Calculus BLUE Multivariable Vol 4: Fields

BLUE 4 INTRO COVER Title Page Table of Contents INSTRUCTIONS LET’S GO! SCENE 23 BLUE 4 PROLOGUE TITLE CHORUS FIELDS CHORUS Scalar & vector fields Flow fields Form fields CHORUS Algebra! Derivatives! Integrals! CHORUS THE BIG THREE CHORUS This is the End BUT SO WHAT? Work & flux Fluid dynamics Electromagnetics Medical imaging data Time series data CHORUS SO MUCH MORE! Chapter 1 - fields TITLE CHORUS What is a -field? CHORUS EXAMPLE: gravitational fields Vector fields vs. flowlines IT’S COMPLICATED CHORUS EXAMPLE: planar vector fields EXAMPLE: radial vector fields BUT SO WHAT? CASE: fluids & gravity CASE: electromagnetic fields CHORUS Derivatives? Matrix fields! Taylor? Polynomial fields! Physics? Tensor fields! RELAX!!! CHORUS Continuous fields The BIG PICTURE PROBLEMS PROBLEMS Chapter 2 - path integrals TITLE CHORUS BUT WHY? CHORUS DEFINITION: scalar path integral The BIG IDEA EXAMPLE: 2-d scalar path integral CHORUS THINK ABOUT IT The FACTS: path integrals CHORUS IMPORTANT! CHORUS EXAMPLE: paths in 3-d The BIG PICTURE PROBLEMS PROBLEMS Chapter 3 - integrating 1-forms TITLE CHORUS Vector path integrals BUT WHY? CHORUS DEFINITION: vector path integral CHORUS IT’S TIME! EXAMPLE: 1-forms EXAMPLE: 1-form fields CHORUS Gradient 1-forms CHORUS DEFINITION: integrating 1-forms EXAMPLE: integrating a 1-form field EXAMPLE: a loop integral Vectors or 1-forms? CHORUS The BIG PICTURE PROBLEMS PROBLEMS Chapter 4 - independence of path TITLE CHORUS A 1-FORM FIELD EXAMPLE: the gravitational 1-form CHORUS EXAMPLE: linear vs. rotational CHORUS THEOREM: Independence of Path Theorem PROOF Hello again, FTIC! CHORUS Detecting gradients EXAMPLE: gradient 1-forms? EXAMPLE: computing potentials CHORUS FTIC = key The BIG PICTURE PROBLEMS PROBLEMS Chapter 5 - circ work and flux TITLE CHORUS Breaking the 4th wall BUT WHY? CHORUS The WORK 1-form EXAMPLE: computing work CHORUS EXAMPLE: circulation EXAMPLE: work & potential CHORUS The FLUX 1-form WORK vs FLUX EXAMPLE: computing flux EXAMPLE: flux across a loop CHORUS The BIG PICTURE PROBLEMS PROBLEMS Chapter 6 - Greens theorem TITLE CHORUS EXAMPLE: an interesting 1-form EXAMPLE: an interesting 1-form CHORUS THEOREM: Green’s Theorem The BIG IDEA EXAMPLE: circulation & flux CHORUS EXAMPLE: differentiability matters EXAMPLE: orientation matters EXAMPLE: orientation matters BEWARE! Orientation CHORUS PROOF PROOF CHORUS BONUS! Applications of Green’s FORESHADOWING The BIG PICTURE PROBLEMS PROBLEMS Please sign... Chapter 7 - grad-curl-div TITLE CHORUS THEOREM: Green’s Theorem CHORUS 2-d curl & divergence Curl vs. Div in 2-d EXAMPLE: curl in 2-d EXAMPLE: divergence in 2-d CHORUS Divergence in 3-d EXAMPLE: divergence and volume Curl spinners in 3-d Curl and circulation densities Curl in 2-d as a vector field EXAMPLE: curl in 3-d NOTATION: del operator CHORUS IMPORTANT! CHORUS SO MUCH MORE! The BIG PICTURE PROBLEMS PROBLEMS Please sign... Chapter 8 - Euclidean forms in 3D TITLE CHORUS LET'S WORK in 3-D! LET’S THINK about 1-forms DEFINITION: basis 2-forms EXAMPLE: Euclidean 2-forms in 3-D CHORUS EXAMPLE: 2-form fields in 3-D EXAMPLE: flux 2-forms EXAMPLE: flux 2-form fields CHORUS The wedge product 3-forms and 3-form fields What lies beyond 3-forms? CHORUS Derivatives of form fields EXAMPLE: derivatives of forms EXAMPLE: the curl as derivative EXAMPLE: divergence as derivative CHORUS IMPORTANT! d^2=0 SUMMARY The BIG PICTURE PROBLEMS PROBLEMS Chapter 9 - integrating 2-forms TITLE CHORUS Integrating planar 2-forms CHORUS This simplifies Green’s Theorem CHORUS Remember... What do 2-forms eat? FORMULA: integrating 2-forms Concerning orientation A field of normals CHORUS EXAMPLE: 2-form field integral EXAMPLE: 2-form field integral CHORUS EXAMPLE: using symmetry CHORUS BUT SO WHAT? Flux 2-form of a vector field FLUX illustrated EXAMPLE: easy flux EXAMPLE: hard flux EXAMPLE: hard flux CHORUS The BIG PICTURE PROBLEMS PROBLEMS Chapter 10 - Gauss theorem TITLE CHORUS Green’s Theorem REDUX 3-d! CHORUS THEOREM: Gauss’s theorem EXAMPLE: flux across a cube EXAMPLE: flux across a cube CHORUS EXAMPLE: I love Gauss! CHORUS I love avocados! EXAMPLE: region between spheres EXAMPLE: region between spheres CHORUS The BIG PICTURE PROBLEMS PROBLEMS Chapter 11 - Stokes theorem TITLE CHORUS Green’s Theorem REDUX THEOREM: Stokes’ Theorem The BIG IDEA CHORUS EXAMPLE: easy Stokes’ Theorem This simplifies Green’s Theorem CHORUS EXAMPLE: different Stokes’ 1 EXAMPLE: different Stokes’ 2 EXAMPLE: different Stokes’ 3 CHORUS IT’S IRRELEVANT!!! CHORUS EXAMPLE: different Stokes’ 4 CHORUS The BIG PICTURE PROBLEMS PROBLEMS Chapter 12 - which theorem when TITLE All the big theorems in a row... CHORUS EXAMPLE: choice of surface EXAMPLE: choice of surface CHORUS EXAMPLE: open surface flux EXAMPLE: open surface flux CHORUS EXAMPLE: path vs loop integrals CHORUS Some advice The BIG PICTURE PROBLEMS PROBLEMS Acknowledgements Chapter 13 - forms and fluids TITLE LET’S HAVE SOME FUN BUT SO WHAT? CHORUS Perfect fluids Velocity fields CHORUS Remember...the material derivative The Euler equations EXAMPLE: a 3-d steady perfect fluid WHAT IS THIS? CIRCULATION THEOREM: Kelvin’s Theorem PROOF of Kelvin CHORUS Vorticity fields CHORUS THEOREM: Helmholtz’s Theorem YOU CAN SEE: Helmholtz PROOF of Helmholtz “Let it go... Let it go...” EXAMPLES: vortex preservation Don’t be discouraged! The BIG PICTURE PROBLEMS PROBLEMS Please sign... Chapter 14 - forms and emag TITLE LET’S HAVE SOME FUN CHORUS Everything is Coupled CHORUS Maxwell’s quadchart Maxwell’s equations: vector version CHORUS Maxwell & the E field Maxwell & the B field CHORUS Faraday & Maxwell 2-forms EXAMPLE: from vectors to forms Maxwell’s equations: forms version BUT SO WHAT? Don’t be discouraged! BONUS! Tensor fields The BIG PICTURE PROBLEMS PROBLEMS ACKNOWLEDGEMENTS Please sign... Chapter 15 - forms and data TITLE LET’S HAVE SOME FUN Computing area from data DETAILS: computing area CHORUS What’s it good for? CHORUS Centroids & moments CHORUS Computing volume from data DETAILS: computing volume DETAILS: computing volume CHORUS The Earth is round! Surface area from data DETAILS: surface area CHORUS But what about...? The BIG PICTURE PROBLEMS PROBLEMS ACKNOWLEDGEMENTS BLUE 4 INTERLUDE THE END...? PAUSE... CHORUS Dimension of "physical" systems CHORUS CASE: dynamical systems CASE: machine learning CHORUS ONWARD! Chapter 16 - differential forms in n-d TITLE CHORUS Let's work in N-D CHORUS DEFINITION: linear k-forms Basis 1/2/3-forms & determinants Basis k-forms & determinants EXAMPLE: basis forms CHORUS Wedge Machine! FACTS: the wedge product EXAMPLE: wedge it up CHORUS EXAMPLE: form fields EXAMPLE: flux forms SUMMARY BUT SO WHAT? CHORUS RECALL: time series data Leading vs. Lagging Parametric curves & oriented area Who's in the lead? So many questions Multiple signals MOTIVATION: forms in n-D CHORUS The BIG PICTURE PROBLEMS PROBLEMS Chapter 17 - calculus of forms TITLE CHORUS The exterior derivative: induction The exterior derivative: intuition EXAMPLE: derivatives of form fields CHORUS RULES: Linearity RULES: Product EXAMPLE: grad, div, & the product rule CHORUS IMPORTANT! d^2=0 PROOF: d^2=0 CHORUS Remember... Integration: 1, 2, 3, ... BONUS! Manifolds Parameterized domains FORMULA: integrating k-forms CHORUS EXAMPLE: a 3-form field integral EXAMPLE: integral on a torus in 4-d CASE: Hamiltonian optics CASE: the brightness 4-form CHORUS The BIG PICTURE PROBLEMS PROBLEMS Please sign... Chapter 18 - Stokes theorem TITLE CHORUS THEOREM: Stokes’ Theorem This is the End THINK! The FTIC is the Key PROOF: d^2=0 REDUX CHORUS RECALL: integration be parts Stokes' & integration by parts CHORUS CASES: Stokes' in advanced math CHORUS It's about time (series) Stokes' FTW The Lead Matrix THINK: why Stokes' helps CASE: discrete time sampling THINK! & don't stop... CHORUS TSSM The BIG PICTURE PROBLEMS PROBLEMS ACKNOWLEDGEMENTS BLUE 4 VISION TITLE un fulgore CHORUS LINEAR ALGEBRA DYNAMICAL SYSTEMS PDEs CHORUS ALGEBRA ANALYSIS COMBINATORICS GEOMETRY TOPOLOGY CHORUS SO MUCH MORE! The BIG PICTURE BLUE 4 CLOSE SCENE 24 COVER About the author REFERENCES Where credit is due Publisher of Beautiful Mathematics