordinate Vectors are How We Do Linear Algebra with "Other" Vectors

202.V8 How are Polynomials and Matrices "Vectors"?

202.V8 How are Polynomials and Matrices "Vectors"?

202.V7 The Dimension of a Vector (Sub)Space

202.V7 The Dimension of a Vector (Sub)Space

202.V7 "Casting Out" Redundant Vectors to Form a Basis

202.V7 "Casting Out" Redundant Vectors to Form a Basis

202.V6 Identifying a Basis

202.V6 Identifying a Basis

202.V6 What is a Basis?

202.V6 What is a Basis?

202.V6 From Subspace to Basis - Example of a Plane

202.V6 From Subspace to Basis - Example of a Plane

202.V4 Proving a Subset is a Vector Subspace

202.V4 Proving a Subset is a Vector Subspace

261.12.9.5 Using Stokes' Theorem, and a (Bonus!) Green's Theorem appearance

261.12.9.5 Using Stokes' Theorem, and a (Bonus!) Green's Theorem appearance

261.12.9.3 Going from Boundary Curve to (Oriented) Surface Patch

261.12.9.3 Going from Boundary Curve to (Oriented) Surface Patch

261.12.9.2 Parametrizing the Boundaries of Surfaces

261.12.9.2 Parametrizing the Boundaries of Surfaces

261.12.10 Teeing Up Stokes' Theorem

261.12.10 Teeing Up Stokes' Theorem

261.12.8.3 Directly Computing a Flux Integral through a Cylinder

261.12.8.3 Directly Computing a Flux Integral through a Cylinder

261.12.8.2 The Idea of a Flux Integral

261.12.8.2 The Idea of a Flux Integral

261.12.8 Normal Vector to a Parametrized Surface

261.12.8 Normal Vector to a Parametrized Surface

261.12.10 All About Stokes' Theorem and its Relatives

261.12.10 All About Stokes' Theorem and its Relatives

202.2.5.3 How does RREF Tell Us About Linear Independence?

202.2.5.3 How does RREF Tell Us About Linear Independence?

202.2.5.1 Warmup for Linear Independence

202.2.5.1 Warmup for Linear Independence

202.2.1.3 Proving Properties of Vector Space (Long but Worth It!)

202.2.1.3 Proving Properties of Vector Space (Long but Worth It!)

202.2.1.2 V1 Axioms for a Vector Field

202.2.1.2 V1 Axioms for a Vector Field

202.V1-V4-V5-V6 Agenda for Today

202.V1-V4-V5-V6 Agenda for Today

261.12.10 A "Grand Unified Theory" of Vector Integrals and Stokes' Theorems

261.12.10 A "Grand Unified Theory" of Vector Integrals and Stokes' Theorems

261.12.10 Applying the Divergence Theorem to a 3D Flux Integral

261.12.10 Applying the Divergence Theorem to a 3D Flux Integral

261.12.10 Divergence Theorem and Flux Integrals: Agenda for Today

261.12.10 Divergence Theorem and Flux Integrals: Agenda for Today

261.12.5 Summary So Far: Green's Theorem and the Divergence Theorem

261.12.5 Summary So Far: Green's Theorem and the Divergence Theorem

261.12.5.3 Applying the Divergence Theorem in Two Dimensions (and Three?)

261.12.5.3 Applying the Divergence Theorem in Two Dimensions (and Three?)

261.12.5.2B Discovering the Divergence Theorem

261.12.5.2B Discovering the Divergence Theorem

261.12.5.2 Macro to Micro: Flux and Flux Density

261.12.5.2 Macro to Micro: Flux and Flux Density

261.12.5.1B The Sign of a Flux Integral

261.12.5.1B The Sign of a Flux Integral

261.12.5.1B The Difference Between Circulation and Flux

261.12.5.1B The Difference Between Circulation and Flux

261.12.5.1 What is a Flux Integral in Two Dimensions?

261.12.5.1 What is a Flux Integral in Two Dimensions?

261.12.5 Flux and Divergence: Agenda for Today

261.12.5 Flux and Divergence: Agenda for Today

261.12.7.2 Using Green's Theorem and Summary of Path-Independence and Circulation Integrals

261.12.7.2 Using Green's Theorem and Summary of Path-Independence and Circulation Integrals

261.12.7.1 Introducing Green's Theorem for Circulation Integrals

261.12.7.1 Introducing Green's Theorem for Circulation Integrals

261.12.6.1 Discovering the Scalar Curl (Circulation Density) of a Vector Field

261.12.6.1 Discovering the Scalar Curl (Circulation Density) of a Vector Field

261.12.6 Introducing the Scalar Curl of a Vector Field

261.12.6 Introducing the Scalar Curl of a Vector Field

261.12.4.4 Path Independence, Circulation, and Conservative Vector Fields

261.12.4.4 Path Independence, Circulation, and Conservative Vector Fields

261.12.3.3 Using the Fundamental Theorem of Line Integrals

261.12.3.3 Using the Fundamental Theorem of Line Integrals

261.12.4.2 The Fundamental Theorem of Line Integrals

261.12.4.2 The Fundamental Theorem of Line Integrals

261.12.4-7 Path-Independence, Scalar Curl, and Green's Theorem: Agenda for Today

261.12.4-7 Path-Independence, Scalar Curl, and Green's Theorem: Agenda for Today

202.0.3 Writing a Well Structured Proof

202.0.3 Writing a Well Structured Proof

202.0.3 Principles for Effective Proofs - Activity

202.0.3 Principles for Effective Proofs - Activity

202.V3 Spanning Sets

202.V3 Spanning Sets

202.V2 Linear Combinations and Span of a Set of Vectors

202.V2 Linear Combinations and Span of a Set of Vectors

261.12.3.2 Evaluating a Line Integral Directly

261.12.3.2 Evaluating a Line Integral Directly

261.12.3.1 Building Intuition: Evaluating a Line Integral

261.12.3.1 Building Intuition: Evaluating a Line Integral

261.12.2.5 Circulation Positive or Negative?

261.12.2.5 Circulation Positive or Negative?

261.12.2.4 Is the Circulation Positive or Negative?

261.12.2.4 Is the Circulation Positive or Negative?

261.12.2.3 Is the Line Integral Negative or Positive?

261.12.2.3 Is the Line Integral Negative or Positive?

261.12.2.2 Refresher: Parametrizing Oriented Curves and Line Segments

261.12.2.2 Refresher: Parametrizing Oriented Curves and Line Segments

261.12.2.1 Building Intuition: What is a Line Integral? Work, Wind, and Flight

261.12.2.1 Building Intuition: What is a Line Integral? Work, Wind, and Flight

261.12.2-3 Introducing Line Integrals and Circulation: Agenda for Today

261.12.2-3 Introducing Line Integrals and Circulation: Agenda for Today

261.12.1.1 Vector Fields as Data Tables and Visualization

261.12.1.1 Vector Fields as Data Tables and Visualization

261.12.1 Weather, Pressure, and Wind: Gradient Vectors and Contour Plots

261.12.1 Weather, Pressure, and Wind: Gradient Vectors and Contour Plots

261.12.1.3 Gradient Vector Fields, Scalar Potential, and the Gradient Screening Test

261.12.1.3 Gradient Vector Fields, Scalar Potential, and the Gradient Screening Test

261.12.1.2 Vector Fields, Integral Curves, and Gradients

261.12.1.2 Vector Fields, Integral Curves, and Gradients

261.12.1 Matthias Kawski's Vector Field Analyzer II

261.12.1 Matthias Kawski's Vector Field Analyzer II

261.12.1 Vector Field Calculus: Overview and Agenda for Today

261.12.1 Vector Field Calculus: Overview and Agenda for Today

261 11 7 2 Triple Integral over a Box and a Cone

261 11 7 2 Triple Integral over a Box and a Cone

261 11 7 Triple Integrals: Intro & Motivation

261 11 7 Triple Integrals: Intro & Motivation

261 11 6 4 Surface Area of a Graph and a Sphere

261 11 6 4 Surface Area of a Graph and a Sphere

261 11 6 Surface Area of Sphere via Parametrization

261 11 6 Surface Area of Sphere via Parametrization

261 11 6 3 Surface Area of a Parametrized Cylinder

261 11 6 3 Surface Area of a Parametrized Cylinder

261 11 6 Surface Area of a Parametrized Surface

261 11 6 Surface Area of a Parametrized Surface

261 11 6 Surface Area and Triple Integrals Agenda for Today

261 11 6 Surface Area and Triple Integrals Agenda for Today

202.2.2.1-4 V2 Definitions and Activity Prep for Linear Combinations and Span

202.2.2.1-4 V2 Definitions and Activity Prep for Linear Combinations and Span

202.2.1.1 V1 Algebraic Properties and Activity 2.1.1

202.2.1.1 V1 Algebraic Properties and Activity 2.1.1

202.V2 Introduction to Linear Combinations and Span

202.V2 Introduction to Linear Combinations and Span

202.V1 Introduction to Section V1: Vector Spaces

202.V1 Introduction to Section V1: Vector Spaces

261.11.5.4 Double Integrals in Polar Coordinates

261.11.5.4 Double Integrals in Polar Coordinates

261.11.4.4 Double Integrals can Compute Centers of Mass and Centroids

261.11.4.4 Double Integrals can Compute Centers of Mass and Centroids

261.11.5 Polar Coordinates and "Tape-Measured" Regions in (r,θ)

261.11.5 Polar Coordinates and "Tape-Measured" Regions in (r,θ)

261.11.4.6 Double Integrals can Compute Joint Probabilities

261.11.4.6 Double Integrals can Compute Joint Probabilities

261.11.4.2 Double Integrals can Compute Mass

261.11.4.2 Double Integrals can Compute Mass

261.11.4.3 Double Integrals can Compute Area

261.11.4.3 Double Integrals can Compute Area

261.11.4-5 Applications and Polar Coordinate Double Integrals: Agenda for Today

261.11.4-5 Applications and Polar Coordinate Double Integrals: Agenda for Today

261.10.8 On the Two-Constraint Lagrange Multiplier Condition

261.10.8 On the Two-Constraint Lagrange Multiplier Condition

261.11.3.1 Double Integral over a Triangular Region

261.11.3.1 Double Integral over a Triangular Region

261.11.2.1 Double Integrals over Rectangular Regions

261.11.2.1 Double Integrals over Rectangular Regions

261.10.8.1 Constrained Optimization: Lagrange Multipliers and the Cross Product Option

261.10.8.1 Constrained Optimization: Lagrange Multipliers and the Cross Product Option

261.10.7.6 Optimization on a Closed and Bounded Region

261.10.7.6 Optimization on a Closed and Bounded Region

261.10.7 EXTRA: The Hessian, or Why the Second Derivative Test Works

261.10.7 EXTRA: The Hessian, or Why the Second Derivative Test Works

261.10.7.3 Classifying Regular Critical Points as Local Max, Local Min, and Saddle

261.10.7.3 Classifying Regular Critical Points as Local Max, Local Min, and Saddle

261.11.3.2-3 "Tape-Measure" Method and Reversing Order of Double Integrals

261.11.3.2-3 "Tape-Measure" Method and Reversing Order of Double Integrals

261.10.8-11.3 Constrained Optimization and Double Integrals: Agenda for Today

261.10.8-11.3 Constrained Optimization and Double Integrals: Agenda for Today

261.10.7.2 Critical Points and the Gradient Vector

261.10.7.2 Critical Points and the Gradient Vector

261.10.7.1 Critical Points and Directional Derivatives

261.10.7.1 Critical Points and Directional Derivatives

261.10.7 Optimization: Agenda for Today

261.10.7 Optimization: Agenda for Today

261.10.6.3 Gradient Vector ∇f(x,y) and Contour Plots

261.10.6.3 Gradient Vector ∇f(x,y) and Contour Plots

261.10.6.5 Using Gradient to Compute Directional Derivatives

261.10.6.5 Using Gradient to Compute Directional Derivatives

261.10.6.4 Gradient as the Direction of Greatest Increase

261.10.6.4 Gradient as the Direction of Greatest Increase

261.10.6 Remember: ∇f(x,y) Lives in the Plane -- not Space!

261.10.6 Remember: ∇f(x,y) Lives in the Plane -- not Space!

261.10.6 Recap: What Does a Gradient Vector ∇f(x,y) Tell Us?

261.10.6 Recap: What Does a Gradient Vector ∇f(x,y) Tell Us?

261.10.6 What Can a Three-Variable Gradient ∇F(x,y,z) Tell Us?

261.10.6 What Can a Three-Variable Gradient ∇F(x,y,z) Tell Us?

261.10.6.2 Using the Directional Derivative Formula

261.10.6.2 Using the Directional Derivative Formula

261.10.6.1 Using a Parametrized Line to Find a Directional Derivative

261.10.6.1 Using a Parametrized Line to Find a Directional Derivative

261.10.6 Directional Derivatives and Gradient: Agenda for Today

261.10.6 Directional Derivatives and Gradient: Agenda for Today

202.1.3.5-6 Solution Sets, Free Parameters, and Pivot Columns

202.1.3.5-6 Solution Sets, Free Parameters, and Pivot Columns

202.1.2.10 Reduced Row Echelon Form and Using Sage

202.1.2.10 Reduced Row Echelon Form and Using Sage

202.1.2.5 Reduced Row Echelon Form and Pivots

202.1.2.5 Reduced Row Echelon Form and Pivots

202.1.2.5 Row Operations, Pivots, and Strategy

202.1.2.5 Row Operations, Pivots, and Strategy

202.1.2.2 Which Row and Column Operations are Valid?

202.1.2.2 Which Row and Column Operations are Valid?

202.1.1.3 Revisiting our First Solution Set of Linear Systems Example

202.1.1.3 Revisiting our First Solution Set of Linear Systems Example

202.E2-E3 Solution Sets of Linear Systems: Agenda for Today

202.E2-E3 Solution Sets of Linear Systems: Agenda for Today

261.10.5 EXTRA: A "Sophisticated Understanding" of the Chain Rule

261.10.5 EXTRA: A "Sophisticated Understanding" of the Chain Rule

261.10.5.4 The "Tree Diagram" Method for Multivariable Chain Rule

261.10.5.4 The "Tree Diagram" Method for Multivariable Chain Rule

261.10.5.1 A First Peek at the Multivariable Chain Rule

261.10.5.1 A First Peek at the Multivariable Chain Rule

261.10.4.3 Linearizations of Multivariable Functions

261.10.4.3 Linearizations of Multivariable Functions

261.10.4.1 Discovering the Tangent Plane / Linearization Equation

261.10.4.1 Discovering the Tangent Plane / Linearization Equation

261.10.4 Just Linearization Things: Tangent Planes and Chain Rule, Agenda

261.10.4 Just Linearization Things: Tangent Planes and Chain Rule, Agenda

261.10.4 Overview of Second Partial Derivatives

261.10.4 Overview of Second Partial Derivatives

261.10.3.6 Contour Plots and Partial Derivatives

261.10.3.6 Contour Plots and Partial Derivatives

261.10.3 Partial Derivatives: Agenda for Today

261.10.3 Partial Derivatives: Agenda for Today

261.10.3.4 Interpreting Partial Derivatives in Context

261.10.3.4 Interpreting Partial Derivatives in Context

261.10.3.3 Computing Partial Derivatives

261.10.3.3 Computing Partial Derivatives

261.0 The Hypatia Equation Editor Add-On

261.0 The Hypatia Equation Editor Add-On

261.10.3.2 Partial Derivatives as Derivatives of Traces

261.10.3.2 Partial Derivatives as Derivatives of Traces

261.10.3.1 Setting the Table: Tangent Lines to Traces of a Graph

261.10.3.1 Setting the Table: Tangent Lines to Traces of a Graph

261.10.1.3 Limits of Multivariable Functions, Second Example

261.10.1.3 Limits of Multivariable Functions, Second Example

261.10.1 About Continuity of Multivariable Functions

261.10.1 About Continuity of Multivariable Functions

261.10.1.1 Limits of Multivariable Functions, First Example

261.10.1.1 Limits of Multivariable Functions, First Example

261.11.6.3 Parametrizing Surfaces of Revolution & Spherical Coordinates

261.11.6.3 Parametrizing Surfaces of Revolution & Spherical Coordinates

261.11.6.2 Parametrizing a Cylinder & Cylindrical Coordinates

261.11.6.2 Parametrizing a Cylinder & Cylindrical Coordinates

261.11.6 Parametrizing the Graph of a Function

261.11.6 Parametrizing the Graph of a Function

261.11.6 What Does It Mean to Parametrize a Curve or Surface?

261.11.6 What Does It Mean to Parametrize a Curve or Surface?

261.N.B-4 Showing a Parametrized Curve Has Constant Speed

261.N.B-4 Showing a Parametrized Curve Has Constant Speed

261.N.B-7 The "Beehive" Parameterized Curve Problem

261.N.B-7 The "Beehive" Parameterized Curve Problem

202.1.1.1 E1 Inconsistent Systems have Contradiction

202.1.1.1 E1 Inconsistent Systems have Contradiction

202.1.1.3 Writing Solution Sets of Systems with Infinitely Many Solutions

202.1.1.3 Writing Solution Sets of Systems with Infinitely Many Solutions

202.E Solving an Inconsistent System via Elimination

202.E Solving an Inconsistent System via Elimination

202.0 Introduction, Syllabus, Grading

202.0 Introduction, Syllabus, Grading

261.9.7.4 Velocity, Speed, and Acceleration Vectors

261.9.7.4 Velocity, Speed, and Acceleration Vectors

261.9.7.1 Derivative of a Vector Valued Function

261.9.7.1 Derivative of a Vector Valued Function

261.9.6.4 Parametrizing Traces and Cross-Sections of Graphs

261.9.6.4 Parametrizing Traces and Cross-Sections of Graphs

261.9.6.2 One Curve, Many Parametrizations

261.9.6.2 One Curve, Many Parametrizations

261.9.6.1 Parametrized Curves and Cartesian Equations

261.9.6.1 Parametrized Curves and Cartesian Equations

261.9.6-7 Parametrization Agenda for Today

261.9.6-7 Parametrization Agenda for Today

261.9.5 Equations for Lines and Planes in ℝ³

261.9.5 Equations for Lines and Planes in ℝ³

261.9.4 Vector Cross Product in ℝ³

261.9.4 Vector Cross Product in ℝ³

261.9.4 The "Scalar Cross Product" of Vectors in ℝ²

261.9.4 The "Scalar Cross Product" of Vectors in ℝ²