Канал: Andrew McCrady

How You Should Think About Infinite Cartesian Products

How You Should Think About Infinite Cartesian Products

Kepler's Second Law as Understood by a Mathematician

Kepler's Second Law as Understood by a Mathematician

The Subgroup Tests in Under 3 Minutes! Fast Abstract Algebra Help

The Subgroup Tests in Under 3 Minutes! Fast Abstract Algebra Help

A Precursor to Group Theory

A Precursor to Group Theory

Partial Sum Formulas and Asymptotic Analysis

Partial Sum Formulas and Asymptotic Analysis

The complex nth roots of unity

The complex nth roots of unity

How to think about the parameters in a homotopy #maths #topology

How to think about the parameters in a homotopy #maths #topology

A Crucial Step to Prove Kepler's Laws of Planetary Motion #maths

A Crucial Step to Prove Kepler's Laws of Planetary Motion #maths

Homotopy Classes, the Path Product, and Associativity

Homotopy Classes, the Path Product, and Associativity

Can you always win this game?

Can you always win this game?

Homotopy Intro

Homotopy Intro

Paths and the Path Product

Paths and the Path Product

ElGamal Encryption and Elliptic Curve Cryptography

ElGamal Encryption and Elliptic Curve Cryptography

Why is the orbit of a planet in a plane?

Why is the orbit of a planet in a plane?

Visual Calculus: Fubini's Theorem for Iterated Double Integrals

Visual Calculus: Fubini's Theorem for Iterated Double Integrals

The Alexander Subbase Theorem: help understanding the definitions and the proof

The Alexander Subbase Theorem: help understanding the definitions and the proof

Understand The Baire Category Theorem: Dense Sets, Nowhere Dense Sets, & Infinity

Understand The Baire Category Theorem: Dense Sets, Nowhere Dense Sets, & Infinity

Piecewise Continuous Linear Functions are Dense Among Continuous Functions

Piecewise Continuous Linear Functions are Dense Among Continuous Functions

The Axiom of Choice: History, Intuition, and Conflict

The Axiom of Choice: History, Intuition, and Conflict

Cards, Marriage, and Python: an Introduction to Graph Theory

Cards, Marriage, and Python: an Introduction to Graph Theory

Can you solve these three tricky counting problems?

Can you solve these three tricky counting problems?

How many squares are on a chessboard? And more!

How many squares are on a chessboard? And more!

How many triangles are there in an n by n grid?

How many triangles are there in an n by n grid?

The Complex Logarithm: Multivalued Functions?!?

The Complex Logarithm: Multivalued Functions?!?

How Harmonic Functions Relate to Holomorphic Functions

How Harmonic Functions Relate to Holomorphic Functions

The Polar Form of the C R Equations

The Polar Form of the C R Equations

Harmonic Functions: why they are cool!

Harmonic Functions: why they are cool!

The Complex Trig Functions

The Complex Trig Functions

The Complex Exponential

The Complex Exponential

The Cross Ratio

The Cross Ratio

The Extended Complex Plane (Riemann Sphere)

The Extended Complex Plane (Riemann Sphere)

The Stereographic Projection: Learn it FAST!

The Stereographic Projection: Learn it FAST!

Euclidean Space, Locally Euclidean Space, and Manifolds

Euclidean Space, Locally Euclidean Space, and Manifolds

Homeomorphism and Connectedness

Homeomorphism and Connectedness

Homeomorphism and The Hausdorff Property

Homeomorphism and The Hausdorff Property

Help with Double Integrals

Help with Double Integrals

Quick Intro to Homeomorphisms

Quick Intro to Homeomorphisms

Visualizing Directional Derivatives and the Gradient

Visualizing Directional Derivatives and the Gradient

limit as z to infinity of z is infinity?!

limit as z to infinity of z is infinity?!

The Finite Complement Topology

The Finite Complement Topology

Translations, Dilations, and Inversions: What do Möbius Transformations do to Circles and Lines?

Translations, Dilations, and Inversions: What do Möbius Transformations do to Circles and Lines?

Linear Fractional Transformations and Möbius Transformations

Linear Fractional Transformations and Möbius Transformations

Distance from a point to a set is Lipschitz

Distance from a point to a set is Lipschitz

Conformal Maps

Conformal Maps

Using the Derivative to Detect Constant Functions Part 2

Using the Derivative to Detect Constant Functions Part 2

Using the Derivative to Detect Constant Functions Part 1

Using the Derivative to Detect Constant Functions Part 1

The Cauchy-Riemann Equations and the Complex Derivative

The Cauchy-Riemann Equations and the Complex Derivative

Partial Derivatives and the Cauchy Riemann Equations

Partial Derivatives and the Cauchy Riemann Equations

What's the Difference Between Complex Differentiable and Holomorphic?

What's the Difference Between Complex Differentiable and Holomorphic?

Continuity for Functions of a Complex Variable

Continuity for Functions of a Complex Variable

Limits of Functions of a Complex Variable

Limits of Functions of a Complex Variable

Functions of a Complex Variable

Functions of a Complex Variable

Paths in the Complex Plane

Paths in the Complex Plane

Basic Topology of the Complex Plane

Basic Topology of the Complex Plane

DeMoivre's Theorem and its Extension

DeMoivre's Theorem and its Extension

Polar Coordinates for Complex Numbers

Polar Coordinates for Complex Numbers

Geometry of Complex Numbers

Geometry of Complex Numbers

Algebra of Complex Numbers

Algebra of Complex Numbers

Intro to Complex Numbers

Intro to Complex Numbers

Linear Equations and Rational Equations

Linear Equations and Rational Equations

Special subsets of the power set

Special subsets of the power set

The Power Set of a Set (and a TrIcKy example!)

The Power Set of a Set (and a TrIcKy example!)

Proving A=B

Proving A=B

Operations with sets

Operations with sets

Proving A is a subset of B

Proving A is a subset of B

What is a subset?

What is a subset?

What is a set?

What is a set?

Exploring Continuity and Differentiability

Exploring Continuity and Differentiability

An Easy Quotient Topology Problem

An Easy Quotient Topology Problem

A Problem Concerning the Quotient Topology and Closed, Continuous, Surjective Maps

A Problem Concerning the Quotient Topology and Closed, Continuous, Surjective Maps

Product Topology on an Infinite Product of Topological Spaces

Product Topology on an Infinite Product of Topological Spaces

The Elliptic Curve Method to Factor Integers

The Elliptic Curve Method to Factor Integers

Pollard’s p-1 Method for Factoring Integers

Pollard’s p-1 Method for Factoring Integers

Undetermined coefficients intro and example

Undetermined coefficients intro and example

The Elliptic Curve Group Law (with examples)

The Elliptic Curve Group Law (with examples)

Intro to Elliptic Curves

Intro to Elliptic Curves

Lipschitz Functions: Intro and Simple Explanation for Usefulness in Machine Learning

Lipschitz Functions: Intro and Simple Explanation for Usefulness in Machine Learning

Quadratic Irrationals and Period Continued Fractions

Quadratic Irrationals and Period Continued Fractions

The Continued Fraction Procedure

The Continued Fraction Procedure

Every Nonzero Rational Number Has A Simple Continued Fraction Representation

Every Nonzero Rational Number Has A Simple Continued Fraction Representation

The Sequence of Partial Convergents

The Sequence of Partial Convergents

Partial Convergents

Partial Convergents

Neighborhood Space (Topology)

Neighborhood Space (Topology)

Axioms for a Complete System of Neighborhoods for a Point in a Topological Space

Axioms for a Complete System of Neighborhoods for a Point in a Topological Space

Complete System of Neighborhoods of a Point

Complete System of Neighborhoods of a Point

Open sets and Neighborhoods of a Point

Open sets and Neighborhoods of a Point

Combinations of Heaviside Functions Part 2

Combinations of Heaviside Functions Part 2

Combination of Heaviside Functions Part 1

Combination of Heaviside Functions Part 1

An inhomogeneous system of differential equations

An inhomogeneous system of differential equations

Unions and Intersections of Interiors

Unions and Intersections of Interiors

Unions and Intersections of Closures

Unions and Intersections of Closures

Second Countable Spaces, and the relation to First Countable Spaces

Second Countable Spaces, and the relation to First Countable Spaces

Local Basis at a Point and First Countable Spaces

Local Basis at a Point and First Countable Spaces

A Characterization of T_1 Spaces

A Characterization of T_1 Spaces

T_1 Spaces: definition and relation to T_0 and T_2 (Hausdorff) Spaces

T_1 Spaces: definition and relation to T_0 and T_2 (Hausdorff) Spaces

T_0 spaces: definition and examples

T_0 spaces: definition and examples

Every Singleton in a Hausdorff Space is Closed

Every Singleton in a Hausdorff Space is Closed

Subbasis for Some Topology on R^2 that is NOT a Basis or ANY Topology on R^2

Subbasis for Some Topology on R^2 that is NOT a Basis or ANY Topology on R^2

Basis for the Usual Topology on R?

Basis for the Usual Topology on R?

Basis for the Lower Limit Topology?

Basis for the Lower Limit Topology?

The Discrete Metric Generates the Discrete Topology

The Discrete Metric Generates the Discrete Topology

The Taxicab Metric on R^n

The Taxicab Metric on R^n

Proof that the Discrete Metric is a metric

Proof that the Discrete Metric is a metric

Quotient Topology for Equivalence Relations (Identification Spaces)

Quotient Topology for Equivalence Relations (Identification Spaces)

Functions (Commutative Diagrams)

Functions (Commutative Diagrams)

Functions (inverses and preimages/inverse images)

Functions (inverses and preimages/inverse images)

Functions (Well-defined, Injective, Surjective, Bijective)

Functions (Well-defined, Injective, Surjective, Bijective)

Functions (Domain, Codomain, Range, Image, and Graph)

Functions (Domain, Codomain, Range, Image, and Graph)

Indexed Families of Sets

Indexed Families of Sets

Euler’s phi(pq) for primes p and q

Euler’s phi(pq) for primes p and q

Intro to Continued Fractions

Intro to Continued Fractions

Jupyter notebook example

Jupyter notebook example

Jupyter notebook example

Jupyter notebook example

Euler’s Proposition

Euler’s Proposition

Gauss’s Lemma

Gauss’s Lemma

Euler’s Criterion

Euler’s Criterion

The Legendre Symbol and Quadratic Reciprocity with Examples

The Legendre Symbol and Quadratic Reciprocity with Examples

Intro to the Legendre Symbol

Intro to the Legendre Symbol

Intro to Quadratic Reciprocity

Intro to Quadratic Reciprocity

Attacking RSA when p and q are close

Attacking RSA when p and q are close

Attacking RSA by Factoring n Given phi(n)

Attacking RSA by Factoring n Given phi(n)

Encoding, RSA encryption/decryption, Decoding Example

Encoding, RSA encryption/decryption, Decoding Example

Encoding and Decoding Phrases and Numbers

Encoding and Decoding Phrases and Numbers

Intro to RSA with an Example

Intro to RSA with an Example

More Diffie-Hellman, the Man in the Middle Attack, and finding primitive roots for certain primes p

More Diffie-Hellman, the Man in the Middle Attack, and finding primitive roots for certain primes p

The Diffie-Hellman Key Exchange

The Diffie-Hellman Key Exchange

Intro to Public Key Cryptography with an example

Intro to Public Key Cryptography with an example

Recap of Primitive Elements in (Z/nZ)^* with examples in Sage

Recap of Primitive Elements in (Z/nZ)^* with examples in Sage

The Existence of Primitive Roots mod p

The Existence of Primitive Roots mod p

Polynomials over Z/pZ

Polynomials over Z/pZ

Intro to cyclic groups and the group of units mod p

Intro to cyclic groups and the group of units mod p

Miller-Rabin and Lucas-Lehmer Primality Tests

Miller-Rabin and Lucas-Lehmer Primality Tests

Pseudoprimality

Pseudoprimality

Computing Huge Powers Mod n

Computing Huge Powers Mod n

Writing a number in binary, by hand and using Sage

Writing a number in binary, by hand and using Sage

GCD(a,b) as a linear combination of a and b, and computing inverse Modulo n

GCD(a,b) as a linear combination of a and b, and computing inverse Modulo n

Multiplicative functions and Euler’s phi function revisited

Multiplicative functions and Euler’s phi function revisited

The Chinese Remainder Theorem

The Chinese Remainder Theorem

Wilson’s Theorem

Wilson’s Theorem

Euler’s phi function and Euler’s Theorem

Euler’s phi function and Euler’s Theorem

The group of units mod n, and the order of an element in a group

The group of units mod n, and the order of an element in a group

Linear equations mod n

Linear equations mod n

Groups, rings, and congruence mod n

Groups, rings, and congruence mod n

A few words on the prime number theorem and the Riemann hypothesis

A few words on the prime number theorem and the Riemann hypothesis

Primes of the form ax+b

Primes of the form ax+b

Prime Number Sieve and the Definition of a Mersenne Prime

Prime Number Sieve and the Definition of a Mersenne Prime

Proof that there are infinitely many primes

Proof that there are infinitely many primes

Proof of the Fundamental Theorem of Arithmetic

Proof of the Fundamental Theorem of Arithmetic

Factoring into Products of Primes, and Quick Notes About RSA Challenges

Factoring into Products of Primes, and Quick Notes About RSA Challenges

The existence of sine and cosine as uniform limits of sequences of functions

The existence of sine and cosine as uniform limits of sequences of functions

Euclid’s Lemma (GCD Part 2)

Euclid’s Lemma (GCD Part 2)

Greatest Common Divisors (Part 1)

Greatest Common Divisors (Part 1)

Prime Factorization for Natural Numbers

Prime Factorization for Natural Numbers

Defining the exponential function and deriving its properties

Defining the exponential function and deriving its properties

Unique function that’s its own derivative with y intercept 1

Unique function that’s its own derivative with y intercept 1

Existence of a function that’s its own derivative

Existence of a function that’s its own derivative

Interchange of limits for Riemann integrable functions

Interchange of limits for Riemann integrable functions

Differentiability and uniform convergence

Differentiability and uniform convergence

Transfer of continuity for uniform convergence

Transfer of continuity for uniform convergence

Intro to interchange of limits for uniformly convergent sequences of functions

Intro to interchange of limits for uniformly convergent sequences of functions

The uniform/sup norm of a sequence of bounded functions

The uniform/sup norm of a sequence of bounded functions

Uniform convergence of sequences of functions

Uniform convergence of sequences of functions

Sequences of functions and pointwise convergence

Sequences of functions and pointwise convergence

The Fundamental theorem of calculus (second form)

The Fundamental theorem of calculus (second form)

The Fundamental theorem of calculus (first form)

The Fundamental theorem of calculus (first form)

Additivity theorem for Riemann integrals

Additivity theorem for Riemann integrals

Continuous functions are Riemann integrable

Continuous functions are Riemann integrable

Squeeze theorem for Riemann integrable functions

Squeeze theorem for Riemann integrable functions

Cauchy criterion for Riemann integrable functions

Cauchy criterion for Riemann integrable functions

Properties of the Riemann integral

Properties of the Riemann integral

Riemann integrable functions on a closed interval

Riemann integrable functions on a closed interval

Partitions and Riemann sums

Partitions and Riemann sums

Taylor’s Theorem

Taylor’s Theorem

Intro to Taylor polynomials

Intro to Taylor polynomials

L’Hospital’s Rule part 2 for inf/inf

L’Hospital’s Rule part 2 for inf/inf

Example of continuous function between two metric spaces

Example of continuous function between two metric spaces

L’Hospital’s Rule part 1

L’Hospital’s Rule part 1

Cauchy Mean Value Theorem and build-up to L’Hospital’s Rule

Cauchy Mean Value Theorem and build-up to L’Hospital’s Rule

The first derivative test

The first derivative test

The Mean Value Theorem with some applications

The Mean Value Theorem with some applications

Function x/(x-1) tends to -inf as x approaches 1 from the left.

Function x/(x-1) tends to -inf as x approaches 1 from the left.

Relative extrema and Rolle’s theorem

Relative extrema and Rolle’s theorem

Derivative of the inverse of a function

Derivative of the inverse of a function

Using the limit definitions of continuity and the derivative

Using the limit definitions of continuity and the derivative

Carathéodory’s Theorem and the chain rule

Carathéodory’s Theorem and the chain rule

Basic derivative rules and notation

Basic derivative rules and notation

Intro to the derivative

Intro to the derivative

Continuity of inverse functions

Continuity of inverse functions

Jump discontinuities

Jump discontinuities

Monotone functions

Monotone functions

Gauges and continuous functions in intervals

Gauges and continuous functions in intervals

The existence of delta-fine partitions

The existence of delta-fine partitions

Partitions, tagged partitions, and gauge functions

Partitions, tagged partitions, and gauge functions

Lipschitz functions

Lipschitz functions

1.1.13 (a) from office hours

1.1.13 (a) from office hours

Criterion for nonuniform continuity, and continuity on closed bounded intervals

Criterion for nonuniform continuity, and continuity on closed bounded intervals

Uniform Continuity

Uniform Continuity

Preservation of intervals under continuous functions

Preservation of intervals under continuous functions

Location of roots and Bolzano’s intermediate value theorem

Location of roots and Bolzano’s intermediate value theorem

Max-Min theorem for continuous functions on closed, bounded intervals

Max-Min theorem for continuous functions on closed, bounded intervals