Канал: Mu Prime Math

Does the Gaussian integral trick work for other functions?

Does the Gaussian integral trick work for other functions?

Proof: Uniqueness of the Tensor Product

Proof: Uniqueness of the Tensor Product

Tensor Product Basis With the Universal Property

Tensor Product Basis With the Universal Property

Complete Derivation: Universal Property of the Tensor Product

Complete Derivation: Universal Property of the Tensor Product

How to Define Homomorphisms on Quotient Groups

How to Define Homomorphisms on Quotient Groups

Why can we change lim n→∞ to lim x→∞?

Why can we change lim n→∞ to lim x→∞?

A Natural Proof of the First Isomorphism Theorem (Group Theory)

A Natural Proof of the First Isomorphism Theorem (Group Theory)

Integral Formula for Natural Log (without knowing the derivative)

Integral Formula for Natural Log (without knowing the derivative)

Supremum, Infimum: Definition and Explanation

Supremum, Infimum: Definition and Explanation

Topology Definitions: Connected and Path-Connected

Topology Definitions: Connected and Path-Connected

Does 1+2+3+...=-1/12?

Does 1+2+3+...=-1/12?

Proof: Orthogonal Matrices Satisfy A^TA=I

Proof: Orthogonal Matrices Satisfy A^TA=I

My favorite proof of the n choose k formula!

My favorite proof of the n choose k formula!

Epsilon-Delta proofs: Can't we make the limit equal anything?

Epsilon-Delta proofs: Can't we make the limit equal anything?

Why can't we group the terms in 1-1+1-1+... ?

Why can't we group the terms in 1-1+1-1+... ?

When can we switch the limit and function?

When can we switch the limit and function?

Topology Definitions: Closure, Boundary, Interior

Topology Definitions: Closure, Boundary, Interior

Why can we do this to find inverse functions?

Why can we do this to find inverse functions?

Bijective Functions Have a Two-Sided Inverse

Bijective Functions Have a Two-Sided Inverse

Proof: Two-Sided Inverse Functions are Unique

Proof: Two-Sided Inverse Functions are Unique

Sum of natural numbers equals n+1 choose 2!

Sum of natural numbers equals n+1 choose 2!

Representation Theory: Irreducible Characters Are an Orthonormal Basis

Representation Theory: Irreducible Characters Are an Orthonormal Basis

Matrix Invertibility With F[x]-Modules

Matrix Invertibility With F[x]-Modules

Difference Between Normalizer, Centralizer, and Stabilizer

Difference Between Normalizer, Centralizer, and Stabilizer

A Concrete Introduction to Tensor Products

A Concrete Introduction to Tensor Products

Euler's Identity With a Limit!

Euler's Identity With a Limit!

Why Are Odd Numbers Not Even?

Why Are Odd Numbers Not Even?

The Cayley-Hamilton Theorem is Easy with F[x]-Modules

The Cayley-Hamilton Theorem is Easy with F[x]-Modules

How the Matrix Characteristic Polynomial is Connected to F[x]-Modules

How the Matrix Characteristic Polynomial is Connected to F[x]-Modules

F[x]-Module Derivation of Rational and Jordan Canonical Forms

F[x]-Module Derivation of Rational and Jordan Canonical Forms

Proof & Intuition: Similar Matrices are Isomorphic F[x]-Modules

Proof & Intuition: Similar Matrices are Isomorphic F[x]-Modules

Introduction and Derivation of F[x]-Modules

Introduction and Derivation of F[x]-Modules

What do Matrices Represent? - Learning Linear Algebra

What do Matrices Represent? - Learning Linear Algebra

Proof: Structure Theorem for Finitely Generated Torsion Modules Over a PID

Proof: Structure Theorem for Finitely Generated Torsion Modules Over a PID

Splitting Homomorphism of R-Modules

Splitting Homomorphism of R-Modules

Proof: Prime Ideals are Maximal in a PID

Proof: Prime Ideals are Maximal in a PID

Proof: Prime Ideal iff R/P is Integral Domain; Maximal iff R/M is Field

Proof: Prime Ideal iff R/P is Integral Domain; Maximal iff R/M is Field

Why Simultaneity MUST Be Relative

Why Simultaneity MUST Be Relative

Length Contraction, Time Dilation, & Relativity of Simultaneity Using Lorentz Transformation

Length Contraction, Time Dilation, & Relativity of Simultaneity Using Lorentz Transformation

Why ct² - x² is Invariant under Lorentz Transformation

Why ct² - x² is Invariant under Lorentz Transformation

Linear Algebra Derivation of Lorentz Transformation

Linear Algebra Derivation of Lorentz Transformation

Intuition: Why the Lorentz Transformation is Linear

Intuition: Why the Lorentz Transformation is Linear

Special Relativity Intro: Reference Frames & Spacetime Diagrams

Special Relativity Intro: Reference Frames & Spacetime Diagrams

Derivative of Area is Perimeter?

Derivative of Area is Perimeter?

Angle Sum Formula Proof with Linear Algebra!

Angle Sum Formula Proof with Linear Algebra!

Inner Semidirect Product Example: Dihedral Group

Inner Semidirect Product Example: Dihedral Group

Every Group is a Quotient of a Free Group

Every Group is a Quotient of a Free Group

Inner & Outer Semidirect Products Derivation - Group Theory

Inner & Outer Semidirect Products Derivation - Group Theory

Proof & Example: Orbit-Stabilizer Theorem - Group Theory

Proof & Example: Orbit-Stabilizer Theorem - Group Theory

Cayley's Theorem Explanation: Every Group is a Permutation Group

Cayley's Theorem Explanation: Every Group is a Permutation Group

Kernel and First Isomorphism Theorem - Group Theory

Kernel and First Isomorphism Theorem - Group Theory

Why Normal Subgroups are Necessary for Quotient Groups

Why Normal Subgroups are Necessary for Quotient Groups

Simple Explanation of Conjugation in the Symmetric Group

Simple Explanation of Conjugation in the Symmetric Group

Proof: Cosets are Disjoint and Equal Size

Proof: Cosets are Disjoint and Equal Size

Functions with a Two-Sided Inverse are Bijective

Functions with a Two-Sided Inverse are Bijective

Quadratic Reciprocity using Gauss's Lemma

Quadratic Reciprocity using Gauss's Lemma

Proof & Explanation: Gauss's Lemma in Number Theory

Proof & Explanation: Gauss's Lemma in Number Theory

Euler's Criterion: Proof and Example

Euler's Criterion: Proof and Example

Proof of Multiplication Rule for Legendre Symbol

Proof of Multiplication Rule for Legendre Symbol

Legendre Symbol Definition and Example

Legendre Symbol Definition and Example

Square Root Problem: Solving the Quartic Equation!

Square Root Problem: Solving the Quartic Equation!

Quadratic Residues Always Have Even Index

Quadratic Residues Always Have Even Index

What is a Quadratic Residue? - Definition and Example

What is a Quadratic Residue? - Definition and Example

How to use "Log" in Modular Arithmetic: Intro to Indices

How to use "Log" in Modular Arithmetic: Intro to Indices

The Awesome Connection between Primitive Roots and their Powers

The Awesome Connection between Primitive Roots and their Powers

Proof: Every Prime has a Primitive Root

Proof: Every Prime has a Primitive Root

What is the Order of x mod p? - Definition and Properties

What is the Order of x mod p? - Definition and Properties

Explicit Formula for Euler's Totient Function!

Explicit Formula for Euler's Totient Function!

Proof that the Totient Function is Multiplicative

Proof that the Totient Function is Multiplicative

Surjections and Bijections, and why they are useful.

Surjections and Bijections, and why they are useful.

Euler's Totient Theorem and Fermat's Little Theorem - Complete Proof & Intuition

Euler's Totient Theorem and Fermat's Little Theorem - Complete Proof & Intuition

Introduction to Euler's Totient Function!

Introduction to Euler's Totient Function!

Coprime Numbers and Reducing mod n

Coprime Numbers and Reducing mod n

When can we cancel a factor in a congruence?

When can we cancel a factor in a congruence?

Divisibility by 11: Proof with Modular Arithmetic

Divisibility by 11: Proof with Modular Arithmetic

Properties of Even/Odd Numbers with Modular Arithmetic

Properties of Even/Odd Numbers with Modular Arithmetic

3 Proofs of the Multiplication Rule in Modular Arithmetic!

3 Proofs of the Multiplication Rule in Modular Arithmetic!

Modular Arithmetic Basics: Congruence mod n

Modular Arithmetic Basics: Congruence mod n

What is an Injective Function? Definition and Explanation

What is an Injective Function? Definition and Explanation

Properties of Dirichlet Convolution with Prof. Omar!

Properties of Dirichlet Convolution with Prof. Omar!

27: Stokes' Theorem - Valuable Vector Calculus

27: Stokes' Theorem - Valuable Vector Calculus

26: Divergence Theorem - Valuable Vector Calculus

26: Divergence Theorem - Valuable Vector Calculus

Derivation of Divergence and Curl Formulas

Derivation of Divergence and Curl Formulas

25: Curl - Valuable Vector Calculus

25: Curl - Valuable Vector Calculus

24: Divergence - Valuable Vector Calculus

24: Divergence - Valuable Vector Calculus

23.5: Useful Surface Integral Shortcuts - Valuable Vector Calculus

23.5: Useful Surface Integral Shortcuts - Valuable Vector Calculus

23: Scalar and Vector Field Surface Integrals - Valuable Vector Calculus

23: Scalar and Vector Field Surface Integrals - Valuable Vector Calculus

An Ultraconvex Putnam Problem.

An Ultraconvex Putnam Problem.

22: Gradient Theorem - Valuable Vector Calculus

22: Gradient Theorem - Valuable Vector Calculus

Proof of the Exact Error of Taylor Series!

Proof of the Exact Error of Taylor Series!

21: Vector Field Line Integrals - Valuable Vector Calculus

21: Vector Field Line Integrals - Valuable Vector Calculus

Explicación del Método Tabular para Integración Por Partes

Explicación del Método Tabular para Integración Por Partes

Mass of a Wire with Line Integral - Vector Calculus Application

Mass of a Wire with Line Integral - Vector Calculus Application

20: Scalar Field Line Integrals - Valuable Vector Calculus

20: Scalar Field Line Integrals - Valuable Vector Calculus

See the c² in the Pythagorean Theorem Proof

See the c² in the Pythagorean Theorem Proof

Power Series of a Rational Number - 2017 Putnam problem B3

Power Series of a Rational Number - 2017 Putnam problem B3

Proof: Repeating Decimal Means Rational (with calculus)

Proof: Repeating Decimal Means Rational (with calculus)

When does f(x)=f⁻¹(x) mean f(x)=x? - Cube Root Coffin Problem Update

When does f(x)=f⁻¹(x) mean f(x)=x? - Cube Root Coffin Problem Update

Coffin Problem ft. blackpenredpen: Cube and Cube Root

Coffin Problem ft. blackpenredpen: Cube and Cube Root

Why are the Product & Sum for 1/(1-x) equal?

Why are the Product & Sum for 1/(1-x) equal?

b^0=1 and Euclidean Distance: Definition vs. Theorem

b^0=1 and Euclidean Distance: Definition vs. Theorem

Complex Finite/Infinite Product Example

Complex Finite/Infinite Product Example

How to write 1/(1-x) as an infinite PRODUCT!

How to write 1/(1-x) as an infinite PRODUCT!

19: Vector Fields - Valuable Vector Calculus

19: Vector Fields - Valuable Vector Calculus

AIME Problem: Double Quadratic with 4 Roots

AIME Problem: Double Quadratic with 4 Roots

Moments of Inertia with Double Integrals - Vector Calculus Application

Moments of Inertia with Double Integrals - Vector Calculus Application

0 ⋅ ∞ Limit Without L'Hospital's Rule

0 ⋅ ∞ Limit Without L'Hospital's Rule

18: Jacobian Determinant - Valuable Vector Calculus

18: Jacobian Determinant - Valuable Vector Calculus

Symmetric Square Root Integral 2 Ways!

Symmetric Square Root Integral 2 Ways!

17: Changing Order of Integration - Valuable Vector Calculus

17: Changing Order of Integration - Valuable Vector Calculus

Epsilon Delta Limits in 4 Minutes

Epsilon Delta Limits in 4 Minutes

Competition Problem - Product Rule with 16 Factors

Competition Problem - Product Rule with 16 Factors

16: Double Integrals - Valuable Vector Calculus

16: Double Integrals - Valuable Vector Calculus

Integral with some Very Big Powers

Integral with some Very Big Powers

15.5: Lagrange Multipliers Example - Valuable Vector Calculus

15.5: Lagrange Multipliers Example - Valuable Vector Calculus

15: Lagrange Multipliers - Valuable Vector Calculus

15: Lagrange Multipliers - Valuable Vector Calculus

Understanding Harmonic Addition Theorem with Vectors!

Understanding Harmonic Addition Theorem with Vectors!

The Gradient Vector: A Geometric Approach

The Gradient Vector: A Geometric Approach

14: Directional Derivatives and Gradient - Valuable Vector Calculus

14: Directional Derivatives and Gradient - Valuable Vector Calculus

Basic log question from the AIME

Basic log question from the AIME

13: Second Partial Derivative Test Derivation - Valuable Vector Calculus

13: Second Partial Derivative Test Derivation - Valuable Vector Calculus

An integral starring Gauss!

An integral starring Gauss!

12: Finding Max/Min in a Bounded Region - Valuable Vector Calculus

12: Finding Max/Min in a Bounded Region - Valuable Vector Calculus

Arcsin Integral

Arcsin Integral

Doing an infinite sum two different ways!

Doing an infinite sum two different ways!

Rational Function Integral: Partial Fractions? Maybe Not

Rational Function Integral: Partial Fractions? Maybe Not

2D Determinants: Proof with Polar Coordinates & Complex Numbers

2D Determinants: Proof with Polar Coordinates & Complex Numbers

11: Clairaut's Theorem Intuition - Valuable Vector Calculus

11: Clairaut's Theorem Intuition - Valuable Vector Calculus

A functional equation from the IMO shortlist!

A functional equation from the IMO shortlist!

10: Partial Derivatives - Valuable Vector Calculus

10: Partial Derivatives - Valuable Vector Calculus

Learning the Shoelace Method with an AMC problem

Learning the Shoelace Method with an AMC problem

9: Surfaces and Contour Plots - Valuable Vector Calculus

9: Surfaces and Contour Plots - Valuable Vector Calculus

log₂(log₁₆n) = log₄(log₄n)

log₂(log₁₆n) = log₄(log₄n)

Taking The AMC Be Like

Taking The AMC Be Like

2020 AMC Problem: How Many Solutions?

2020 AMC Problem: How Many Solutions?

8: Tangent and Normal Vectors - Valuable Vector Calculus

8: Tangent and Normal Vectors - Valuable Vector Calculus

7: Arc Length of Curves - Valuable Vector Calculus

7: Arc Length of Curves - Valuable Vector Calculus

University of Cambridge Taylor series question

University of Cambridge Taylor series question

6: Derivative of Dot and Cross Products - Valuable Vector Calculus

6: Derivative of Dot and Cross Products - Valuable Vector Calculus

A basic integral - know your strategies!

A basic integral - know your strategies!

5: Equations of Planes - Valuable Vector Calculus

5: Equations of Planes - Valuable Vector Calculus

Integral full of eˣ - Look for the Substitution!

Integral full of eˣ - Look for the Substitution!

AMC Problem: Circles and Prime Factors

AMC Problem: Circles and Prime Factors

Blackpenredpen & Dr. Peyam Shirt Unboxing!

Blackpenredpen & Dr. Peyam Shirt Unboxing!

4.5: Proof the Cross Product is Distributive - Valuable Vector Calculus

4.5: Proof the Cross Product is Distributive - Valuable Vector Calculus

4: Cross Product - Valuable Vector Calculus

4: Cross Product - Valuable Vector Calculus

3.5: Determinants in 3D - Valuable Vector Calculus

3.5: Determinants in 3D - Valuable Vector Calculus

3: Determinant Formula Intuition - Valuable Vector Calculus

3: Determinant Formula Intuition - Valuable Vector Calculus

2: Dot Product - Valuable Vector Calculus

2: Dot Product - Valuable Vector Calculus

Divisible by 3 & 9: Why add the digits?

Divisible by 3 & 9: Why add the digits?

1: How do vectors work? - Valuable Vector Calculus

1: How do vectors work? - Valuable Vector Calculus

0: Intro - Valuable Vector Calculus

0: Intro - Valuable Vector Calculus

We didn't learn this in class! - Diff EQ competition problem

We didn't learn this in class! - Diff EQ competition problem

A creative infinite sum AMC problem

A creative infinite sum AMC problem

Diff EQ Battle 6: Variation of Parameters

Diff EQ Battle 6: Variation of Parameters

An AMAZING infinite product!

An AMAZING infinite product!

Solving second-order differential equations a different way!

Solving second-order differential equations a different way!

9: Variation of Parameters - Dissecting Differential Equations

9: Variation of Parameters - Dissecting Differential Equations

A nice little trig integral

A nice little trig integral

Diff EQ Battle 5: Systems!

Diff EQ Battle 5: Systems!

8: Eigenvalue Method for Systems - Dissecting Differential Equations

8: Eigenvalue Method for Systems - Dissecting Differential Equations

Calculus competition problem: the 100th derivative?

Calculus competition problem: the 100th derivative?

7: Wronskian Determinant - Dissecting Differential Equations

7: Wronskian Determinant - Dissecting Differential Equations

RLC Circuits - Differential Equation Application

RLC Circuits - Differential Equation Application

Proof that the square root of ANY integer is irrational (besides perfect squares)

Proof that the square root of ANY integer is irrational (besides perfect squares)

Functions as Sums of Exponentials

Functions as Sums of Exponentials

Convolution Theorem and ℒ {Integral} Explanation

Convolution Theorem and ℒ {Integral} Explanation

Simple Guide to Series Convergence Tests

Simple Guide to Series Convergence Tests

Diff EQ Battle 4: Nonconstant Coefficients!

Diff EQ Battle 4: Nonconstant Coefficients!

Laplace transform of tf(t) and f(t)/t

Laplace transform of tf(t) and f(t)/t

Interesting probability question from the AMC

Interesting probability question from the AMC

Laplace transform of any derivative

Laplace transform of any derivative

Laplace transform rules: Linearity and Shifting

Laplace transform rules: Linearity and Shifting

AIME, the smart way and the stupid way

AIME, the smart way and the stupid way

6: Laplace Transforms - Dissecting Differential Equations

6: Laplace Transforms - Dissecting Differential Equations

Diff EQ Battle 3: Add or Multiply?

Diff EQ Battle 3: Add or Multiply?

Diff EQ Battle 2: Tricky Double Root

Diff EQ Battle 2: Tricky Double Root

Integral of ln(x) with Feynman's trick!

Integral of ln(x) with Feynman's trick!

Diff EQ BATTLE 1 ft. blackpenredpen

Diff EQ BATTLE 1 ft. blackpenredpen

5: Undetermined Coefficients - Dissecting Differential Equations

5: Undetermined Coefficients - Dissecting Differential Equations

Integral of cos(ax)cos(bx) WITHOUT integration by parts

Integral of cos(ax)cos(bx) WITHOUT integration by parts

Product to sum formulas for sin and cos - PROOF

Product to sum formulas for sin and cos - PROOF

4: First-Order Linear Differential Equations - Dissecting DEs

4: First-Order Linear Differential Equations - Dissecting DEs

Blackpenredpen Interview with Mu Prime - Highlights

Blackpenredpen Interview with Mu Prime - Highlights

A fun limit - Sum in the denominator?

A fun limit - Sum in the denominator?

3: Complex Roots - Dissecting Differential Equations

3: Complex Roots - Dissecting Differential Equations

Mu Prime Outtakes

Mu Prime Outtakes

Proving Euler's Identity FAST

Proving Euler's Identity FAST

2: Repeated Roots - Dissecting Differential Equations

2: Repeated Roots - Dissecting Differential Equations

A BEAST of a series - are you ready?

A BEAST of a series - are you ready?

Proving an Integral Formula for the Beta Function

Proving an Integral Formula for the Beta Function

Binomial Coefficients Identity with Induction

Binomial Coefficients Identity with Induction

1: Homogeneous Differential Equations - Dissecting DEs

1: Homogeneous Differential Equations - Dissecting DEs

0: Intro - Dissecting Differential Equations

0: Intro - Dissecting Differential Equations

Trig sub won't save you now

Trig sub won't save you now

Crazy limit - This looks familiar!

Crazy limit - This looks familiar!

9: Gershgorin Circle Theorem - Learning Linear Algebra

9: Gershgorin Circle Theorem - Learning Linear Algebra

8: Shifted Inverse Power Method - Learning Linear Algebra

8: Shifted Inverse Power Method - Learning Linear Algebra

7: Power Method for Eigenvalues - Learning Linear Algebra

7: Power Method for Eigenvalues - Learning Linear Algebra

6: Eigenvalues: Why det(A-λI) = 0 - Learning Linear Algebra

6: Eigenvalues: Why det(A-λI) = 0 - Learning Linear Algebra