This online textbook provides an overview of Calculus in clear, easy to understand language designed for the non-mathematician.

Our writing is based on three premises. First, life sciences students are motivated by and respond well to actual data related to real life sciences problems. Second, the ultimate goal of calculus in the life sciences primarily involves modeling living systems with difference and differential equations. Understanding the concepts of derivative and integral are crucial, but the ability to compute a large array of derivatives and integrals is of secondary importance. Third, the depth of calculus for life sciences students should be comparable to that of the traditional physics and engineering calculus course; else life sciences students will be short changed and their faculty will advise them to take the 'best' (engineering) course.

Our writing is based on three premises. First, life sciences students are motivated by and respond well to actual data related to real life sciences problems. Second, the ultimate goal of calculus in the life sciences primarily involves modeling living systems with difference and differential equations. Understanding the concepts of derivative and integral are crucial, but the ability to compute a large array of derivatives and integrals is of secondary importance. Third, the depth of calculus for life sciences students should be comparable to that of the traditional physics and engineering calculus course; else life sciences students will be short changed and their faculty will advise them to take the 'best' (engineering) course.

In our text, mathematical modeling and difference and differential equations lead, closely follow, and extend the elements of calculus. Chapter one introduces mathematical modeling in which students write descriptions of some observed processes and from these descriptions derive first order linear difference equations whose solutions can be compared with the observed data. In chapters in which the derivatives of algebraic, exponential, or trigonometric functions are defined, biologically motivated differential equations and their solutions are included. The chapter on partial derivatives includes a section on the diffusion partial differential equation. There are two chapters on non-linear difference equations and on systems of two difference equations and two chapters on differential equations and on systems of differential equation.

This books covers the following topics: Geometry of Rn; functions from R to Rn; functions from Rn to R; functions from Rm to Rn.

Calculus with Theory, covers the same material as 18.01 (Single Variable Calculus), but at a deeper and more rigorous level. It emphasizes careful reasoning and understanding of proofs. The course assumes knowledge of elementary calculus.

Table of Contents:

Chapter 0 -- Review and Preview
Chapter 1 -- Functions, Graphs, Limits and Continuity
Chapter 2 -- The Derivative
Chapter 3 -- Derivatives and Graphs
Chapter 4 -- The Integral
Chapter 5 -- Applications of Definite Integrals
Chapter 6 -- Introduction to Differential Equations
Chapter 7 -- Inverse Trigonometric Functions
Chapter 8 -- Improper Integrals and Integration Techniques
Chapter 9 -- Polar, Parametric & Conics
Chapter 10 -- Infinite Series & Power Series
Chapter 11 -- Vectors, Lines and Planes in 3D
Chapter 12 -- Vector-Valued Functions
Chapter 13 -- Functions of Several Variables
Chapter 14 -- Double and Triple Integrals
Chapter 15 -- Vector Calculus

This text for Analytic Geometry and Calculus I, II, and III is a Dalton State College remix of APEX Calculus 3.0.

Calculus arose as a tool for solving practical scientific problems through the centuries. However, it is often taught as a technical subject with rules and formulas (and occasionally theorems), devoid of its connection to applications. In this textbook, the applications form an important focal point, with emphasis on life sciences. This places the techniques and concepts into practical context, as well as motivating quantitative approaches to biology taught to undergraduates. While many of the examples have a biological flavour, the level of biology needed to understand those examples is kept at a minimum. The problems are motivated with enough detail to follow the assumptions, but are simplified for the purpose of pedagogy.

Table of Contents:
1 Power functions as building blocks
2 Average rates of change, average velocity and the secant line
3 Three faces of the derivative: geometric, analytic, and computational
4 Differentiation rules, simple antiderivatives and applications
5 Tangent lines, linear approximation, and Newton’s method
6 Sketching the graph of a function using calculus tools
7 Optimization
8 Introducing the chain rule
9 Chain rule applied to related rates and implicit differentiation
10 Exponential functions
11 Differential equations for exponential growth and decay
12 Solving differential equations
13 Qualitative methods for differential equations
14 Periodic and trigonometric functions
15 Cycles, periods, and rates of change

This textbook covers calculus of a single variable, suitable for a year-long (or two-semester) course. Chapters 1-5 cover Calculus I, while Chapters 6-9 cover Calculus II. The book is designed for students who have completed courses in high-school algebra, geometry, and trigonometry. Though designed for college students, it could also be used in high schools. The traditional topics are covered, but the old idea of an infinitesimal is resurrected, owing to its usefulness (especially in the sciences).

There are 943 exercises in the book, with answers and hints to selected exercises.

Table of Contents
1 The Derivative
2 Derivatives of Common Functions
3 Topics in Differential Calculus
4 Applications of Derivatives
5 The Integral
6 Methods of Integration
7 Analytic Geometry and Plane Curves
8 Applications of Integrals
9 Infinite Sequences and Series

This textbook covers calculus of a single variable, suitable for a year-long (or two-semester) course. Chapters 1-5 cover Calculus I, while Chapters 6-9 cover Calculus II. The book is designed for students who have completed courses in high-school algebra, geometry, and trigonometry. Though designed for college students, it could also be used in high schools. The traditional topics are covered, but the old idea of an infinitesimal is resurrected, owing to its usefulness (especially in the sciences).

This subject provides an introduction to fluid mechanics. Students are introduced to and become familiar with all relevant physical properties and fundamental laws governing the behavior of fluids and learn how to solve a variety of problems of interest to civil and environmental engineers. While there is a chance to put skills from Calculus and Differential Equations to use in this subject, the emphasis is on physical understanding of why a fluid behaves the way it does. The aim is to make the students think as a fluid. In addition to relating a working knowledge of fluid mechanics, the subject prepares students for higher-level subjects in fluid dynamics.

This open-source book by Crowell, Robbin, and Angenent is a spin-off of a previous open-source book by Robbin and Angenent. It covers the first semester of a freshman calculus course.

A rigorous introduction designed for mathematicians into perturbative quantum field theory, using the language of functional integrals. Basics of classical field theory. Free quantum theories. Feynman diagrams. Renormalization theory. Local operators. Operator product expansion. Renormalization group equation. The goal is to discuss, using mathematical language, a number of basic notions and results of QFT that are necessary to understand talks and papers in QFT and string theory.

This course focuses on an in-depth reading of Principia Mathematica Philosophiae Naturalis by Isaac Newton, as well as several related commentaries and historical philosophical texts.

This book is an approachable introduction to calculus with applications to biology and environmental science. For example, one application in the book is determining the volume of earth moved in the 1959 earthquake that created Quake Lake. Another application uses differential equations to model various biological examples, including moose and wolf populations at Isle Royale National Park, ranavirus in amphibians, and competing species of protozoa. The text focuses on intuitive understanding of concepts, but still covers most of the algebra and calculations common in a survey of calculus course.

Table of Contents
Algebra Tips and Tricks: Part I
Algebra Tips and Tricks Part I (Combining Terms, Distributing, Functions, Graphing)
Homework for Algebra Tips and Tricks: Part I
Derivative Introduction
Ball Toss Project
Position to Velocity
Homework: Position to Velocity
Algebra Tips and Tricks Part II (Piecewise Defined Functions)
Limits
Homework: Limits
Algebra Tips and Tricks Part III (Factoring)
Algebraic Limits
Homework: Algebraic Limits
Instantaneous Velocity
Homework: Instantaneous Velocity
Algebra Tips and Tricks IV (Tips for dealing with fractions)
Definition of Derivative Examples
Project: Hard Definition of Derivative Problems
Homework: Examples of the Definition of the Derivative
Project: Killdeer Migration Speed
Rules for Derivatives
Algebra Tips and Tricks Part V (Exponents)
Power Rule
Homework: Power Rule
Algebra Tips and Tricks Part VI (Logarithms)
Exponentials, Logarithms, and Trig Functions
Homework: Exponents, Logs, Trig Functions
Product Rule
Homework: Product Rule
Quotient Rule
Homework: Quotient Rule
Chain Rule
Homework: Chain Rule
Multirule Derivatives
Homework: Multirule Derivatives
Anti-derivatives
Homework: Anti-derivatives
More Derivative Intuition
Derivatives and Graphs
Homework: Derivative Graphs
Second Derivatives and Interpreting the Derivative
Homework: Second Derivatives and Interpreting the Derivative
Optimization
Homework: Optimization
Derivatives in Space
Homework: Derivatives in Space
Differential Equations
Recurrance Relations
Homework: Recurrence Relations
Introduction to Differential Equations
Homework: Introduction to Differential Equations
Understanding Differential Equations
Homework: Understanding Differential Equations
Initial Value Problems
Homework: Initial Value Problems
Growth and Decay
Homework: Growth and Decay
Exploring Graphs of Differential Equations
Project: Modelling with Differential Equations
Intuition for Integration
Introduction to Integrals
Numeric Integration Techniques
Homework: Numeric Integration Techniques
Fundamental Theorem of Calculus
Homework: The Fundamental Theorem of Calculus
Project: Measuring Streamflow
Project: Quake Lake
Rules for Integration
Power, exponential, trig, and logarithm rules
Homework: Power, exponential, trig, and logarithmic rules
u-substitution
Homework: u-substitution
Integral Applications
Homework: Integral Applications
Integration by Parts
Homework: Integration by Parts
Acknowledgements

Analysis I in its various versions covers fundamentals of mathematical analysis: continuity, differentiability, some form of the Riemann integral, sequences and series of numbers and functions, uniform convergence with applications to interchange of limit operations, some point-set topology, including some work in Euclidean n-space. MIT students may choose to take one of three versions of 18.100: Option A (18.100A) chooses less abstract definitions and proofs, and gives applications where possible. Option B (18.100B) is more demanding and for students with more mathematical maturity; it places more emphasis from the beginning on point-set topology and n-space, whereas Option A is concerned primarily with analysis on the real line, saving for the last weeks work in 2-space (the plane) and its point-set topology. Option C (18.100C) is a 15-unit variant of Option B, with further instruction and practice in written and oral communication.

This textbook, or really a “coursebook” for a college freshman-level class, has been updated for Spring 2014 and provides an introduction to programming and problem solving using both Matlab and Mathcad. We provide a balanced selection of introductory exercises and real-world problems (i.e. no “contrived” problems). We include many examples and screenshots to guide the reader. We assume no prior knowledge of Matlab or Mathcad.

This is a text for a two-term course in introductory real analysis for junior or senior mathematics majors and science students with a serious interest in mathematics. Prospective educators or mathematically gifted high school students can also benefit from the mathematical maturity that can be gained from an introductory real analysis course.

The book is designed to fill the gaps left in the development of calculus as it is usually presented in an elementary course, and to provide the background required for insight into more advanced courses in pure and applied mathematics. The standard elementary calculus sequence is the only specific prerequisite for Chapters 1–5, which deal with real-valued functions. (However, other analysis oriented courses, such as elementary differential equation, also provide useful preparatory experience.) Chapters 6 and 7 require a working knowledge of determinants, matrices and linear transformations, typically available from a first course in linear algebra. Chapter 8 is accessible after completion of Chapters 1–5.

This course covers the mathematical techniques necessary for understanding of materials science and engineering topics such as energetics, materials structure and symmetry, materials response to applied fields, mechanics and physics of solids and soft materials. The class uses examples from the materials science and engineering core courses (3.012 and 3.014) to introduce mathematical concepts and materials-related problem solving skills. Topics include linear algebra and orthonormal basis, eigenvalues and eigenvectors, quadratic forms, tensor operations, symmetry operations, calculus of several variables, introduction to complex analysis, ordinary and partial differential equations, theory of distributions, and fourier analysis. Users may find additional or updated materials at Professor Carter's 3.016 course Web site.

A math based economics course designed to provide the skills needed to solve fundamental problems in both macroeconomics and microeconomics, by covering concepts in precalculus and calculus.