Calculus BLUE Multivariable Calculus Vol II Derivatives

✍ Scribed by Robert Ghrist

• Publisher: Agenbyte Press
• Year: 2019
• Tongue: English
• Leaves: 475
• Series: 2
• Edition: 3
• Category: Library

✦ Table of Contents

BLUE 2 INTRO
COVER
Title Page
Table of Contents
Instructions
LET’S GO!
SONG 19
BLUE 2 PROLOGUE
TITLE
CHORUS
We got it…
CHORUS
Change is…
How it all works
CASE: Gradients
CASE: Linearization
CASE: Optimization
CASE: Lagrange
BUT SO WHAT?
CASE: Approximation
CASE: the IFT
CASE: Statistics
CHORUS
SO MUCH MORE!
ONWARD!
Chapter 1 - multivariate functions
TITLE
CALCULUS!
CHORUS
Graphing functions 1-D
Graphing functions 2-D
Functions!
CHORUS
CASE: curves
CASE: surfaces
CASE: level set curves
CASE: level set surfaces
CASE: color, time, & more
CHORUS
CASE: weather
CASE: weather
CASE: weather
CHORUS
CASE: coordinate change
CASE: coordinate change
CASE: coordinate change
CHORUS
CASE: market equilibrium
CASE: market equilibrium
CASE: epidemic models
CHORUS
CASE: robot kinematics
CASE: data
CHORUS
Spin the Dials!
CHORUS
The BIG PICTURE
PROBLEMS
PROBLEMS
ACKNOWLEDGEMENTS
Chapter 2 - partial derivatives
TITLE
Remember…
The derivative
DEFINITION: partial derivative
DEFINITION: partial derivative
CHORUS
EXAMPLE: partial & implicit differentiation
NOTATION: partial derivatives
CHORUS
Slopes & derivatives
LET'S SEE: Slopes & derivatives
CHORUS
EXAMPLE: inputs & outputs
CASE: managing a bakery
CHORUS
It’s the question that drives you
The BIG PICTURE
PROBLEMS
PROBLEMS
Chapter 3 - the derivative
TITLE
What is the derivative?
It’s a matrix!
BUT SO WHAT?
IMPORTANT!
CHORUS
Curves & rates of change
Surfaces & rates of change
NOTATION: partial derivatives
EXAMPLE: rates of change
EXAMPLE: polar coordinates
Seeing derivatives
CHORUS
EXAMPLE: sensitivity
SUMMARY
CAUTION
The BIG PICTURE
PROBLEMS
PROBLEMS
Chapter 4 - differentiation
TITLE
DEFINITION: derivative
The Derivative
EXAMPLE: linear functions
CHORUS
An unusual annoyance
OK, so what? Oh wait.
YOU CAN SEE: the wrinkle
PANIC!!!
CHORUS
EXAMPLE: squaring square matrices
EXAMPLE: AHHA...
CHORUS
THINK ABOUT IT
Like, seriously, think!
CHORUS
The BIG PICTURE
PROBLEMS
PROBLEMS
ACKNOWLEDGEMENTS
Chapter 5 - the chain rule
TITLE
Remember…
1-d chain rule
In other words…
CHORUS
It’s time…
THEOREM: Chain Rule
CHORUS
EXAMPLE: evaluation
EXAMPLE: higher dimensions
CHORUS
EXAMPLE: chain rule
FALSE CHORUS
Classical chain rules
Classical chain rules
CHORUS
The BIG PICTURE
PROBLEMS
PROBLEMS
Chapter 6 - differentiation rules
TITLE
CHORUS
Linearity
EXAMPLE: linear & affine functions
CHORUS
EXAMPLE: dot products
CHORUS
The product rule, REDUX
CHORUS
EXAMPLE: quadratic forms
…continued
CHORUS
The material derivative
Material derivative derivation
EXAMPLE: BOOP!
FORESHADOWING
The BIG PICTURE
PROBLEMS
PROBLEMS
Chapter 7 - the inverse function theorem
TITLE
CHORUS
Recall: inverse rule for derivatives
CHORUS
DEFINITION: inverse functions
EXAMPLE: local nonlinear inverses
The BIG IDEA
CHORUS
The inverse rule for derivatives
EXAMPLE: derivative of an inverse
CHORUS
THEOREM: Inverse Function Theorem
CHORUS
EXAMPLE: polar coordinates, redux
CHORUS
EXAMPLE: solving nonlinear equations
CHORUS
EXAMPLE: inverse kinematics
The BIG PICTURE
PROBLEMS
PROBLEMS
Chapter 8 - the implicit function theorem
TITLE
Remember…
The classical case
EXAMPLE: classical case
CHORUS
EXAMPLE: implicit function
CHORUS
THEOREM: implicit function theorem
EXAMPLE: implicit function theorem
CHORUS
EXAMPLE: price equilibria
EXAMPLE: price equilibria
CHORUS
EXAMPLE: GPS accuracy
EXAMPLE: GPS accuracy
EXAMPLE: GPS accuracy
CHORUS
The BIG PICTURE
PROBLEMS
PROBLEMS
ACKNOWLEDGEMENTS
Chapter 9 - gradients
TITLE
CHORUS
Level sets in 2-d…
…are curves
Level sets in 3-d?
Nonlinear classifiers
CHORUS
Vectors or Matrices?
DEFINITION: gradient
The Gradient
CHORUS
EXAMPLE: computing gradients
CHORUS
Gradients & level sets
You can see this in 2-d
You can see this in 3-d
EXAMPLE: spheres
CHORUS
FALSE CHORUS
DEFINITION: vectors
Dictionary
BONUS: stochastic gradient descent
FORESHADOWING: gradients fields
CHORUS
Remember this…
The BIG PICTURE
PROBLEMS
PROBLEMS
Chapter 10 - tangent spaces
TITLE
CHORUS
Tangent planes
EXAMPLE: tangent planes
CHORUS
Parametrized curves
CHORUS
Parametrized surfaces
EXAMPLE: parametrized tangent planes
EXAMPLE: continued…
But what about…?
CHORUS
BONUS! Kernels & images
Tangent spaces: implicit
Tangent spaces: parametric
The BIG PICTURE
PROBLEMS
PROBLEMS
Please sign…
Chapter 11 - linearization
TITLE
CHORUS
Linearization to approximate curves
Linearization to approximate surfaces
CHORUS
Differentials or derivatives?
CHORUS
EXAMPLE: numerical approximation
EXAMPLE: tolerances and error
CHORUS
Relative rates
EXAMPLE: beam deflection error
EXAMPLE: beam deflection error
Do you see a pattern?
CHORUS
Accuracy of linear approximation
Taylor time!
The BIG PICTURE
PROBLEMS
PROBLEMS
ACKNOWLEDGEMENTS
Chapter 12 - taylor series
TITLE
Remember…
Let’s recall: Taylor series
What a Taylor polynomial means
CHORUS
What a Taylor polynomial means 2
The FORMULA
RELAX!!!
CHORUS
NOTATION: variables
NOTATION: powers
EXAMPLE: multi-index monomials
NOTATION: factorials
CHORUS
NOTATION: derivatives
EXAMPLE: higher derivatives
IMPORTANT!
CHORUS
FORMULA: Taylor series about 0
FORMULA: Taylor series about a
CHORUS
Remarks
The BIG PICTURE
PROBLEMS
PROBLEMS
Chapter 13 - computing taylor series
TITLE
Let’s see some…
CHORUS
EXAMPLE: 2-d Taylor series
CHORUS
EXAMPLE: Taylor the hard way
EXAMPLE: Taylor the easy way
CHORUS
EXAMPLE: Using the chain rule
BUT SO WHAT?
EXAMPLE: local solutions to equations
CHORUS
WHY?
DEFINITION: Hessian
BONUS!
CAUTION!
The BIG PICTURE
PROBLEMS
Chapter 14 - critical points and optimization
TITLE
Remember…
CHORUS
Time for a definition
DEFINTION: critical points
LEMMA: critical points
CHORUS
EXAMPLE: finding critical points
Graphing critical points
A saddle!
CHORUS
The second derivative
Trace-determinant method
The second derivative test
EXAMPLE: classification
Degenerate critical points
CHORUS
CAUTION: boundaries
CHORUS
The BIG PICTURE
PROBLEMS
Please sign…
Chapter 15 - optimization - linear regression
TITLE
Linear regression
CHORUS
Least squares problem
Take a partial derivative
Take another partial derivative
Solve for the critical point
CHORUS
Solve for the critical point
The solution!
CHORUS
Check minimality
BONUS!
Linear regression
Nonlinear regression?
Topological data analysis?!
The BIG PICTURE
PROBLEMS
Chapter 16 - optimization - nash equilibria
TITLE
It's time for some
CHORUS
Payoff matrices
EXAMPLE: rock, scissors, paper
EXAMPLE: even-odd
EXAMPLE: Mendelsohn
CHORUS
Playing at random
Expected payoffs from random play
CHORUS
EXAMPLE: even-odd mixed strategy
…continued
A saddle point equilibrium
Nash equilibria
LET'S SEE: Nash equilibrium
BONUS!
EXAMPLE: Nash equilibria
CHORUS
THINK!
The BIG PICTURE
PROBLEMS
ACKNOWLEDGEMENTS
Chapter 17 - constrained optimization
TITLE
CHORUS
CASES: constraints
CHORUS
RECALL: 1-d bounded optimization
BUT SO WHAT?
CHORUS
EXAMPLE: a simple constraint
…continued
…continued
CHORUS
EXAMPLE: an optimal box
…continued
CHORUS
EXAMPLE: a parametrized boundary
…continued
…continued
Higher dimensions? Yikes!
The BIG PICTURE
PROBLEMS
Chapter 18 - the lagrange multipler
TITLE
CHORUS
EXAMPLE: a simple constraint
CHORUS
EXAMPLE: a simple constraint
Think gradient…
CHORUS
You can see it…3D
You can see it…2D
THIS WORKS!
CHORUS
THEOREM: Lagrange multiplier
EXAMPLE: Lagrange
The BIG IDEA
The Lagrange equations
WHAT IS IT?
Interpretations of Lagrange
CHORUS
BONUS! Lagrange & rates of change
Lagrange multiplier: A PICTURE!
CHORUS
The BIG PICTURE
PROBLEMS
Chapter 19 - using the lagrange equation
TITLE
The Lagrange equations
CHORUS
EXAMPLE: supply chains
CHORUS
EXAMPLE: inscribing a cone
CHORUS
EXAMPLE: hard & in 3-d
…continued
…continued
CHORUS
EXAMPLE: resource allocation
CHORUS
EXAMPLE: algebraic-geometric inequality
…continued
FORESHADOWING
The BIG PICTURE
PROBLEMS
PROBLEMS
BLUE 2 EPILOGUE
TITLE
SO MUCH MORE!
CHORUS
You should…
OPTIMIZATION
Multi-Lagrange
…continued
CHORUS
EIGENVALUES
CHORUS
DYNAMICAL SYSTEMS
Linear ODE systems
The matrix exponential
Classification of equilibria
CHORUS
Nonlinear dynamics
CHORUS
OW MY HEAD!
SO MUCH MORE!
BLUE 2 FORESHADOW
TITLE
CHORUS
FAIL!!!
The definite integral
CHORUS
One weird trick…
CHORUS
Take it to the limit
CHORUS
APPLICATION: mass
APPLICATION: probability
CHORUS
the BIG THEOREM
CHORUS
Coordinate systems
APPLICATION: surface area
Higher dimensions…
The BIG PICTURE
LET’S GO!
BLUE 2 CLOSE
SONG 20
COVER
About the Author
REFERENCES
Where credit is due
Publisher of Beautiful Mathematics