Engineering Math

Introduction 1ptSome Prerequisite Topics Sets And Set Notation Well Ordering And Induction The Complex Numbers Polar Form Of Complex Numbers Roots Of Complex Numbers The Quadratic Formula The Complex Exponential Dividing Polynomials The Fundamental Theorem Of Algebra Exercises Integrals, Functions of One Variable Properties of the Integral Uniform Convergence of Continuous Functions Uniform Convergence And The Integral Some Important Improper Integrals Gamma Function Laplace Transforms I 1ptLinear Algebra And Multivariable Calculus Fundamentals Fn Algebra in Rn Geometric Meaning Of Vector Addition In R3 Lines Distance in Rn Geometric Meaning Of Scalar Multiplication In R3 Exercises Physical Vectors Exercises Vector Products The Dot Product The Geometric Significance Of The Dot Product The Angle Between Two Vectors Work And Projections The Dot Product And Distance In Cn Exercises The Cross Product The Box Product Proof of the distributive law Torque Center Of Mass Angular Velocity Vector Identities And Notation Planes Exercises Systems Of Equations Systems Of Equations, Algebraic Procedures Elementary Operations Gauss Elimination Balancing Chemical Reactions Dimensionless Variables MATLAB And Row Reduced Echelon Form Exercises Matrices Addition And Scalar Multiplication Of Matrices Multiplication of Matrices Linear Transformations and Matrices Multiplication of Matrices The Transpose Some Examples of Linear Functions on Rn Rotations in R2 Projections Rotations About A Particular Vector The Inverse of a Matrix The Identity And Inverses Finding The Inverse Of A Matrix MATLAB And Matrix Arithmetic Exercises Subspaces Spans and Bases Subspaces Exercises Eigenvalues and Eigenvectors Definition of Eigenvalues An Introduction to Determinants Cofactors and 22 Determinants The Determinant of a Triangular Matrix Properties of Determinants Finding Determinants Using Row Operations Applications A Formula For The Inverse Finding Eigenvalues Using Determinants Matrices and The Inner Product Eigenvalues and Eigenvectors Using Matlab Distance and Unitary Matrices Schur's Theorem Diagonalization Approximations Fredholm Alternative Least Squares Regression lines Identifying the Closest Point Using MATLAB The Singular Value Decomposition Exercises Vector Valued Functions Vector Valued Functions Vector Fields Exercises Continuous Functions Sufficient Conditions For Continuity Limits Of A Function Properties Of Continuous Functions Exercises Open And Closed Sets Exercises Some Fundamentals Combinations Of Continuous Functions The Nested Interval Lemma Convergent Sequences, Sequential Compactness Continuity And The Limit Of A Sequence The Extreme Value Theorem And Uniform Continuity Convergence of Functions Root Test Convergence of Sums Connected Sets Exercises Vector Valued Functions Of One Variable Limits Of A Vector Valued Function Of One Real Variable The Derivative And Integral Geometric And Physical Significance Of The Derivative Differentiation Rules Leibniz's Notation Exercises Line Integrals Arc Length And Orientations Line Integrals And Work Another Notation For Line Integrals Exercises Independence Of Parametrization Hard Calculus Independence Of Parametrization Motion On A Space Curve Space Curves Some Simple Techniques Geometry Of Space Curves Exercises Functions Of Many Variables Review Of Limits Exercises The Directional Derivative And Partial Derivatives The Directional Derivative Partial Derivatives Exercises Mixed Partial Derivatives Partial Differential Equations Exercises The Derivative Of A Function Of Many Variables The Derivative Of Functions Of One Variable Exercises The Derivative Of Functions Of Many Variables Exercises C1 Functions The Chain Rule The Chain Rule For Functions Of One Variable The Chain Rule For Functions Of Many Variables Exercises Related Rates Problems The Derivative Of The Inverse Function Proof Of The Chain Rule Exercises The Gradient The Gradient And Tangent Planes Exercises Optimization Local Extrema Exercises The Second Derivative Test Exercises Lagrange Multipliers Exercises Proof Of The Second Derivative Test* The Riemannn Integral On Rn Methods For Double Integrals Density And Mass Exercises Methods For Triple Integrals Definition Of The Integral Iterated Integrals Exercises Mass And Density Exercises The Integral In Other Coordinates Polar Coordinates Exercises Cylindrical And Spherical Coordinates Volume and Integrals in Cylindrical Coordinates Volume And Integrals in Spherical Coordinates Exercises The General Procedure Exercises The Moment Of Inertia And Center Of Mass Exercises The Integral on Two Dimensional Surfaces In R3 The Two Dimensional Area In R3 Surfaces Of The Form z=f( x,y) MATLAB and Graphing Surfaces Piecewise Defined Surfaces Flux Integrals Exercises Calculus Of Vector Fields Divergence And Curl Of A Vector Field Vector Identities Vector Potentials The Weak Maximum Principle Exercises The Divergence Theorem Coordinate Free Concept Of Divergence Some Applications Of The Divergence Theorem Hydrostatic Pressure Archimedes Law Of Buoyancy Equations Of Heat And Diffusion Balance Of Mass Balance Of Momentum The Reynolds Transport Formula Frame Indifference Bernoulli's Principle The Wave Equation A Negative Observation Volumes Of Balls In Rn Electrostatics Exercises Stokes And Green's Theorems Green's Theorem Exercises Stoke's Theorem From Green's Theorem The Normal and the Orientation The Mobeus Band A General Green's Theorem Conservative Vector Fields Some Terminology Moving Coordinate Systems The Acceleration In Polar Coordinates Planetary Motion The Equal Area Rule, Kepler's Second Law Inverse Square Law, Kepler's First Law Kepler's Third Law The Angular Velocity Vector Angular Velocity Vector on Earth Coriolis Force and Centripetal Force Coriolis Force on the Rotating Earth The Foucault Pendulum Exercises Curvilinear Coordinates Basis Vectors Exercises Curvilinear Coordinates Exercises Transformation of Coordinates. Differentiation and Christoffel Symbols Gradients and Divergence Exercises Curl and Cross Products Implicit Function Theorem* More Continuous Partial Derivatives The Method Of Lagrange Multipliers The Local Structure Of C1 Mappings II Linear Algebra and Differential Equations Determinants Basic Techniques And Properties Cofactors And 22 Determinants The Determinant Of A Triangular Matrix Properties Of Determinants Finding Determinants Using Row Operations Applications A Formula For The Inverse Finding Eigenvalues Using Determinants Cramer's Rule MATLAB And Determinants The Cayley Hamilton Theorem Exercises The Mathematical Theory Of Determinants The Function sgn The Determinant The Definition Permuting Rows Or Columns A Symmetric Definition The Alternating Property Of The Determinant Linear Combinations And Determinants The Determinant Of A Product Cofactor Expansions Formula For The Inverse Cramer's Rule Methods and Recipes for First Order ODE First Order Linear Equations Bernouli Equations Separable Differential Equations, Stability Homogeneous Equations Exact Equations The Integrating Factor The Case Where M,N Are Affine Linear Linear and Nonlinear Differential Equations Computer Algebra Methods Maple Mathematica MATLAB Scientific Notebook Exercises First Order Linear Systems The Theory of the Fundamental Matrix Laplace Transform Methods Using a Computer Algebra System Using Computer Algebra to Find Fundamental Matrix Homogeneous Particular and General Solutions Picard Iteration Higher Order Scalar Linear Equations Numerical Solutions For Systems A Few Numerical Methods Using MATLAB to Find Solutions Stability of Equilibrium Points Periodic Orbits, Poincare Bendixon Theorem Exercises Solutions Near a Regular Singular Point The Euler Equations Some Simple Observations on Power Series Regular Singular Points Abel's Formula Finding the Solution Method of Frobenius The Bessel Equations The Case where 0=x"0117=0 The Case of 0=x"0117 Not an Integer Case Where 0=x"0117 is an Integer Other Properties of Bessel Functions Exercises Boundary Value Problems, Fourier Series Boundary Value Problems Eigenvalue Problems Fourier Series Mean Square Approximation Pointwise Convergence of Fourier Series Explanation of Pointwise Convergence Theorem Mean Square Convergence Integrating and Differentiating Fourier Series Odd and Even Extensions Exercises Some Partial Differential Equations Laplacian in Orthogonal CurvilinearCoordinates Heat and Wave Equations Heat Equation The Wave Equation Nonhomogeneous Problems Laplace Equation Rectangles Circular Disks Exercises III Fundamentals of Complex Analysis Analytic Functions Cauchy Riemann Equations The Cauchy Riemann Equations Contour Integrals Cauchy Integral Theorem Primitives and Cauchy Goursat Theorem Functions Differentiable on a Disk, Zeros Liouville's Theorem Riemann Sphere Exercises Isolated Singularities and Analytic Functions Open Mapping Theorem for Complex Valued Functions Functions Analytic on an Annulus Isolated Singularities Meromorphic Functions The Residue Theorem Evaluation of Improper Integrals Exercises Some Fundamental Functions and Transforms Gamma Function Laplace Transform Fourier Transform The Inversion of Laplace Transforms The Bromwich Integral Exercises IV Probability and Statistics Probability Improper Integrals Combinations The Binomial Theorem Exercises Counting and Basic Probability Exercises General Considerations Probability Moment Generating Functions Independence and Conditional Probability Statistical Tests The Distribution of nS2/0=x"011B2 Confidence Intervals for Variance The T and F Distributions The T Distribution Confidence Intervals for the Mean Testing For Two Different Means The F Distribution Confidence Intervals for the Ratio of Two Variances Maximum Likelihood Estimates Quadratic Forms Linear Regression Goodness of Fit Contingency Tables The Theory Of The Riemannn Integral An Important Warning Basic Definition Basic Properties Which Functions Are Integrable? Iterated Integrals The Change Of Variables Formula Some Observations A Rigid Body Rotating About a Point Lagrangian Mechanics The Spinning Top and the Euler Angles