The Shape Operator

The Basics

Introduction

Basic Concepts

Represetations and structure theory

Associated Vector Bundles

Group Actions

Fiber Bundles

Connections and Curvature

Spinors

Spinors

Summary

A linear transformation maps point $(x, y)$ to $(x^\prime, y^\prime)$. Linear transformations are denoted by $T$.

linear transformaton

coordinate system

cartesian plane

function graph

map

function composition

inverse function

intercept

square root

absolute value

sine

periodic function

cosine

tangent

exponential

natural logarithm

polynomial functions

vertical translation

horizontal translation

vertical scaling

horizontal scaling

general quadratic functions

right-angle triangle

the unit circle

the unit circle and the sine function

the unit circle and angles

non-unit circles

triangles

sum of angles

sphere

cylinder

circle

polar coordinates

circumference and arc length

union

intersection

vectors

vector addition

vector subtraction

scaling

vector representation

relative motion

the right-hand rule

complex numbers in polar representation

polar representation example

fundamental theorem of algebra

the fundamental idea of linear algebra

vector operations

matrix times vector 1

matrix times vector 2

matrix times vector 3

matrix times vector 4

matrix times matrix 1

matrix times matrix 2

matrix times matrix 3

matrix times matrix variant 1

matrix times matrix variant 2

matrix times matrix variant 3

matrix multiplication 1

matrix multiplication 2

matrix multiplication rules

determinant

area of a parallelogram

volume of a parallelepiped

quadratic polynomial in lambda

cavalieri's principle

swap rows

multiply a row by a constant

points

three-dimensional points

lines 1

lines 2

lines as intersection of planes

planes 1

planes 2

distance between a line and the origin

distance between a plane and the origin

projection onto a line

projection onto a plane

the rank-nullity theorem

the rank-nullity theorem proof 1

the rank-nullity theorem proof 2

the rank-nullity theorem proof 3

linear transformations

key properties of a linear transformation

linearity

pictorial kernel and image

x projection

y projection

x reflection

y reflection

reflection through a line

reflection through a plane

rotations

vector space structure

orthogonal 3

the action of a linear transformation T

the osculating circle

computing principal curvatures

not intrinsic

signed curvature

departure from tangent

the rate of turning of the tangent 1

the rate of turning of the tangent 2

the rate of turning of the normal

the force needed to hold an object in a circular orbit

Newton's curvature formula

travelling at unit speed

the Frenet frame

The Frenet-Serret equations

the principal normal

Euler's curvature formula

the principal curvatures of a surface

Newton's tratrix 1665

pseudosphere

surfaces of revolution

curvature contraction

net acceleration

Meusnier's theorem

geodesic curvature

Clairaut's theorem

the proof of Kepler's second law

proof of Clairaut's theorem 1

proof of Clairaut's theorem 2

proof of Clairaut's theorem 3

Directional derivatives

the shape operator

the geometric effect of the shape operator

singular value decomposition

the inverse matrix

the transpose matrix

the geometric interpretation of the shape operator

the curvature of a normal section of a surface

References

Sir Roger Penrose, The Road to Reality, (2004).
Tristan Needham, Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts, Princeton University Press, 2021.
Ivan Savov, No Bullshit Guide to Linear Algebra, published by Minireference Co., 2016.