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$Field with 9 Elements: Gaussian Integers Mod 3 (ℤ3[i]={a+bi | a,b ∈ ℤ3, i^2 = -1} w/ mod 3 + and *)$

Multivariable Second Derivative Test

Bill Kinney

Bill Kinney

Multivariable Second Derivative Test

TOPOLOGY! Global Extrema of Continuous z=f(x,y) over a Closed and Bounded (Compact) Domain

Gradient Vector in Polar Coordinates ||∇u||^2 = (∂u/∂r)^2 + (∂u/∂θ)^2/r^2 | Geometric Interpretation

Implicit Differentiation Using Partial Derivatives (Implicit Function Theorem Formula)

Gradient Vectors, Directional Derivatives, & Chain Rule in Multivariable Calculus

Multivariable Calculus Problem-Solving Review for Exam 2 (w/ Introduction to Directional Derivative)

Linear Approximations, Differentiability, Total Differentials, Gradient Vectors, Tangent Planes

Multi Calc, Lec 15: Partial Derivatives (Notation & Geometric Interpretation)

Limits and Continuity of Multivariable Functions

Multi Calc, Lec 13B: Functions of 2 and 3 Variables (Domain, Range, and Graphs)

Multi Calc, Lec 13A: Curvature of a Helix is CONSTANT! (Use Wolfram Mathematica to Compute It)

Multi Calc, Lec 12B: Curvature (Definition, Meaning, and Formulas)

Multi Calc, Lec 12A: Unit Tangent & Normal Vectors, Tangential & Normal Components of Acceleration

Multivariable Calculus, 20 Minute Review of Topics on Exam 1

The SURPRISING Derivative of ||r(t)||^4

Multi Calc, Lec 11A: Arc Length Parameterization Example

In-Depth Parametric Curve Example with Mathematica (Velocity, Acceleration, Speed, & Distance)

Multi Calc, Lec 9: Differential Calculus of Vector-Valued Functions (Velocity Vector is Tangent)

How To Find The Equation of a Plane Given Three Points in ℝ^3

Multi Calc, Lec 8A: Describing Lines in ℝ^3 (including Symmetric Equations)

Multi Calc, Lec 7: Vector Cross Product (includes 3x3 Determinant Shortcut)

Why does v∙w = ||v|| ||w|| cos(θ)? The proof is fun!

Why Does proj_w(v)=((v∙w)/(v∙w))w??? (Projection Vector Formula Derivation)

Multi Calc, Lec 6A: The Dot Product and Angle Between Two Vectors

Multi Calc, Lec 5B: Vector Geometry and Vector Algebra in ℝ^3

Multi Calc, Lec 5A: 3-Dimensional Space ℝ^3 (including Distance Formula, Spheres, and Hyperboloids)

Multi Calc, Lec 4: Polar Coordinates

Multi Calc, Lec 3B: Surface Area of Solid of Revolution (from Parametric Curve)

Multi Calc, Lec 3A: Position & Velocity Vectors, Speed, Distance (Arc Length) for Parametric Eqs

Multivariable Calculus, Lec 2: Using Mathematica for Parametric Equations & Curves in 2D

Multivariable Calculus, Lecture 1: Parametric Curves in the Plane ℝ^2

Review Abstract Algebra in 30 Minutes

A Galois Group Isomorphic to the Symmetric Group S3

Find the Subfield Lattice of ℚ(√3,√5) over ℚ by Using Galois Theory!

Review All of Calculus 2 in 30 Minutes

Solve dy/dx = y^2 in TWO WAYS!! (Separation of Variables and Power Series Solution)

Damped Harmonic Oscillator with Complex Eigenvalues (Use Euler's Formula, Diff Eq for Calc 2, Pt 8A)

The Galois Group Gal(ℚ(∜2,i)/ℚ) has 8 Elements!

ℚ(∛2) is NOT a Galois Extension of ℚ (Fundamental Theorem of Galois Theory Does NOT Apply)

Galois Theory: Fundamental Definitions and a Basic Example

A Sneak Peek at Fourier Series by Playing Sound Waves in Wolfram Mathematica

How to Model Simple Harmonic Motion for Mass on a Spring (Diff Eqs for Calculus 2, Part 7B)

WOW! 4 Equilibrium Points! Logistic Growth Competing Species Model (Diff Eqs for Calc 2, Part 7A)

Galois Field GF(2^3) = GF(8)

Two Simple Finite Field Examples: ℤ2 and Galois Field GF(2^2) = GF(4)

Differential Equations for Calculus 2, Pt 6: Intro to Competing Species Model (use Mathematica)

Find a Splitting Field of x^3-1 over ℚ

ℚ(√2 + √3) = ℚ(√2,√3)...But WHY??? 🤔

ℚ(∛5) is an Algebraic Field Extension of ℚ of Degree 3 (so it's a finite degree extension too!)

Differential Equations for Calculus 2, Part 5B: Predator-Prey Model (Robins and Worms)

Differential Equations for Calculus 2, Part 5A: Logistic Model for Population Growth

Can a Polynomial Have Two Splitting Fields over ℚ???

Field Extensions and Kronecker's Theorem (Fundamental Theorem of Field Theory), including Examples

Differential Equations for Calculus 2, Part 4B: Free Fall with Air Resistance

Differential Equations for Calculus 2, Part 4A: Mixing Problem (Compartmental Analysis)

How to Avoid Going BROKE in Retirement with CALCULUS!

A Simple Field Extension! ℚ(√2) is a splitting field for f(x)=x^2-2 over the rational numbers ℚ

Vector Spaces in Linear Algebra vs Abstract Algebra

Differential Equations for Calculus 2, Part 3: Newton's Law of Cooling

The Calculus of Drug Dosing

Abstract Algebra Review for Exam 3

Finite Field Constructions and Calculations are FUN 🎉 in Abstract Algebra!

Differential Equations for Calculus 2, Part 2: Separation of Variables, Slope Fields, Euler's Method

Why is x^4+x+1 irreducible over the rationals ℚ? Use the Mod p Irreducibility Test

Eisenstein's Criterion is Weird!

Are Irreducible Polynomials Like Prime Numbers?!?

Polynomial Division Algorithm over a Field and Corollaries (Remainder Theorem, Factor Theorem)

Differential Equations for Calculus 2, Part 1: Basic Models and Separation of Variables

ℤ[x], ℚ[x], & ℝ[x] Contain ℤ as a Subring while ℤ(x), ℚ(x), & ℝ(x) Contain ℚ as a Minimal Subfield

Idempotent Elements in a Ring Help Us to Study Ring Homomorphisms

Natural Projection Homomorphisms Have Kernels That Are Easy to Find!

Use Evaluation Homomorphism ℤ[x] → ℤ by f(x) ↦ f(0) to Show ＜x＞ is Prime but Not Maximal in ℤ[x]

Win $1,000,000 by Solving the Hardest Math Problem in the World!

How to Visualize z^2, 1/z, & e^z as Complex Mappings of the Complex Plane (use Wolfram Mathematica)

Discover the Geometry of Complex Number Multiplication by Example

Ring Homomorphism Definition | Preimage φ^-1(K) of a Ring Ideal K (Use Ideal Test)

Maximal Ideal Definition, Examples, and Relationship to Fields in Abstract Algebra