Calculus BLUE Vol I Vectors & Matrices Vol 1

Finally, a math book that looks GREAT on a phone/tablet screen! This volume is the first in a 4-volume set of CALCULUS BLUE books on multivariable calculus and is part of a revolutionary series of graphical Mathematics texts optimized for reading on phones/tablets/laptops. Not only is the format novel, the curricular approach is a contemporary take on the classic core, with a particular emphasis on preparation for modern data science. Calculus BLUE Multivariable Volume 1: Vectors & Matrices introduces vector and matrix algebra in the context of preparing for multivariable calculus. Topics covered in this text * curves and surfaces * vectors and coordinates * matrices and matrix algebra * matrix equations and linear systems * row reduction and inverse matrix computation * coordinate changes and bases * determinants and their interpretation and computation * a preview of linear algebra and matrices of partial derivatives Throughout, the text is filled with contemporary applications to robotics, data, mechanics, economics, probability, networks, and biology. Theory and application are brought to life in full eye-scorching color (though a lot of it is blue). This volume consists of 18 chapters (plus a prologue, epilogue, and foreshadowing of volume 2), each containing exercises for practice, for a total of 425 pages of mathematics and color and joy and hard labor. The 2nd edition includes a new chapter, an interlude, updated graphics throughout, and more efficient compression for faster navigation. NOTE! This is graphics intensive, and the file size is quite large. You will want to download it over a wireless connection rather than cellular. Also, although Amazon says this is a "print replica", there is no printed text this is based on. This book was hand-crafted for electronic format. It looks phenomenal on a modern smart-phone. You can pinch-zoom, use bookmarks, build custom flashcards, and more. Enjoy! BLUE 1 INTRO BLUE 1 COVER Title Page Table of Contents Instructions LET’S GO! SONG 17 BLUE 1 PROLOGUE PROLOGUE title CALCULUS! CHORUS Multivariate machines CHORUS Surfaces in 3-d Coordinates in n-d CHORUS LINEAR ALGEBRA! Vectors & matrices BUT SO WHAT? CASE: machine learning CASE: statistics CASE: geometry CASE: linear systems CHORUS SO MUCH MORE! ONWARD! Chapter 1 - lines and planes Chapter 1 – lines & planes CHORUS Remember... FORMULAE: lines EXAMPLE: lines CHORUS FORMULAE: planes EXAMPLE: planes CHORUS Parametrized lines EXAMPLE: parametrized lines CHORUS I wonder... The BIG IDEA BONUS! Hyperplanes BONUS! Hyperplane classifiers BONUS! Nonlinear classifiers The BIG PICTURE PROBLEMS PROBLEMS Chapter 2 - curves surfaces Chapter 2 – curves & surfaces CHORUS Planar curves CHORUS Put on your 3-d glasses! CHORUS Surfaces in 3-d EXAMPLE: surface parametrizations CHORUS Let’s see some... EXAMPLE: spheres EXAMPLE: ellipsoids EXAMPLE: hyperboloids EXAMPLE: hyperboloids EXAMPLE: elliptic paraboloids EXAMPLE: hyperbolic paraboloids EXAMPLE: cones EXAMPLE: cylinders RELAX!!! CHORUS The BIG PICTURE PROBLEMS PROBLEMS Please sign... Chapter 3 - coordinates Chapter 3 – coordinates & points CHORUS Coordinates in 2-d Coordinates in 3-d But what about... Coordinates in n-d BUT SO WHAT? CHORUS CASE: robot kinematics CASE: wireless signals CASE: customer profiles CHORUS Distance in n-d EXAMPLE: distance in 3-d EXAMPLE: distance in 8-d EXAMPLE: distance in 49-d CHORUS The BIG PICTURE PROBLEMS PROBLEMS Chapter 4 - vectors Chapter 4 - vectors CHORUS Vectors DEFINITION: vectors Vector components NOTATION: vectors CHORUS Vector algebra Vector visualization Vector lengths CHORUS Vector parametrizations CHORUS Basis vectors EXAMPLE: basis vectors EXAMPLE: vector length CAUTION CHORUS FORESHADOWING: vector calculus The BIG PICTURE PROBLEMS PROBLEMS Please sign... Chapter 5 - dot products Chapter 5 – dot product ALGEBRA! CHORUS DEFINITION: dot product CHORUS Dot product geometry EXAMPLE: dot products Orthogonality CHORUS Orthogonal projection +/- dot products & orientation EXAMPLE: projections CHORUS Hyperplane equations Hyperplane classifiers Hyperplane classifiers CHORUS LOVE & the dot product Computational love theory BONUS! FORESHADOWING: Fourier theory The BIG PICTURE PROBLEMS PROBLEMS Chapter 6 - cross products Chapter 6 – cross product CHORUS DEFINITION: cross product CHORUS Orthogonality Use the RIGHT hand! EXAMPLE: cross product & planes CHORUS Cyclic diagram Basis vector cross products EXAMPLE: computing cross product Cross product geometry Distances to lines & planes CHORUS DEFINITION: scalar triple product That’s how I spell it... CHORUS BONUS! The BIG PICTURE PROBLEMS PROBLEMS Chapter 7 - intro to vector calculus Chapter 7 – vector calculus CHORUS EXAMPLE: vectors in physics HEY WAIT! CHORUS Calculus & curves The derivative as a vector Tangent vectors EXAMPLE: velocity & acceleration CHORUS EXAMPLE: arclength integrals EXAMPLE: arclength computation CHORUS Vector differentiation rules EXAMPLE: gravitation CHORUS The BIG PICTURE PROBLEMS PROBLEMS Chapter 8 - vector calculus and motion Chapter 8 – vector calculus & motion CHORUS EXAMPLE: integrating for position CHORUS Acceleration: tangents & normals EXAMPLE: unit tangents & unit normals Decomposition of acceleration CHORUS Curvature: definition & computation The osculating circle EXAMPLE: curvature in 2-d CHORUS The binormal vector & torsion BONUS! CHORUS YOU NEED MORE! The BIG PICTURE PROBLEMS PROBLEMS Acknowledgements BLUE 1 INTERLUDE THE BEGINNING? PAUSE... CHORUS Curves in the nth dimension Time series data ...in the nth dimension Dimension of "physical" systems CHORUS It's not obvious Rates of change? It's the question ONWARD! Chapter 9 - matrices Chapter 9 - matrices What is the matrix? DEFINITION: matrix Matrix sizes Matrix forms Zero & identity matrices CHORUS CASE: stress matrix CASE: digital images CASE: social networks CASE: genetic correlations CASE: Markov chains CASE: darn you autocomplete! CHORUS DEFINITION: transpose Symmetry of transpose The BIG PICTURE PROBLEMS PROBLEMS Chapter 10 - matrix algebra Chapter 10 – matrix algebra ALGEBRA! Matrix addition Scalar multiplication CHORUS Matrix multiplication EXAMPLE: matrix vector multiplication CHORUS Rotation matrix J A complex representation CHORUS Noncommutativity First underpants... EXAMPLE: transpose & multiplication The BIG PICTURE PROBLEMS PROBLEMS Chapter 11 - matrix equations Chapter 11 – matrix equations CHORUS Linear systems Ax = b CHORUS CASE: chemical reactions CASE: circuit mesh analysis CASE: network traffic flow CASE: commodity production CASE: spring-mass systems CHORUS How to solve? The BIG PICTURE PROBLEMS PROBLEMS Chapter 12 - row reduction Chapter 12 – row reduction Ax = b How hard is it? CHORUS EXAMPLE: back-substitution CHORUS Augmented matrix The BIG IDEA Row operation 1 Row operation 2 Row operation 3 EXAMPLE: 3-by-3 system ...continued CHORUS Tips & tricks: pivoting The BIG PICTURE PROBLEMS PROBLEMS Acknowledgements Chapter 13 - inverse matrices Chapter 13 – inverse matrices FAKE!!! DEFINITION: inverse Matrix inverse CHORUS Noninvertibility EXAMPLE: 2-by-2 inverse BUT SO WHAT? Solving equations CHORUS Computing inverses ...continued Complete row reduction CHORUS EXAMPLE: the geometric series! Hooray! The BIG PICTURE PROBLEMS PROBLEMS Chapter 14 - linear transformations Chapter 14 – linear transformations CHORUS The BIG IDEA Linear transformations Basis vectors suffice Let’s see some... CASE: rescaling CASE: projection CASE: shearing CASE: rotation CHORUS IMPORTANT! Composition is matrix multiplication CHORUS EXAMPLE: Euler angles EXAMPLE: Euler angles EXAMPLE: a 4-d transform BONUS! The BIG PICTURE PROBLEMS PROBLEMS Acknowledgements Chapter 15 - coordinate transformations Chapter 15 – coordinate transforms CHORUS Where’s the best deli? Nonlinear coordinates CHORUS Basis vectors The BIG IDEA EXAMPLE: changing coordinates CHORUS EXAMPLE: change of basis Coordinate changes Coordinate changes Coordinate changes CHORUS EXAMPLE: rotations EXAMPLE: rotations FORESHADOWING: nonlinear coordinates The BIG PICTURE PROBLEMS PROBLEMS Please sign... Acknowledgements Chapter 16 - algebraic determinants Chapter 16 – algebraic determinants DEFINITION: determinant Determinants: 2-by-2 CHORUS Determinants: 3-by-3 CHORUS Scalar triple & cross products CHORUS Minor expansion ...continued EXAMPLE: 3-by-3 determinant CHORUS EXAMPLE: 4-by-4 determinant Sign convention CHORUS TEASER The BIG PICTURE PROBLEMS PROBLEMS Chapter 17 - geometric determinants Chapter 17 – geometric determinants CHORUS Determinants & area Determinants & volume CHORUS Cross product & determinants Cross product & areas CHORUS Determinants & n-volumes WATCH THIS! Determinant = change in n-volume CHORUS Composition & determinants CHORUS Theorems on determinants The BIG PICTURE PROBLEMS PROBLEMS Chapter 18 - computing determinants Chapter 18 – computing determinants Computing determinants: O(n!) ? CHORUS Triangular is better... The BIG IDEA CHORUS Row operation 1 Row operation 2 Row operation 3 SUMMARY EXAMPLE: row reduction & determinants EXAMPLE: determinants & row exchange CHORUS ALGORITHM: determinants CHORUS Transpose & determinant BONUS! The BIG PICTURE PROBLEMS PROBLEMS BLUE 1 EPILOGUE BLUE 1 EPILOGUE SO MUCH MORE! CHORUS LEARN LINEAR ALGEBRA! EIGENVALUES DEFINITION: eigenvalues What eigenvalues mean CHORUS VECTOR SPACES DEFINITION: vector spaces BUT SO WHAT? EXAMPLES: vector spaces CHORUS SO MUCH MORE! BLUE 1 FORESHADOW BLUE 1 FORESHADOW CHORUS How many rates? CHORUS It’s a matrix! CHORUS Chain Rule! CHORUS Gradients! CHORUS Taylor series! CHORUS Max/min! CHORUS Lagrange! The BIG PICTURE LET’S GO! BLUE 1 CLOSE SONG 18 BLUE 1 COVER About the Author REFERENCES Where Credit is Due Publisher of Beautiful Mathematics