Advanced Calculus using Mathematica
Advanced Calculus using Mathematica: NoteBook Edition is a complete text on calculus of several variables written in Mathematica NoteBooks. The eText has large movable figures and interactive programs to illustrate things like “zooming in” to see “local linearity.” In addition to lots of traditional style exercises, the eText also has sections on computing with Mathematica. Solutions to many exercises are in closed cells of the eText
Contents
You need a copy of Mathematica to use the text. Many schools have site licenses that allow students to get Mathematica free.
Chapter 0: An Introduction to Mathematica & Calculus
A Mathematica Primer
This electronic section gets you started computing with Mathematica.
Mathematica help
This electronic section shows you how to get help with Mathematica, including plain English input with Wolfram Alpha
An Interactive Preview
This electronic section gives an example of interactive multivariable calculus plotting a surface and tangent that you can move
Chapter 1: Basic Equations and Graphs
01 Explicit Line, Parabola, and Circle
02 Implicit Line in 2-D and Implicit Circle
03 Parametric Line in 2-D and Parametric Circles
04 Sliding and Squashing Equations in 2-D
05 Mathematica Plots in 2D
06 Graphing in 3-D by Slicing
06 Graphing in 3-D by Slicing
07 Mathematica Plots in 3D
Chapter 1 Lab 1, Lab 2
Chapter 2: Vector Geometry
01 Position and Displacement Vectors
02 Geometry of Vector Sum
03 Displacement Vectors and Differences
04 Geometry of Scalar Multiplication
05 Angles, Perpendiculars, and Projections
06 Cross Product
07 A Lexicon for Translation and Mathematica
08 Abstract Vector Algebra
09 Just in Time Algebra & Geometry (area, volume, and determinants)
Chapter 2 Lab1
Chapter 3: Derivatives and Graphs of Explicit Functions
01 Explicit Linear Functions: Graphs, Slopes, and Gradient
02 Practical Functions of Several Variables
03 Derivatives & Linear Approximation, The Explicit Tangent
04 Differentiation Skills and Mathematica
05 Nonlinear Gradients & Directional Derivatives
06 Gradients and Tangents with Mathematica
07 Approximation by Differentials
08 More Variables and Abstract Differentiation Rules
09 Advanced Examples of Continuity, Tangency, and Smoothness
Chapter 3 Lab1, Lab2, Lab3
Chapter 4: Implicit Curves, Surfaces, and Contour Plots
01 Implicit Linear Functions
02 Contour Graphs, Gradients, and Tangents
03 Mathematica Contour Plots
04 Implicit Planes
05 Implicit Surfaces
06 Mathematica Implicit Tangent Planes
07 Differentiation Review
Chapter 4 Lab1, Lab2, Lab3, Lab4
Chapter 5: Inverse and Implicit Functions
00 Introduction
01 m Linear Equations in n Unknowns
02 Local Solution of Implicit Equations
03 The Inverse Function Theorem
04 Differentiation with Constraints
05 Local Flows and Inverse Functions
06 Functional Dependence
Chapter 6: Taylor’s Formula in Several Variables
01 Second Order Derivatives, Symmetry
02 Principal Axes for 2×2 Symmetric Matrices
03 Local Convexity and Taylor's Second Order Formula in 2D
04 Mathematica and Taylor's Formula
05 Second Order Taylor's Formula in nD
06 Higher Order Taylor's Formula
07 Constrained Second Order Derivatives
Chapter 7: Max-min in Several Variables
01 A First Look at Theory
02 Interior Critical Points
03 Compact Domains
04 Boundary Extrema & Lagrange Multipliers in 2D
05 MAX-min with Mathematica
06 Boundary Extrema & Lagrange Multipliers in 3D
07 Combined Tools in MAX-min
Chapter 8: Multiple Integrals in Cartesian Coordinates
01 Riemann Sums in 1-D
02 Riemann Sums in 2-D
03 Iterated Integrals
04 Integration in 3-D
05 Integration with Mathematica
06 Improper Integrals
Chapter 8 Lab1, Lab2, Lab3, Lab4
Chapter 9: Parametric Curves
01 Parametric Lines
02 Parametric Curves in 2D
03 Mathematica 2D Parametric Curves
04 Parametric Curves in 3D
05 Mathematica 3D Parametric Curves
06 Parametric Tangents
07 Arclength
08 Product Rules, Curvature
09 Parametrization by Arclength
Chapter 9 Lab1, Lab2, Lab3, Lab4
Chapter 10: Motion along Curves: Gas, Brakes, & Tires
01 Speed, Velocity, Acceleration
02 Gas, Brakes, and Turns
Lab for Chapter 10
Chapter 11: Vector Fields and Velocity Flows in 2D
01 Vector Fields
02 Vector Fields with Mathematica
03 Velocity Flows
04 Flow Along and Across a Curve
05 Path Independence and The Chain Rule
06 Gradient Flows and Max-min Revisited
Chapter 11 Lab1, Lab2
Chapter 12: Green’s Theorem, 2-D Divergence and Swirl
01 Introduction to the Three Forms of Green's Theorem
02 Formal Divergence and Swirl
03 Swirl & Geometry of Domains in 2D
04 The Microscopic View of Divergence and Swirl
Chapter 12 Lab1
Chapter 13: Coordinate Systems in 2 Dimensions
01 Oriented Area of a Parallelogram & Linear Coordinates
02 Area Integrals in Polar Coordinates
03 Area Integrals for General Coordinates in 2D
04 Differentiation in Coordinates
Chapter 14: Path Integrals & Vector Fields in 3D
01 Vector Fields in 3 D
02 Path Integrals in 3 D and nD
03 The Chain Rule and Gradient (again)
04 Vector Field Identities
Chapter 15: Parametric Surfaces
01 Parametric Planes
02 Parametric Surfaces
03 Parametric Surface Tangents
04 Area of Parametric Surfaces
05 Orientation and Flux Integrals
Chapter 16: Curvature of Surfaces
01 Surface Curvature
02 Motion Constrained to a Surface
03 Differentiation on a Surface
Chapter 17: Coordinate Systems in 3 Dimensions
01 Oriented Volume of a Parallelepiped & Linear Coordinates
02 Volume Integrals in Cylindrical Coordinates
03 Volume Integrals in Spherical Coordinates
04 Volume Integrals for General Coordinates in 3D
05 Derivatives in Different Coordinate Systems
Chapter 18: Differential Forms
01 Differential Forms in 2D & 3D
02 Exterior Derivatives of Differential Forms
03 Path Integrals of 1-Forms
05 Surface Flux Integrals of 2-Forms
05 Volume Integrals of 3-Forms
06 Differential Forms in n-Dimensions
Chapter 19: Flow in 3D, Divergence & Curl
01 Gradient Fields in 3D
02 Stokes' Theorem and Curl in 3D
03 Simply Connected Domains in 3D & Gradient Fields
04 Gauss's Theorem and Divergence in 3D
05 Handles and Bubbles
06 Microscopic Divergence and Curl
Chapter 20: Partial Differential Equations
01 The Heat Equation and Divergence
02 Laplace's Equation
03 Maxwell's Equations
Chapter 20 Lab1
Chapter 21: Infinite Series
01 Geometric Series
02 Comparison and Weierstrass' M-test
03 The Classical Power Series
04 Convergence and the Ratio Test
05 Integration and Differentiation of Series
06 Series with Mathematica
07 Fourier Series