Calculus BLUE Multivariable Volume 3: Integrals

✍ Scribed by Robert Ghrist

• Tongue: English
• Leaves: 469
• Series: Calculus BLUE Multivariable
• Edition: 3
• Category: Library

✦ Table of Contents

BLUE 3 INTRO
COVER
Title Page
Table of Contents
Instructions
LET’S GO!
CANTO 21
BLUE 3 PROLOGUE
TITLE
CHORUS
INTEGRALS!
CHORUS
The BIG IDEA
BUT SO WHAT?
CASE: surface areas
CASE: volumes
CASE: averages
CASE: mass & moments
CASE: solid body mechanics
CASE: probability
CASE: coordinate systems
CASE: data & spheres
CHORUS
SO MUCH MORE!
Chapter 1 - defining integrals
TITLE
CHORUS
Two types of integrals
CHORUS
Indefinite FAIL!!!
CHORUS
Riemann sums
CHORUS
How to think
Discretize!
DEFINITION: the integral
SEE IT: a Riemann sum
CHORUS
Bounded domains of integration
SEE IT: converging cubes
NOTATION: integrals
WHO CARES?
CHORUS
EXAMPLE: a computation?
Com-Pli-Ca-Ted
BONUS! luh-BAYG
The BIG PICTURE
PROBLEMS
Please sign…
Chapter 2 - fubini theorem
TITLE
CHORUS
How to compute?
Remember…
CHORUS
The double sum
THEOREM: Fubini
What Fubini looks like
CHORUS
Partial integration practice
EXAMPLE: simple double integral
CHORUS
EXAMPLE: area between curves
REMARKS: the Fubini Theorem
CHORUS
The BIG PICTURE
PROBLEMS
PROBLEMS
Chapter 3 - double integrals
TITLE
Double integrals
CHORUS
CHORUS
EXAMPLE: area of an ellipse
EXAMPLE: a simple mass computation
CHORUS
EXAMPLE: order of integration matters
CHORUS
EXAMPLE: additivity and integrals
CHORUS
EXAMPLE: improper double integrals
CHORUS
Gaussians!
The BIG PICTURE
PROBLEMS
PROBLEMS
Chapter 4 - triple integrals
TITLE
CHORUS
SEE IT: triple to double to single
Let’s see some…
CHORUS
EXAMPLE: a 3-d simplex
EXAMPLE: a 3-d simplex
CHORUS
EXAMPLE: fill in the blanks
CHORUS
EXAMPLE: visualizing in 3-d
EXAMPLE: visualizing in 3-d
YOU HAVE TO TRY!
The MORAL
CHORUS
Complex shapes are…
The BIG PICTURE
PROBLEMS
PROBLEMS
Chapter 5 - averages
TITLE
CHORUS
Remember…
The classical case
DEFINITION: average
DEFINITION: average
CHORUS
EXAMPLE: average vs extremal
EXAMPLE: average vs extremal
CHORUS
EXAMPLE: average area in 4-d
EXAMPLE: average area in 4-d
CHORUS
The root-mean-square
EXAMPLE: a high dimensional RMS
The BIG PICTURE
PROBLEMS
PROBLEMS
Chapter 6 - centroids and centers
TITLE
CHORUS
The classical case
The classical case
Centroids in 3-d
EXAMPLE: centroid in 3-d
EXAMPLE: centroid in 3-d
CHORUS
DEFINITION: center of mass
EXAMPLE: center of mass
EXAMPLE: center of mass
EXAMPLE: center of mass
CHORUS
EXAMPLE: center by parts
FOR FUN!
Is there a monostatic solid?
The GOMBOC
The BIG PICTURE
PROBLEMS
PROBLEMS
Acknowledgements
Chapter 7 - moments of inertia
TITLE
Remember…
Moment of inertia
LET'S RACE!
CHORUS
EXAMPLE: planar moment
CHORUS
EXAMPLE: solid cone moment
CHORUS
The Parallel Axis Theorem
EXAMPLE: parallel axis theorem
CHORUS
EXAMPLE: hollow cube moment
EXAMPLE: hollow cube moment
CHORUS
Radius of gyration
CASES: Mass distributions
CASES: Mass distributions
The BIG PICTURE
PROBLEMS
PROBLEMS
Chapter 8 - inertia matrix
TITLE
HEY WAIT!
CHORUS
The inertia matrix
EXAMPLE: inertia of a prism
EXAMPLE: inertia of a prism
CHORUS
EXAMPLES: mixed moments & symmetry
BUT SO WHAT?
CHORUS
Rotation about a skew axis
EXAMPLE: rotating cube
EXAMPLE: rotating cube
EXAMPLE: rotating prism
CAUTION!
BONUS!
FORESHADOWING
The BIG PICTURE
PROBLEMS
PROBLEMS
Chapter 9 - solid body mechanics
TITLE
CHORUS
Let’s spin!
CHORUS
Rotating elements
DEFINITION: angular velocity
CHORUS
DEFINITION: angular momentum
Momentum and Moments
EXAMPLE: inertia matrix of a prism
CHORUS
Torque and angular acceleration
Conservation
CHORUS
DEFINITION: kinetic energy
SUMMARY
The BIG PICTURE
PROBLEMS
PROBLEMS
Chapter 10 - probability and integration
TITLE
Quincunx!
1-D probability density
CHORUS
1-D probability: expectation and variance
CHORUS
Mass vs. Probability
EXAMPLE: a Pareto density
Probability densities
CHORUS
n-D probability density
n-D probability: expectation and variance
WHY? n-D random variables
Tracking, that’s why…
EXAMPLE: 2-D probability
EXAMPLE: 2-D probability
RELAX!!!
The BIG PICTURE
PROBLEMS
PROBLEMS
Chapter 11 - independence and covariance
TITLE
CHORUS
Multiple random variables
CHORUS
Marginal probability density
Marginalization
EXAMPLE: marginal probability densities
DEFINITION: independence
Independent random variables
Contours of Independence
Contours of Dependence
EXAMPLE: independent random variables
CHORUS
Expectation of linear combination
Variance of linear combination
CHORUS
EXAMPLE: portfolio risk
Yes, I said Yes.
CHORUS
Covariance and Correlation
Examples of correlation
CHORUS
The BIG PICTURE
PROBLEMS
PROBLEMS
Chapter 12 - covariance matrices
TITLE
CHORUS
ALGEBRA!
Covariance matrix
CHORUS
HEY WAIT!
CHORUS
Linear transformations
EXAMPLE: independence of combinations
CHORUS
CHORUS
EXAMPLE: portfolio risk, redux
CHORUS
Tracking & Prediction
The Motion Model
The Prediction Step
EXAMPLE: state update
EXAMPLE: state update
CHORUS
Uncertainty growth
Just Think!
BONUS!
The BIG PICTURE
PROBLEMS
PROBLEMS
Chapter 13 - cylindrical coordinates
TITLE
Integrals are hard…
CHORUS
Polar coordinates
CHORUS
The polar area element
EXAMPLE: polar double integral
CHORUS
Integrating a Gaussian
Told you it was tricksy
Integrating a Gaussian 2
CHORUS
Cylindrical coordinates
Cylindrical slices
CHORUS
EXAMPLE: solid cone moment
EXAMPLE: inertia matrix of cylinder
EXAMPLE: inertia matrix of cylinder
This would have been tough…
CHORUS
Higher dimensional polar?
The BIG PICTURE
PROBLEMS
PROBLEMS
Chapter 14 - spherical coordinates
TITLE
CHORUS
Spherical to Euclidean
Euclidean to Spherical
CAUTION!
CHORUS
Spherical slices
Spherical shapes
CHORUS
The Volume Element
CHORUS
EXAMPLE: spherical averages
EXAMPLE: spherical averages
EXAMPLE: a harder integral
EXAMPLE: solid ball moment
CHORUS
Solid Angle Form
The BIG PICTURE
PROBLEMS
PROBLEMS
Chapter 15 - changes of variables
TITLE
Volume elements: WHY?
CHORUS
Volume elements are the key!
CHORUS
Theorems on determinants
Determinants & n-volumes
Determinant = change in n-volume
CHORUS
Linear change of variables
EXAMPLE: area of an ellipse
CHORUS
LINEARIZE!
Determinants are the key!
LEMMA: volume elements
THIS WORKS!
CHORUS
THEOREM: change of variables
CHORUS
The BIG PICTURE
PROBLEMS
PROBLEMS
Please sign…
Chapter 16 - choosing coordinates
TITLE
CHORUS
EXAMPLE: u-subs redux
CHORUS
EXAMPLE: choosing coordinates
CHORUS
EXAMPLE: a 4-cycle engine
EXAMPLE: thermo & work
EXAMPLE: thermo & work
EXAMPLE: thermo & work
CHORUS
EXAMPLE: the Basel problem
EXAMPLE: the Basel problem
EXAMPLE: the Basel problem
EXAMPLE: the Basel problem
CHORUS
The BIG PICTURE
PROBLEMS
PROBLEMS
ACKNOWLEDGEMENTS
Chapter 17 - surface integrals
TITLE
CHORUS
Remember…
CHORUS
Parametrized surfaces REDUX
Remember the cross product?
The surface area element
EXAMPLE: surface area element of a sphere
…continued
EXAMPLE: surface area of a graph
EXAMPLE: area of a hyperbolic paraboloid
CHORUS
CASES: surface integrals in the wild
EXAMPLE: center of mass
CHORUS
The general surface area element
Higher dimensions…?
CHORUS
The BIG PICTURE
PROBLEMS
Chapter 18 - gaussians redux
TITLE
CHORUS
Probability distributions
Recall: 1-d Gaussians
CHORUS
Standard Gaussians
General Gaussians
CHORUS
Gaussian Covariance matrices
CHORUS
The Measurement Step
Data Fusion
Idea of the Kalman Filter
That’s a track, Jack!
CHORUS
Gaussian products
Gaussian products, ugh!
CHORUS
The Kalman Filter
To the moon…
BONUS! Fusion
CHORUS
The BIG PICTURE
PROBLEMS
PROBLEMS
ACKNOWLEDGEMENTS
Chapter 19 - data and dimension
TITLE
CHORUS
The Bell Curve
CHORUS
2-D Gaussian statistics
CHORUS
Consider the unit ball…
Volumes of spheres
Volumes of balls
CHORUS
Music of the Spheres
CHORUS
The n-D Gaussian mystery
CHORUS
The Magic Sphere
OW MY HEAD!
The BIG PICTURE
PROBLEMS
PROBLEMS
ACKNOWLEDGEMENTS
BLUE 3 EPILOGUE
TITLE
SO MUCH MORE!
CHORUS
You should…
CHORUS
INTEGRAL TRANSFORMS
Kernels
Examples of transforms
CHORUS
NUMERICAL ANALYSIS
Sampling on a mesh
Examples of weights
CHORUS
High dimensions, sigh
CHORUS
MONTE CARLO INTEGRATION
A Monte Carlo method
A Monte Carlo method
CHORUS
SO MUCH MORE!
BLUE 3 FORESHADOW
TITLE
CHORUS
Fields forever…
CHORUS
EXAMPLE: planar vector fields
Isn’t it ironic?
The BIG IDEA
CHORUS
CASE: gradient
CASE: divergence
CASE: curl
CHORUS
THE BIG THREE
CHORUS
A 1-FORM FIELD
CHORUS
Stokes’ Theorem
CHORUS
The Beatific Vision
The BIG PICTURE
LET’S GO!
BLUE 3 CLOSE
SCENE 22
COVER
About the author
REFERENCES
Where credit is due
Publisher of Beautiful Mathematics