حساب دیفرانسیل و انتگرال آبی، جلد دوم: مشتقات ۲

Finally, a math book that looks GREAT on a phone/tablet screen!This volume is the second 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 2: Derivatives uses vector and matrix algebra to understand derivatives and their applications to multivariable calculus. Topics covered in this text include: * multivariate functions with arbitrary numbers of inputs & outputs * partial derivatives * the derivative as a linear transformation on vectors of rates of change * differentiation rules, including the chain rule as matrix multiplication * the inverse and implicit function theorems, with applications * gradients and directional derivatives * linearization and Taylor expansion of multivariate functions * optimization of multivariate functions * applications to linear regression, game theory, & Nash equilibria * constrained optimization and the Lagrange multiplier * a hint of deeper applications in optimization theory, dynamical system, & moreThroughout, 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 3), each containing exercises for practice, for over 425 pages of mathematics and color and joy and hard work. The 2nd 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, search, highlight, build flashcards, and more. Enjoy! BLUE 2 INTRO COVER Title Page Table of Contents Instructions LET’S GO! SCENE 19 BLUE 2 PROLOGUE TITLE CHORUS You got it! CALCULUS! CHORUS Change is... How it all works CHORUS SO MUCH MORE! Chapter 1 - multivariate functions TITLE CALCULUS CHORUS Functions Let’s see some... Graphing functions 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: linear regression 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 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? EXAMPLE: ideal gas equation ...continued CHORUS The domain... ...maps to... ...the range CHORUS Linear combinations NOTATION: partial derivatives EXAMPLE: rates of change EXAMPLE: polar coordinates Seeing derivatives SUMMARY CAUTION The BIG PICTURE PROBLEMS PROBLEMS Chapter 4 - differentiation TITLE Slide Number 2 Slide Number 3 Slide Number 4 Slide Number 5 Slide Number 6 Slide Number 7 Slide Number 8 Slide Number 9 Slide Number 10 Slide Number 11 Slide Number 12 Slide Number 13 Slide Number 14 Slide Number 15 Slide Number 16 Slide Number 17 Slide Number 18 Slide Number 19 Chapter 5 - the chain rule TITLE Remember... 1-d chain rule In other words... CHORUS It’s time... THEOREM: Chain Rule CHORUS EXAMPLE: chain rule EXAMPLE: evaluation EXAMPLE: higher dimensions 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: quadratic forms ...continued CHORUS EXAMPLE: dot products CHORUS The product rule, REDUX The BIG PICTURE 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 EXAMPLE: polar coordinates, redux CHORUS EXAMPLE: solving nonlinear equations CHORUS The BIG PICTURE PROBLEMS PROBLEMS Chapter 8 - the implicit function theorem TITLE REMEMBER The classical case EXAMPLE: classical case CHORUS EXAMPLE: implicit function THEOREM: implicit function theorem EXAMPLE: implicit function theorem CHORUS 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 Vectors or Matrices? DEFINITION: gradient The Gradient CHORUS EXAMPLE: computing gradients CHORUS Level sets Gradients & level sets You can see this! You can see this! EXAMPLE: spheres CHORUS DEFINITION: vectors Dictionary CHORUS Remember this... FORESHADOWING: Gradients fields 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 Tangent spaces: implicit Tangent spaces: parametric BONUS! Kernels & images The BIG PICTURE PROBLEMS PROBLEMS Please sign... Chapter 11 - linearization TITLE CHORUS Linearization is approximation CHORUS Differentials or derivatives? EXAMPLE: numerical approximation CHORUS EXAMPLE: tolerances and error CHORUS Relative rates EXAMPLE: beam deflection error EXAMPLE: beam deflection error Do you see a pattern? CHORUS BUT SO WHAT? The BIG PICTURE PROBLEMS ACKNOWLEDGEMENTS Chapter 12 - taylor series TITLE Remember... Let’s recall CHORUS The FORMULA NOTATION: variables NOTATION: powers EXAMPLE: multi-index monomials NOTATION: derivatives EXAMPLE: higher derivatives IMPORTANT! NOTATION: factorials 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! The BIG PICTURE Chapter 16 - optimization - nash equilibria TITLE Game theory! CHORUS Payoff matrices EXAMPLE: rock, scissors, paper EXAMPLE: even-odd EXAMPLE: Mendelsohn CHORUS EXAMPLE: even-odd mixed strategy ...continued A saddle point equilibrium Nash equilibria 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 The BIG PICTURE PROBLEMS Chapter 18 - the lagrange multipler TITLE CHORUS EXAMPLE: a simple constraint CHORUS EXAMPLE: a simple constraint Think gradient... 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 You can see it... 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 The Fubini Theorem CHORUS EXAMPLE: triple integral CHORUS Mass distributions Probability distributions CHORUS The Change of Variables Theorem CHORUS Coordinate systems The BIG PICTURE LET’S GO! BLUE 2 CLOSE SCENE 20 COVER About the author