# Differential and Integral Calculus by Courant

• Calculus
• micromass
In summary, "Differential and Integral Calculus" by Richard Courant is a comprehensive textbook that covers the fundamentals of calculus, including the continuum of numbers, functions, limits, continuity, differentiation, integration, and infinite series. It also includes a section on the theory of functions of several variables and their applications in physics and mechanics. The book is divided into two volumes, with the first volume focusing on the basics of calculus and the second volume delving into more advanced topics such as implicit functions, multiple integrals, and vector analysis. With numerous examples and exercises, this textbook is an essential resource for students and professionals in the field of mathematics.

## For those who have used this book

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Code:
[LIST]
[*] Introductory Remarks
[*] Introduction
[LIST]
[*] The Continuum of Numbers
[*] The Concept of function
[*] More Detailed Study of the Elementary Functions
[*] Functions of an Integral Variable. Sequences of Numbers
[*] The Concept of the Limit of a Sequence
[*] Further Discussion of the Concept of Limit
[*] The Concept of Limit where the Variable is Continuous
[*] The Concept of Continuity
[/LIST]
[*] Appendix
[LIST]
[*] Preliminary Remarks
[*] The Principle of the Point of Accumulation and its Applications
[*] Theorems on Continuous Functions
[*] Some Remarks on the Elementary Functions
[/LIST]
[*] Appendix
[LIST]
[*] Polar Co-ordinates
[*] Remarks on Complex Numbers
[/LIST]
[*] The Fundamental Ideas of the Integral and Differential Calculus
[LIST]
[*] The Definite Integral
[*] Examples
[*] The Derivative
[*] The Indefinite Integral, the Primitive Function, and the Fundamental Theorems of the Differential and Integral Calculus
[*] Simple Methods of Graphical Integration
[*] Further Remarks on the Connexion between the Integral and the
Derivative
[*] The Estimation of Integrals and the Mean Value Theorem of the
Integral Calculus
[/LIST]
[*] Appendix
[LIST]
[*] The Existence of the Definite Integral of a Continuous Function
[*] The Relation between the Mean Value Theorem of the Differential Calculus and the Mean Value Theorem of the Integral Calculus
[/LIST]
[*] Differentiation and Integration of The Elementary Functions
[LIST]
[*] The Simplest Rules for Differentiation and their Applications
[*] The Corresponding Integral Formulae
[*] The Inverse Function and its Derivative
[*] Differentiation of a Function of a Function
[*] Maxima and Minima
[*] The Logarithm and the Exponential Function
[*] Some Applications of the Exponential Function
[*] The Hyperbolic Functions
[*] The Order of Magnitude of Functions
[/LIST]
[*] Appendix
[LIST]
[*] Some Special Functions
[*] Remarks on the Differentiability of Functions
[*] Some Special Formulae
[/LIST]
[*] Further Development of the Integral Calculus
[LIST]
[*] Elementary Integrals
[*] The Method of Substitution
[*] Further Examples of the Substitution Methods
[*] Integration by Parts
[*] Integration of Rational Functions
[*] Integration of some Other Classes of Functions
[*] Remarks on Functions which are not Integrable in Terms of Elementary Functions
[*] Extension of the Concept Integral. Improper Integrals
[/LIST]
[*] Appendix
[LIST]
[*] The Second Mean Value Theorem of the Integral Calculus
[/LIST]
[*] Applications
[LIST]
[*] Representation of Curves
[*] Applications to the Theory of Plane Curves
[*] Examples
[*] Some Very Simple Problems in the Mechanics of a Particle
[*] Further Applications: Particle Sliding down a Curve
[*] Work
[/LIST]
[*] Appendix
[LIST]
[*] Properties of the Evolute
[*] Area bounded by Closed Curves
[/LIST]
[*] Taylor's Theorem and the Approximate Expression of Functions by Polynomials
[LIST]
[*] The Logarithm and the Inverse Tangent
[*] Taylor's Theorem
[*] Applications. Expansions of the Elementary Functions
[*] Geometrical Applications
[/LIST]
[*] Appendix
[LIST]
[*] Example of a Function which cannot be expanded in a Taylor
Series
[*] Proof that e is Irrational
[*] Proof that the Binomial Series Converges
[*] Zeros and Infinities of Functions, and So-called Indeterminate Expressions
[/LIST]
[*] Numerical Methods
[LIST]
[*] Preliminary Remarks
[*] Numerical Integration
[*] Applications of the Mean Value Theorem and of Taylor's Theorem. The Calculus of Errors
[*] Numerical Solution of Equations
[/LIST]
[*] Appendix
[LIST]
[*] Stirling's Formula
[/LIST]
[*]  Infinite Series and Other Limiting Processes
[LIST]
[*] Preliminary Remarks
[*] The Concepts of Convergence and Divergence
[*] Tests for Convergence and Divergence
[*] Sequences and Series of Fnnctions
[*] Uniform and Non-uniform Convergence
[*] Power Series
[*] Expansion of Given Functions in Power Series. Method of Undetermined Coefficients. Examples
[*] Power Series with Complex Terms
[/LIST]
[*] Appendix
[LIST]
[*] Multiplication and Division of Series
[*] Infinite Series and Improper Integrals
[*] Infinite Products
[*] Series involving Bernoulli's Numbers
[/LIST]
[*] Fourier Series
[LIST]
[*] Periodic Functions
[*] Use of Complex Notation
[*] Fourier Series
[*] Examples of Fourier Series
[*] The Convergence of Fourier Series
[/LIST]
[*] Appendix
[LIST]
[*] Integration of Fourier Series
[/LIST]
[*]A Sketch of The Theory of Functions of Several Variables
[LIST]
[*] The Concept of Function in the Case of Several Variables
[*] Continuity
[*] The Derivatives of a Function of Several Variables
[*] The Chain Rule and the Differentiation of Inverse Functions
[*] Implicit Functions
[*] Multiple and Repeated Integrals
[/LIST]
[*] The Differential Equations for the Simplest Types of Vibration
[LIST]
[*] Vibration Problems of Mechanics and Physics
[*] Solution of the Homogeneous Equation. Free Oscillations
[*] The Non-homogeneous Equation. Forced Oscillations
[*] Additional Remarks on Differential Equations
[/LIST]
[*] Summary of Important Theorems and Formulas
[*] Miscellaneous Examples
[*] Index
[/LIST]

Code:
[LIST]
[*] Preliminary Remarks on Analytical Geometry and Vector Analysis
[LIST]
[*] Rectangular Co-ordinates and Vectors
[*] The Area of a Triangle, the Volume of a Tetrahedron, the Vector Multiplication of Vectors
[*] Simple Theorems on Determinants of the Second and Third Order
[*] Affine Transformations and the Multiplication of Determinants
[/LIST]
[*] Functions of Several Variables and Their Derivatives
[LIST]
[*] The Concept of Function in the Case of Several Variables
[*] Continuity
[*] The Derivatives of a Function
[*] The Total Differential of a Function and its Geometrical Meaning
[*] Functions of Functions (Compound Functions) and the Introduction of New Independent Variables
[*] The Mean Value Theorem and Taylor's Theorem for Functions of Several Variables
[*] The Application of Vector Methods
[/LIST]
[*] Appendix
[LIST]
[*] The Principle of the Point of Accumulation in Several Dimensions and its Applications
[*] The Concept of Limit for Functions of Several Variables
[*] Homogeneous Functions
[/LIST]
[*] Developments and Applications of the Differential Calculus
[LIST]
[*] Implicit Functions
[*] Curves and Surfaces in Implicit Form
[*] Systems of Functions, Transformations, and Mappings
[*] Applications
[*] Families of Curves, Families of Surfaces, and their Envelopes
[*] Maxima and Minima
[/LIST]
[*] Appendix
[LIST]
[*] Sufficient Conditions for Extreme Values
[*] Singular Points of Plane Curves
[*] Singular Points of Surfaces
[*] Connexion between Euler's and Lagrange's Representations of the Motion of a Fluid
[*] Tangential Representation of a Closed Curve
[/LIST]
[*] Multiple Integrals
[LIST]
[*] Ordinary Integrals as Functions of a Parameter
[*] The Integral of a Continuous Function over a Region of the Plane or of Space
[*] Reduction of the Multiple Integral to Repeated Single Integrals
[*] Transformation of Multiple Integrals
[*] Improper Integrals
[*] Geometrical Applications
[*] Physical Applications
[/LIST]
[*] Appendix
[LIST]
[*] The Existence of the Multiple Integral
[*] General Formula for the Area (or Volume) of a Region bounded by Segments of Straight Lines or Plane Areas (Gukhn's Formula). The Polar Planimeter
[*] Volumes and Areas in Space of any Number of Dimensions
[*] Improper Integrals as Functions of a Parameter
[*] The Fourier Integral
[*] The Eulerian Integrals (Gamma Function)
[*] Differentiation and Integration to Fractional Order. Abel's Integral Equation
[*] Note on the Definition of the Area of a Curved Surface
[/LIST]
[*] Integration over Regions in Several Dimension
[LIST]
[*] Line Integrals
[*] Connexion between Line Integrals and Double Integrals in the Plane. (The Integral Theorems of Gauss, Stokes, and Green)
[*] Interpretation and Applications of the Integral Theorems for the Plane
[*] Surface Integrals
[*] Gauss's Theorem and Green's Theorem in Space
[*] Stokes's Theorem in Space
[*] The Connexion between Differentiation and Integration for Several Variables
[/LIST]
[*] Appendix
[LIST]
[*] Remarks on Gauss's Theorem and Stokes's Theorem
[*] Representation of a Source-free Vector Field as a Curl
[/LIST]
[*] Differential Equations
[LIST]
[*] The Differential Equations of the Motion of a Particle in Three Dimensions
[*] Examples on the Mechanics of a Particle
[*] Further Examples of Differential Equations
[*] Linear Differential Equations
[*] General Remarks on Differential Equations
[*] The Potential of Attracting Charges
[*] Further Examples of Partial Differential Equations
[/LIST]
[*] Calculus of Variations
[LIST]
[*] Introduction
[*] Euler's Differential Equation in the Simplest Case
[*] Generalizations
[/LIST]
[*] Functions of a Complex Variable
[LIST]
[*] Introduction
[*] Foundations of the Theory of Functions of a Complex Variable)
[*] The Integration of Analytic Functions
[*] Cauchy's Formula and its Applications
[*] Applications to Complex Integration (Contour Integration)
[*] Many-valued Functions and Analytic Extension
[/LIST]
[*] Supplement
[LIST]
[*] Real Numbers and the Concept of Limit
[*] Miscellaneous Examples
[*] Summary of Important Theorems and Formulae
[*] Index
[/LIST]
[/LIST]

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it is not the same book but just as good. it is a rewritten version by john of courant's book.

but do not be handicapped by tradition. if some mickey mouse modern explanation of calculus speaks to you, read that, and put courant back on the shelf for later. learn wherever you can.

How is the exposition of the calculus in this book compared to moore modern treatments of the subject? I'm about to study calculus rigorously later (self-study) and I don't really know what a good calculus book should contain. For instance, there seems to be a lack of set theory in this book?

I read this book (by Fritz John) in Undergraduate, the book is great but Rude also, I read Apostol Calculus twice before going through this!

indeed this book is more concrete than modern books that use set theoretic language. that actually makes it more precise and for most people clearer. e.g. in old fashioned language, f is continuous at p if for every number e>0, there is a number d>0 so that whenever |x-p| < d and f is defined at x, then |f(x)-f(p)|.

in modern set language one says: for every neighborhood V of f(p), there is a neighborhood U of p so that whenever x is in the intersection of U with Domain(f), then f(p) is in V.

My students found this language more abstract and harder to grapple with. It also does not ket you actually check by computation that any specific f(x), like x^2, is continuous, since you have nothing to calculate with. Plus you have to understand what a "neighborhood" of q is, namely a set containing some open interval around q.

since |x-p| < e is equivalent to x being in the open interval (p-e, p+e), you can see the modern language is kind of a conceptual simplification but also kind of an obfuscation of the classic approach.

if you want to be kept aware of the use of calculus in physics, courant will have much more of that than a modern abstract approach like spivak, which has essentially none.

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Thank you, mathwonk! I have this book accessible for free at the local library so if the lack of set theory won't handicapp me for further math studies, I will give it a try since I liked what I've read in the book so far.

And out of pure interest, mathwonk, have you ever taught calculus from Courant or John/Courant?

yes once in honors calculus i used courant/john (rather than courant) because of the price break. we were able to find enough used copies at between $10-$30 to supply the class. but i didn't really "teach out of it" in the sense of following it. I made up my own sequence of topics and wrote my own course notes as usual. This is the way things were always done at harvard when i was there, i.e. you give a recommended book but you teach the course yourself.

## 1. What is the difference between differential and integral calculus?

Differential calculus is concerned with the study of rates of change, while integral calculus is focused on finding the total change or accumulation of a quantity over a given interval.

## 2. How is Calculus by Courant different from other calculus textbooks?

Calculus by Courant emphasizes both theoretical and practical aspects of calculus, providing a comprehensive understanding of the subject. It also includes many real-world applications and examples to aid in understanding the concepts.

## 3. Is this book suitable for beginners in calculus?

While Calculus by Courant is a challenging textbook, it is suitable for students who have a strong foundation in algebra and trigonometry. It is also a great resource for those looking to deepen their understanding of calculus.

## 4. Does this book cover both single and multivariable calculus?

Yes, Calculus by Courant covers both single and multivariable calculus, making it a comprehensive resource for students studying this subject.

## 5. Are there any online resources available to supplement this textbook?

Yes, Courant's Calculus book is accompanied by an online platform with additional resources, including practice problems, video lectures, and interactive quizzes to help students further understand the material.

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