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