The SEVENTH of the seven beauties: the derivation of the integral of signed curvature

The SEVENTH of the seven beauties: The integral of signed curvature over a closed curve

The Polar Form of Complex Numbers

De Moivre's Formula for Complex Numbers: A Precursor to Euler's Formula

The Elementary Properties of Complex Numbers According to Hamilton

Accepting Complex Numbers: Hamilton vs Euler

The Birth of Complex Numbers and Tartaglia's Formula

Tartaglia's Formula in Action and Numbers of the form a+b√5

Tartaglia's Formula in Action: Two Examples

Euler's Derivation of the Tartaglia-Cardano Formula

Polynomial Equations and the Fundamental Theorem of Algebra

The SIXTH of the seven beauties. The integral of position vector dotted with unit normal.

The FIFTH of the seven beauties. The integral of position vector dotted with unit tangent.

The FOURTH of the seven beauties. The integral of position vector dotted curvature normal.

The THIRD of the seven beauties. The integral of unit normal over a curve.

The SECOND of the seven beauties! The integral of curvature normal over a curve.

A brief review of curves and integration over curves.

The FIRST of the seven beauties! The integral of unit tangent over a curve.

The Expression for the Arc Length of a Curve in Cartesian Coordinates

The Geometric Expression for the Arc Length of a Curve

From Physical to Arithmetic Integrals - Achieving Complete Agreement with Intuition

Hate most advanced math books? These words of encouragement are for you

From Physical to Arithmetic Integrals - How to Achieve Invariance

Converting a Physical Integral into an Arithmetic One: Identifying the Challenge

Why Change of Variables Is Often an Inherently Ambiguous Task

The Derivative of the Inverse Function, or How to Reparameterize a Curve

The Derivative of the Inverse Function - Why It Is Confusing (and how to eventually fix it)

Integration by Substitution Is an Application of the Chain Rule

The Need for Arbitrary Parameterizations of Curves

Arithmetic Integrals And the Fundamental Theorem of Calculus

Geometric Integrals - a Fundamental Concept to Be Accepted on an Intuitive Basis

Hypersurfaces, Unit Normals, and Signed Curvature

The Curvature of a Curve, Its Principal Normal, And Its Osculating Plane

The Torsion of a Curve

Linear Decomposition by the Dot Product

The Arc Length Parameterization

The Analysis of the Unit Circle as a Planar Curve in Differential Geometry.

The Closest Point on a Curve to a Given Point: Combing Calculus And Geometry

Kinematic Analysis of a Rigid Bar

Classical Euclidean Geometry Is Limited to Three Dimensions

Constant Speed Motion - Analysis via differentiation of vector-valued functions

The Properties of Vector Differentiation

The Main Intuitive Idea of Calculus (applied to curves in Differential Geometry)

The Geometric Interpretation of the DERIVATIVE of a Vector-Valued Function

The Geometric Interpretation of a Vector-Valued Function: It's a curve - end of story.

Differentiation of Vectors: What It Actually Means to DIfferentiate a Vector-Valued Function!

My Proof of Pythagoras' Theorem is NOT circular!

The Heights of a Triangle Are Concurrent, a Vector-Based Proof That Does Not Require Any Ingenuity!

The Orthogonal Projection Expressed in Terms of The Dot Product

Ceva's Theorem, A Vector Based Proof - No Ingenuity Required! Concurrent Lines in a Triangle

The Bisector Theorem: A Vector-Based Proof - No Ingenuity Required!

The Medians of a Triangle Are Concurrent, A Vector-Based Proof - As Straightforward as It Gets!

Introduction to Affine Spaces: What you can do in Linear Algebra w/o inner products. Less Is More!

A Simple Solution to an Otherwise Challenging Problem - Algebra and Geometry Working Together!

The Vector Equation of a Straight Line. This will help us solve a number of difficult problems soon!

The Diagonals of a Rhombus Intersect at a Right Angle - A Dot Product Proof

The Dot Product Proof of the Pythagorean Theorem - The power of Vectors Is on Full Display!

The Dot Product and the Inner Product: Comparing These Key Concepts in Linear Algebra, Geometry, ℝⁿ

What Is a Vector? It depends! Geometric perspective vs Linear Algebra vs ℝⁿ

Mr. Tibbs, the Queen's butler in BFG made a math mistake. Did the Movie Get It Right? We Investigate