حساب دیفرانسیل و انتگرال آبی: چندمتغیره، جلد سوم: انتگرال‌ها

Finally, a math book that looks GREAT on a phone/tablet screen! This volume is the third 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 3: Integrals continues the journey through multivariable calculus, building on previous material from linear algebra and multivariate derivatives in volumes 1 and 2. Topics covered in this text * multivariate Riemann sums, Riemann integrals, and the Fubini Theorem * double, triple, and higher-dimensional integrals, with motivations & applications * using integrals to compute averages * using integrals to compute mass, centroids, and centers of mass * moments of inertia, the inertia matrix, and basic solid body mechanics * probability in single and multiple variables; independence and expectation * the variance-covariance matrix and its applications * cylindrical & spherical coordinates * the Change of Variables Theorem and its applications * surface integrals, with applications to surface area and more * multivariate Gaussians, the Kalman filter, and high-dimensional geometry 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 19 chapters (plus a prologue, epilogue, and foreshadowing of volume 4), each containing exercises for practice, for over 450 pages of mathematics and color and joy and hard work. The 3rd edition includes a new chapter, updated graphics throughout, and more efficient compression for faster navigation. NOTE! This is graphics intensive, and the file size is quite large. You may 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, flashcards, and more. Enjoy! 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