Transcription of 1 Vector Calculus, Linear Algebra, and Difierential Forms ...
1 1 Vector calculus , Linear algebra , and Di erentialForms: A Uni ed ApproachTable of ContentsPREFACExiCHAPTER 0 Reading Quanti ers and Set Real In nite Complex Numbers26 CHAPTER 1 Vectors, Matrices, and Introducing the Actors: Points and Introducing the Actors: A Matrix as a The Geometry Limits and Four Big Di erential Rules for Computing Mean Value Theorem and Criteria for Di Review Exercises for Chapter 1162 CHAPTER 2 Solving The Main Algorithm: Row Solving Equations Using Row Matrix Inverses and Elementary Linear Combinations, Span, and Linear Kernels, Images, and the Dimension An Introduction to Abstract Vector Newton's The Inverse and Implicit Function Review Exercises for Chapter 2285 CHAPTER 3 Higher Partial Derivatives, Quadratic Forms ,and Tangent Taylor Polynomials in Several Rules for Computing Taylor Quadratic Classifying Critical Points of Constrained Critical Points and Lagrange Geometry of Curves and Review Exercises for Chapter 3394 CHAPTER 4 De ning the Probability and Centers of What Functions Can Be Integrated?
2 Integration and Measure Zero (Optional) Fubini's Theorem and Iterated Numerical Methods of Other Volumes and The Change of Variables Lebesgue Review Exercises for Chapter 4523 CHAPTER 5 Volumes of Parallelograms and their Computing Volumes of Fractals and Fractional Review Exercises for Chapter 5555 CHAPTER 6 Forms and Vector Forms Integrating Form Fields over Parametrized Orientation of Integrating Forms over Oriented Forms and Vector Boundary The Exterior The Exterior Derivative in the Language of Vector The Generalized Stokes's The Integral Theorems of Vector Review Exercises for Chapter 6664 APPENDIX A: Some Harder Arithmetic of Real Cubic and Quartic Two Extra Results in Proof of the Chain Proof of Kantorovich's Proof of Lemma (Superconvergence) Proof of Di erentiability of the Inverse Proof of the Implicit Function Proof of Theorem : Equality of Crossed Proof of Proposition Proof of Rules for Taylor Taylor's Theorem with Proof of Theorem (Completing Squares) Geometry of Curves and Surfaces: Proof of the Central Limit Proof of Fubini's Justifying the Use of Other Existence and Uniqueness of the Rigorous Proof of the Change of Variables Justifying Volume Lebesgue Measure and Proofs for Lebesgue Justifying the Change of Computing the Exterior The Proof of Stokes' Theorem771 APPENDIX B: Monte Carlo Determinant Program786 BIBLIOGRAPHY789 INDEX791
