The Complete Idiot's Guide to Calculus

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  • #1
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Main Question or Discussion Point

The Complete Idiot's Guide to Calculus

INTRODUCTION
I've never really been very good at math and when I found out I had to take a Calculus class I started to panic. Once I gathered myself I went to the local bookstore to see if I could get a book to read so i could get a heads start. We are all familiar with the dummies and idiot books I think. This one caught my eye and I decided to buy it

AUDIENCE
This book is for anyone looking for a first glance at Calculus.

SCOPE
Covers every major topic from Limits to Special Series.

PROS
The author is fantastic at making the math fun. It's not like reading a textbook at all. He throws in some history, fun quotes and applies concepts into real world applications.

Although the book goes through the very basics, I felt I knew enough after reading to take my course head on. There are many examples for each concept and the author includes some check point exercises and quizes.

It's a short read at 300 pages, something you could pick up and finish in a week if your curious about what Calculus is all about.

CONS
As I said the book goes through the very basics and that could be bad for some people who want to know topics in depth. Otherwise I can't really see anything wrong with this book.

CONCLUSION

Find it on amazon:
https://www.amazon.com/dp/0028643658/?tag=pfamazon01-20&tag=pfamazon01-20

It's paperback, very light and I got it for about $20.

Rating: 5/5
 
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Answers and Replies

  • #2
17
0
Well, I saw that this section of the forum was empty and since I have read through plenty of school texts...

Calculus - 3rd Edition
James Stewart
Brooks/Cole Publishing Company

Approximate price: $110 USD (New), $72 USD (used - what I paid many years ago)

The most current is 4th Edition but everything is about the same, chapters are rearranged.

Contents:
Review and preview
1. Limits and rates of change
2. Derivatives
3. The mean value theorem and curve sketching
4. Integrals
5. Applications of integration
6. Inverse funcitons
7. Techniques of integration
8. Further applications of integrations
9. Parametric equations and polar coordinates
10. Infinite sequences and series
11. Three-dimensional analytic geometry and vectors
12. Partial derivatives
13. Multiple integrals
14. Vector calculus
15. Differential Equations

Pros:
This book covers pretty much every aspect of calculus that an aspiring student, or other, would need to know. It has ample examples with pretty thorough explainations of the rules involved for problem solving. And plenty of problems to solve, too!

Cons:
The main downside of this book (IMO) is the tendency to not explain what algebraic (or other) function is used to help solve example problems which leave the reader to wonder "how?" certain numbers 'magically' appeared (a little confused).

Benefits:
The reader will be a much stronger mathematician and problem solver. Not only in math but in regards to other subjects as well; such as physics, statics, dynamics, statistics, etc...

Conclusion:
My overall rating is an enthusiastic two thumbs up. I have probably been through the book 3 or 4 times over throughout my educational career, and it always presents a challenge. I definitely recommend anyone wanting to learn and possibly master calculus to pick up this book and get to work!

Look for more reviews by me to follow!
 
  • #3
7
0
Hello,
I have decided that over the summer I want to get a start on my AP Calculus class for next yr and was thinking a review book would be the way to got. I have narrowed it down to Be Prepared for the AP Calculus Exam by Mark Howell and Martha Montgomery and Master the AP Calculus AB & BC by Mark Wilding. I chose these two because they seem to offer the most exploitation for the topics plus review for the exam. However my question is if anyone has used either and how they liked them. From what I can tell from the reviews both offer a good understanding, but the second seems to have more praise and the first seems to be more up-to-date on technology. However I cannot go to my local bookstore and compare them side by side because they do not carry the first title. So I was wondering what people who have used either felt about them or if you could recommend a better one?
 
  • #4
78
0
So I was in the library one day when I noticed two big, old calculus textbooks sitting on the shelf. One was called Calculus and Analytic Geometry by James E. Shockley. The other was called Calculus by Harley Flanders. Since I didn't have time to look through them, I checked them both out to look at later. Unfortunately, Amazon failed to help me this time since no one had written any reviews on them, so I started reading them on my own.

So far, I've enjoyed the rigor of Shockley's book and the feel of Flanders' book. However, I'm still not sure how these books compare with other modern ones. So if anyone here just so happens to have heard of them, what is your opinion of them?
 
  • #5
17
0
I just got this book and it is outstanding. In my collection of textbooks Stewart is my favorite, and this book by Adrian Banner is now my favorite supplement.

( Okay I have to keep relearning this stuff, it is therapy).

But anyone taking Calculus or someone who needs a refresher, this is the book.

Unlike the outlines, this book offers a style that explains topics in detail, but also provides key points to study.

I highly recommend this book. If you only could choose one, this would be it.
 
  • #6
51
0
Hello, I couldn't post in the "calculus and beyond" area for some reason, so this looks like the next best spot.

I'm an engineering student (well maybe Physics still not entirely sure) finishing up Calc 2 (well another month) and we are using Calculus by Rogawski, which I personally disliked. I often learn the material from websites instead of the text.

I need a good text recommendation for Calc 3, which I'll be taking in the Fall. (I'm also taking Diff Equations this summer so if there is a good text that covers both, then bonus). Two popular ones seem to be Stewart and Spivak. I've read that Stewart is a little lacking in the Calc 3 department, but I've also read that Spivak is pretty proof-heavy and advanced (as a non-math major is that necessary?).
I like books that have a good-sized associated solutions manual, as I seem to learn best by seeing how problems are done and repeatedly writing them out.

Thanks in advance for any help.
 
  • #7
7
0
Hello

I am a physics student and for our first year we have a choice of 2 books for calculas:

Thomas' Calculas 12th edition
George B. Thomas

Calculus : A Complete Course 7th edition
Robert A. Adams

Our teacher says that Thomas' Calculas is simpler, and Amazon reviews for the previous versions of Calculus : A Complete Course say that it is overly simplified and dosen't explain proofs in full, so it is more suited for engineer majors; I don't know about the seventh edition. I have some background in Calculas as well (complex limits, riemann integrals, complex numbers, curves, Discrete Maths etc), but I still hate the words 'obviously' 'it is easy to understand that' etc.
Science and math is based on proof, is it not?

As Einstien once said: "Everything should be made as simple as possible, but not simpler."

Thanks in advance
trujafar
 
  • #8
My inquiry is hard to describe with out some background so bear with me.

I have done coursework through Differential Equations and vector calculus in college. The intuitive parts of my education covered in vector calc make perfect sense to me because most of it is very kinesthetically intuitive, while the more abstract content of Diff-eq's and Linear algebra take serious work on my part.

I feel that I never built the foundation necessary for higher math because I brushed through a lot of coursework back in high school relating to integration and linear algebra. It was easy enough that I didn't have to build a depth of understanding to get through tests.

I'm looking for a few textbooks that cover Statistics and Probability, Linear Algebra and Calculus (primarily integration and proofs) that I can work through from start to finish to reconstruct my foundations.

I'm currently working through Spivak's "Calculus" and Axler's "Linear Algebra Done Right." Though I am not dead-set on them, I had always heard good things.

Any advice would be appreciated.
 
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  • #9
I just started college and at the suggestion of a couple members I went out and bought Serge Lang's "A First Course in Calculus" but I'm finding it a tad difficult to understand. Should I continue to try and understand the material or use an easier textbook to try and get it?

Also I bought Halliday's "Fundamentals of Physics" for my first Physics textbook but I haven't started on it because I want to try to understand Calculus first before I try to understand Physics. Is this still a good book for beginners Physics?
 
  • #10
22,097
3,283

Table of Contents:
Code:
[LIST]
[*] Preface 
[*] To the Student 
[*] To the Instructor 
[*] Acknowledgments 
[*] What Is Calculus? 
[*] Preliminaries 
[LIST]
[*] Real Numbers and the Real Line 
[LIST]
[*] Intervals 
[*] The Absolute Value 
[*] Equations and Inequalities Involving Absolute Values 
[/LIST]
[*] Cartesian Coordinates in the Plane 
[LIST]
[*] Axis Scales 
[*] Increments and Distances 
[*] Graphs 
[*] Straight Lines 
[*] Equations of Lines
[/LIST] 
[*] Graphs of Quadratic Equations 
[LIST]
[*] Circles and Disks 
[*] Equations of Parabolas 
[*] Reflective Properties of Parabolas 
[*] Scaling a Graph 
[*] Shifting a Graph 
[*] Ellipses and Hyperbolas 
[/LIST]
[*] Functions and Their Graphs
[LIST] 
[*] The Domain Convention 
[*] Graphs of Functions 
[*] Even and Odd Functions; Symmetry and Reflections 
[*] Reflections in Straight Lines 
[*] Defining and Graphing Functions with Maple 
[/LIST]
[*] Combining Functions to Make New Functions 
[LIST]
[*] Sums, Differences, Products, Quotients, and Multiples 
[*] Composite Functions 
[*] Piecewise Defined Functions 
[/LIST]
[*] Polynomials and Rational Functions 
[LIST]
[*] Roots and Factors 
[*] Roots and Factors of Quadratic Polynomials 
[*] Miscellaneous Factorings 
[/LIST]
[*] The Trigonometric Functions
[LIST]
[*] Some Useful Identities 
[*] Some Special Angles 
[*] The Addition Formulas 
[*] Other Trigonometric Functions 
[*] Maple Calculations 
[*] Trigonometry Review 
[/LIST]
[/LIST]
[*] Limits and Continuity 
[LIST]
[*]  Examples of Velocity, Growth Rate, and Area 
[LIST]
[*] Average Velocity and Instantaneous Velocity 
[*] The Growth of an Algal Culture 
[*] The Area of a Circle 
[/LIST]
[*]  Limits of Functions 
[LIsT]
[*] One-Sided Limits 
[*] Rules for Calculating Limits 
[*] The Squeeze Theorem 
[/LIST]
[*]  Limits at Infinity and Infinite Limits 
[LIST]
[*] Limits at Infinity 
[*] Limits at Infinity for Rational Functions 
[*] Infinite Limits 
[*] Using Maple to Calculate Limits 
[/LIST]
[*]  Continuity 
[LIST]
[*] Continuity at a Point 
[*] Continuity on an Interval 
[*] There Are Lots of Continuous Functions 
[*] Continuous Extensions and Removable Discontinuities 
[*] Continuous Functions on Closed, Finite Intervals 
[*] Finding Maxima and Minima Graphically 
[*] Finding Roots of Equations 
[/LIST]
[*] The Formal Definition of Limit 
[LIST]
[*] Using the Definition of Limit to Prove Theorems 
[*] Other Kinds of Limits 
[/LIST]
[*] Chapter Review 
[/LIST]
[*] Differentiation 
[LIST]
[*]  Tangent Lines and Their Slopes 
[LIST]
[*] Normals 
[/LIST]
[*] The Derivative 
[LIST]
[*] Some Important Derivatives 
[*] Leibniz Notation 
[*] Differentials 
[*] Derivatives Have the Intermediate-Value Property 
[/LIST]
[*] Differentiation Rules
[LIST] 
[*] Sums and Constant Multiples 
[*] The Product Rule 
[*] The Reciprocal Rule 
[*] The Quotient Rule 
[/LIST]
[*] The Chain Rule 
[LIST]
[*] Finding Derivatives with Maple 
[*] Building the Chain Rule into Differentiation Formulas 
[*] Proof of the Chain Rule (Theorem 6) 
[/LIST]
[*]  Derivatives of Trigonometric Functions
[LIST] 
[*] Some Special Limits 
[*] The Derivatives of Sine and Cosine 
[*] The Derivatives of the Other Trigonometric Functions 
[/LIST]
[*]  The Mean-Value Theorem 
[LIST]
[*] Increasing and Decreasing Functions 
[*] Proof of the Mean-Value Theorem 
[/LIST]
[*] Using Derivatives
[LIST]
[*] Approximating Small Changes 
[*] Average and Instantaneous Rates of Change 
[*] Sensitivity to Change 
[*] Derivatives in Economics 
[/LIST]
[*]  Higher-Order Derivatives 
[*]  Implicit Differentiation 
[LIST]
[*] Higher-Order Derivatives 
[*] The General Power Rule 
[/LIST]
[*]  Antiderivatives and Initial-Value Problems 
[LIST]
[*] Antiderivatives 
[*] The Indefinite Integral 
[*] Differential Equations and Initial-Value Problems 
[/LIST]
[*]  Velocity and Acceleration 
[LIST]
[*] Velocity and Speed 
[*] Acceleration 
[*] Falling Under Gravity 
[/LIST]
[*] Chapter Review 
[/LIST]
[*] Transcendental Functions 
[LIST]
[*] Inverse Functions 
[LIST]
[*] Inverting Non-One-to-One Functions 
[*] Derivatives of Inverse Functions 
[/LIST]
[*] Exponential and Logarithmic Functions 
[LIST]
[*] Exponentials 
[*] Logarithms 
[/LIST]
[*] The Natural Logarithm and Exponential 
[LIST]
[*] The Natural Logarithm 
[*] The Exponential Function 
[*] General Exponentials and Logarithms 
[*] Logarithmic Differentiation 
[/LIST]
[*] Growth and Decay 
[LIST]
[*] The Growth of Exponentials and Logarithms 
[*] Exponential Growth and Decay Models 
[*] Interest on Investments 
[*] Logistic Growth 
[/LIST]
[*] The Inverse Trigonometric Functions 
[LIST]
[*] The Inverse Sine (or Arcsine) Function 
[*] The Inverse Tangent (or Arctangent) Function 
[*] Other Inverse Trigonometric Functions 
[/LIST]
[*]  Hyperbolic Functions 
[LIST]
[*] Inverse Hyperbolic Functions 
[/LIST]
[*] Second-Order Linear DEs with Constant Coefficients
[LIST] 
[*] Recipe for Solving ay" + by' + с у = 0 
[*] Simple Harmonic Motion 
[*] Damped Harmonic Motion 
[/LIST]
[*] Chapter Review 
[/LIST]
[*] Some Applications of Derivatives 
[LIST]
[*] Related Rates 
[LIST]
[*] Procedures for Related-Rates Problems 
[/LIST]
[*] Extreme Values
[LIST] 
[*] Maximum and Minimum Values 
[*] Critical Points, Singular Points, and Endpoints 
[*] Finding Absolute Extreme Values 
[*] The First Derivative Test 
[*] Functions Not Defined on Closed, Finite Intervals 
[/LIST]
[*] Concavity and Inflections 
[LIST]
[*] The Second Derivative Test 
[/LIST]
[*] Sketching the Graph of a Function 
[LIST]
[*] Asymptotes 
[*] Examples of Formal Curve Sketching 
[/LIST]
[*] Extreme-Value Problems
[LIST]
[*] Procedure for Solving Extreme-Value Problems
[/LIST]
[*] Finding Roots of Equations 
[LIST]
[*] Newton's Method 
[*] Fixed-Point Iteration 
[*] "Solve" Routines 
[/LIST]
[*]  Linear Approximations 
[LIST]
[*] Approximating Values of Functions 
[*] Error Analysis 
[/LIST]
[*]  Taylor Polynomials
[LIST] 
[*] Taylor's Formula 
[*] Big-O Notation
[/LIST] 
[*]  Indeterminate Forms 
[LIST]
[*] l'Hopital's Rules 
[/LIST]
[*] Chapter Review 
[/LIST]
[*] Integration 
[LIST]
[*]  Sums and Sigma Notation 
[LIST]
[*] Evaluating Sums 
[/LIST]
[*] Areas as Limits of Sums
[LIST]
[*] The Basic Area Problem 
[*] Some Area Calculations 
[/LIST]
[*]  The Definite Integral
[LIST] 
[*] Partitions and Riemann Sums 
[*] The Definite Integral 
[*] General Riemann Sums 
[/LIST]
[*]  Properties of the Definite Integral
[LIST] 
[*] A Mean-Value Theorem for Integrals 
[*] Definite Integrals of Piecewise Continuous Functions 
[/LIST]
[*] The Fundamental Theorem of Calculus 
[*] The Method of Substitution 
[LIST]
[*] Trigonometric Integrals
[/LIST]
[*] Areas of Plane Regions 
[LIST]
[*] Areas Between Two Curves
[/LIST]
[*] Chapter Review
[/LIST] 
[*] Techniques of Integration
[LIST]
[*] Integration by Parts
[LIST]
[*] Reduction Formulas
[/LIST]
[*] Inverse Substitutions
[LIST]
[*] The Inverse Trigonometric Substitutions
[*] Completing the Square
[*] Other Inverse Substitutions
[*] The tan(theta/2) Substitution
[/LIST]
[*] Integrals of Rational Functions
[LIST]
[*] Linear and Quadratic Denominators 
[*] Partial Fractions
[/LIST]
[*] Integration Using Computer Algebra or Tables
[LIST] 
[*] Using Maple for Integration 
[*] Using Integral Tables
[/LIST] 
[*] Improper Integrals 
[LIST]
[*] Improper Integrals of Type I 
[*] Improper Integrals of Type II 
[*] Estimating Convergence and Divergence
[/LIST] 
6.6 The Trapezoid and Midpoint Rules
[LIST] 
[*] The Trapezoid Rule 
[*] The Midpoint Rule 
[*] Error Estimates
[/LIST] 
[*] Simpson's Rule
[*] Other Aspects of Approximate Integration 
[LIST] 
[*] Approximating Improper Integrals 
[*] Using Taylor's Formula 
[*] Romberg Integration 
[*] Other Methods
[/LIST] 
[*] Chapter Review
[/LIST] 
[*] Applications of Integration 
[LIST] 
[*] Volumes by Slicing — Solids of Revolution
[LIST]  
[*] Volumes by Slicing 
[*] Solids of Revolution 
[*] Cylindrical Shells
[/LIST] 
[*] More Volumes by Slicing 
[*] Arc Length and Surface Area 
[LIST] 
[*] Arc Length 
[*] The Arc Length of the Graph of a Function 
[*] Areas of Surfaces of Revolution
[/LIST] 
[*] Mass, Moments, and Centre of Mass
[LIST]  
[*] Mass and Density 
[*] Moments and Centres of Mass 
[*] Two- and Three-Dimensional Examples 
[/LIST]
[*] Centroids 
[LIST] 
[*] Pappus's Theorem 
[/LIST]
[*] Other Physical Applications 
[LIST] 
[*] Hydrostatic Pressure 
[*] Work 
[*] Potential Energy and Kinetic Energy
[/LIST] 
[*] Applications in Business, Finance, and Ecology
[LIST]  
[*] The Present Value of a Stream of Payments 
[*] The Economics of Exploiting Renewable Resources
[/LIST] 
[*] Probability 
[LIST] 
[*] Discrete Random Variables 
[*] Expectation, Mean, Variance, and Standard Deviation 
[*] Continuous Random Variables 
[*] The Normal Distribution
[/LIST] 
[*] First-Order Differential Equations
[LIST] 
[*] Separable Equations 
[*] First-Order Linear Equations 
[/LIST]
[*] Chapter Review 
[/LIST]
[*] Conics, Parametric Curves, and Polar Curves
[LIST]  
[*] Conics
[LIST]  
[*] Parabolas 
[*] The Focal Property of a Parabola 
[*] Ellipses 
[*] The Focal Property of an Ellipse 
[*] The Directrices of an Ellipse 
[*] Hyperbolas 
[*] The Focal Property of a Hyperbola 
[*] Classifying General Conies 
[/LIST]
[*] Parametric Curves 
[LIST] 
[*] General Plane Curves and Parametrizations 
[*] Some Interesting Plane Curves
[/LIST] 
[*] Smooth Parametric Curves and Their Slopes
[LIST]  
[*] The Slope of a Parametric Curve 
[*] Sketching Parametric Curves
[/LIST] 
[*] Arc Lengths and Areas for Parametric Curves 
[LIST] 
[*] Arc Lengths and Surface Areas 
[*] Areas Bounded by Parametric Curves 
[/LIST]
[*] Polar Coordinates and Polar Curves 
[LIST] 
[*] Some Polar Curves 
[*] Intersections of Polar Curves 
[*] Polar Conics
[/LIST] 
[*] Slopes, Areas, and Arc Lengths for Polar Curves 
[LIST] 
[*] Areas Bounded by Polar Curves 
[*] Arc Lengths for Polar Curves
[/LIST] 
[*] Chapter Review
[/LIST] 
[*] Sequences, Series, and Power Series
[LIST]  
[*] Sequences and Convergence 
[LIST] 
[*] Convergence of Sequences
[/LIST] 
[*] Infinite Series 
[LIST] 
[*] Geometric Series 
[*] Telescoping Series and Harmonic Series 
[*] Some Theorems About Series
[/LIST] 
[*] Convergence Tests for Positive Series 
[LIST] 
[*] The Integral Test 
[*] Using Integral Bounds to Estimate the Sum of a Series 
[*] Comparison Tests 
[*] The Ratio and Root Tests 
[*] Using Geometric Bounds to Estimate the Sum of a Series
[/LIST] 
[*] Absolute and Conditional Convergence
[LIST]  
[*] The Alternating Series Test 
[*] Rearranging the Terms in a Series
[/LIST] 
[*] Power Series 
[LIST] 
[*] Algebraic Operations on Power Series 
[*] Differentiation and Integration of Power Series 
[*] Maple Calculations
[/LIST] 
[*] Taylor and Maclaurin Series
[LIST]  
[*] Maclaurin Series for Some Elementary Functions 
[*] Other Maclaurin and Taylor Series 
[*] Taylor's Formula Revisited
[/LIST] 
[*] Applications of Taylor and Maclaurin Series
[LIST]  
[*] Approximating the Values of Functions 
[*] Functions Defined by Integrals 
[*] Indeterminate Forms
[/LIST] 
[*] The Binomial Theorem and Binomial Series 
[LIST] 
[*] The Binomial Series 
[/LIST]
[*] Fourier Series 
[LIST] 
[*] Periodic Functions 
[*] Fourier Series 
[*] Convergence of Fourier Series 
[*] Fourier Cosine and Sine Series 
[/LIST]
[*] Chapter Review 
[/LIST]
[*] Vectors and Coordinate Geometry in 3-Space 
[LIST] 
[*] Analytic Geometry in Three Dimensions 
[LIST] 
[*] Euclidean n-Space 
[*] Describing Sets in the Plane, 3-Space, and n-Space
[/LIST] 
[*] Vectors 
[LIST] 
[*] Vectors in 3-Space 
[*] Hanging Cables and Chains 
[*] The Dot Product and Projections 
[*] Vectors in n-Space
[/LIST] 
[*] The Cross Product in 3-Space 
[LIST] 
[*] Determinants 
[*] The Cross Product as a Determinant 
[*] Applications of Cross Products
[/LIST] 
[*] Planes and Lines 
[LIST] 
[*] Planes in 3-Space 
[*] Lines in 3-Space 
[*] Distances
[/LIST] 
[*] Quadric Surfaces 
[*] A Little Linear Algebra
[LIST]  
[*] Matrices 
[*] Determinants and Matrix Inverses 
[*] Linear Transformations 
[*] Linear Equations 
[*] Quadratic Forms, Eigenvalues, and Eigenvectors
[/LIST] 
[*] Using Maple for Vector and Matrix Calculations 
[LIST] 
[*] Vectors 
[*] Matrices 
[*] Linear Equations 
[*] Eigenvectors and Eigenfunctions 
[/LIST]
[*] Chapter Review 
[/LIST]
[*] Vector Functions and Curves
[LIST] 
[*] Vector Functions of One Variable
[LIST] 
[*] Differentiating Combinations of Vectors
[/LIST]
[*] Some Applications of Vector Differentiation
[LIST]  
[*] Motion Involving Varying Mass
[*] Circular Motion
[*] Rotating Frames and the Coriolis Effect
[/LIST]
[*] Curves and Parametrizations
[LIST] 
[*] Parametrizing the Curve of Intersection of Two Surfaces 
[*] Arc Length
[*] Piecewise Smooth Curves
[*] The Arc-Length Parametrization
[/LIST]
[*] Curvature, Torsion, and the Frenet Frame
[LIST]  
[*] The Unit Tangent Vector
[*] Curvature and the Unit Normal
[*] Torsion and Binormal, the Frenet-Serret Formulas
[/LIST] 
[*] Curvature and Torsion for General Parametrizations
[LIST]  
[*] Tangential and Normal Acceleration
[*] Evolutes
[*] An Application to Track (or Road) Design
[*] Maple Calculations
[/LIST]
[*] Kepler's Laws of Planetary Motion
[LIST]  
[*] Ellipses in Polar Coordinates
[*] Polar Components of Velocity and Acceleration 
[*] Central Forces and Kepler's Second Law
[*] Derivation of Kepler's First and Third Laws
[*] Conservation of Energy
[/LIST]
[*] Chapter Review
[/LIST]
[*] Partial Differentiation 
[LIST] 
[*] Functions of Several Variables
[LIST] 
[*] Graphical Representations
[*] Using Maple Graphics
[/LIST]
[*] Limits and Continuity
[*] Partial Derivatives
[LIST] 
[*] Tangent Planes and Normal Lines
[*] Distance from a Point to a Surface: A Geometric Example
[/LIST] 
[*] Higher-Order Derivatives
[LIST] 
[*] The Laplace and Wave Equations
[/LIST]
[*] The Chain Rule
[LIST] 
[*] Homogeneous Functions
[*] Higher-Order Derivatives
[/LIST]
[*] Linear Approximations, Differentiability, and Differentials
[LIST]  
[*] Proof of the Chain Rule
[*] Differentials
[*] Functions from n-space to m-space
[/LIST]
[*] Gradients and Directional Derivatives 
[LIST] 
[*] Directional Derivatives
[*] Rates Perceived by a Moving Observer
[*] The Gradient in Three and More Dimensions
[/LIST]
[*] Implicit Functions
[LIST] 
[*] Systems of Equations
[*] Jacobian Determinants
[*] The Implicit Function Theorem
[/LIST]
[*] Taylor Series and Approximations
[LIST] 
[*] Approximating Implicit Functions
[/LIST]
[*] Chapter Review
[/LIST]
[*] Applications of Partial Derivatives
[LIST]  
[*] Extreme Values
[LIST] 
[*] Classifying Critical Points
[/LIST]
[*] Extreme Values of Functions Defined on Restricted Domains 
[LIST] 
[*] Linear Programming
[/LIST]
[*] Lagrange Multipliers
[LIST] 
[*] The Method of Lagrange Multipliers
[*] Problems with More than One Constraint
[*] Nonlinear Programming 
[/LIST]
[*] The Method of Least Squares
[LIST] 
[*] Linear Regression
[*] Applications of the Least Squares Method to Integrals
[/LIST] 
[*] Parametric Problems
[LIST] 
[*] Differentiating Integrals with Parameters
[*] Envelopes
[*] Equations with Perturbations
[/LIST]
[*] Newton's Method 
[LIST] 
[*] Implementing Newton's Method Using a Spreadsheet
[/LIST] 
[*] Calculations with Maple 
[LIST] 
[*] Solving Systems of Equations
[*] Finding and Classifying Critical Points
[/LIST]
[*] Chapter Review
[/LIST]
[*] Multiple Integration 
[LIST] 
Double Integrals 
[LIST] 
[*] Double Integrals over More General Domains
[*] Properties of the Double Integral
[*] Double Integrals by Inspection
[/LIST]
[*] Iteration of Double Integrals in Cartesian Coordinates 
[*] Improper Integrals and a Mean-Value Theorem 
[LIST] 
[*] Improper Integrals of Positive Functions 
[*] A Mean-Value Theorem for Double Integrals
[/LIST] 
[*] Double Integrals in Polar Coordinates
[LIST]  
[*] Change of Variables in Double Integrals 
[/LIST]
[*] Triple Integrals 
[*] Change of Variables in Triple Integrals 
[LIST] 
[*] Cylindrical Coordinates 
[*] Spherical Coordinates 
[/LIST]
[*] Applications of Multiple Integrals 
[LIST] 
[*] The Surface Area of a Graph 
[*] The Gravitational Attraction of a Disk 
[*] Moments and Centres of Mass 
[*] Moment of Inertia 
[/LIST]
[*] Chapter Review 
[/LIST]
[*] Vector Fields 
[LIST] 
[*] Vector and Scalar Fields
[LIST]  
[*] Field Lines (Integral Curves) 
[*] Vector Fields in Polar Coordinates 
[/LIST]
[*] Conservative Fields
[LIST]  
[*] Equipotential Surfaces and Curves 
[*] Sources, Sinks, and Dipoles 
[/LIST]
[*] Line Integrals 
[LIST] 
[*] Evaluating Line Integrals
[/LIST] 
[*] Line Integrals of Vector Fields
[LIST]  
[*] Connected and Simply Connected Domains 
[*] Independence of Path 
[/LIST]
[*] Surfaces and Surface Integrals
[LIST]  
[*] Parametric Surfaces 
[*] Composite Surfaces 
[*] Surface Integrals 
[*] Smooth Surfaces, Normals, and Area Elements 
[*] Evaluating Surface Integrals 
[*] The Attraction of a Spherical Shell
[/LIST] 
[*] Oriented Surfaces and Flux Integrals
[LIST]  
[*] Oriented Surfaces 
[*] The Flux of a Vector Field Across a Surface 
[/LIST]
[*] Chapter Review 
[/LIST]
[*] Vector Calculus
[LIST]  
[*] Gradient, Divergence, and Curl
[LIST] 
[*] Interpretation of the Divergence
[*] Distributions and Delta Functions
[*] Interpretation of the Curl
[/LIST]
[*] Some Identities Involving Grad, Div, and Curl
[LIST]  
[*] Scalar and Vector Potentials
[*] Maple Calculations
[/LIST]
[*] Green's Theorem in the Plane
[LIST] 
[*] The Two-Dimensional Divergence Theorem
[/LIST]
[*] The Divergence Theorem in 3-Space
[LIST]  
[*] Variants of the Divergence Theorem
[/LIST]
[*] Stokes's Theorem
[*] Some Physical Applications of Vector Calculus
[LIST]  
[*] Fluid Dynamics
[*] Electromagnetism
[*] Electrostatics
[*] Magnetostatics
[*] Maxwell's Equations
[/LIST]
[*] Orthogonal Curvilinear Coordinates
[LIST]  
[*] Coordinate Surfaces and Coordinate Curves
[*] Scale Factors and Differential Elements
[*] Grad, Div, and Curl in Orthogonal Curvilinear Coordinates 
[/LIST]
[*] Chapter Review 
[/LIST]
[*] Ordinary Differential Equations
[LIST]  
[*] Classifying Differential Equations
[*] Solving First-Order Equations 
[LIST] 
[*] Separable Equations 
[*] First-Order Homogeneous Equations 
[*] Exact Equations 
[*] Integrating Factors 
[*] First-Order Linear Equations
[/LIST] 
[*] Existence, Uniqueness, and Numerical Methods 
[LIST] 
[*] Existence and Uniqueness of Solutions
[*] Numerical Methods 
[/LIST]
[*] Differential Equations of Second Order 
[LIST] 
[*] Equations Reducible to First Order 
[*] Second-Order Linear Equations
[/LIST] 
[*] Linear Differential Equations with Constant Coefficients 
[LIST] 
[*] Constant-Coefficient Equations of Higher Order 
[*] Euler (Equidimensional) Equations
[/LIST] 
[*] Nonhomogeneous Linear Equations
[LIST]  
[*] Resonance 
[*] Variation of Parameters 
[*] Maple Calculations
[/LIST] 
[*] Series Solutions of Differential Equations 
[*] Chapter Review 
[/LIST]
[*] Appendix: Complex Numbers
[LIST]  
[*] Definition of Complex Numbers 
[*] Graphical Representation of Complex 
[*] Numbers 
[*] Complex Arithmetic 
[*] Roots of Complex Numbers 
[/LIST]
[*] Appendix: Complex Functions
[LIST] 
[*] Limits and Continuity
[*] The Complex Derivative
[*] The Exponential Function
[*] The Fundamental Theorem of Algebra
[/LIST]
[*] Appendix: Continuous Functions
[LIST] 
[*] Limits of Functions 
[*] Continuous Functions 
[*] Completeness and Sequential Limits 
[*] Continuous Functions on a Closed, Finite Interval
[/LIST] 
[*] Appendix: The Riemann Integral 
[LIST] 
[*] Uniform Continuity 
[/LIST]
[*] Appendix: Doing Calculus with Maple 
[LIST] 
[*] List of Maple Examples and Discussion
[/LIST] 
[*] Answers to Odd-Numbered 
[*] Exercises 
[*] Index 
[/LIST]
 
Last edited by a moderator:
  • #11
22,097
3,283

Table of Contents:
Code:
[LIST]
[*] Mappings from Numbers to Vectors and Vectors to Numbers
[LIST]
[*] Vectors
[LIST]
[*] Definition of points in n-space
[*] Located vectors
[*] Scalar product
[*] The norm of a vector
[*] Lines and planes
[*] The cross product
[/LIST]
[*] Differentiation of Vectors
[LIST]
[*] Derivative
[*] Length of curves
[*] The chain rule and applications
[/LIST]
[*] Functions of Several Variables
[LIST]
[*] Graphs and level curves
[*] Partial derivatives
[*] Differentiability and gradient
[/LIST]
[*] The Chain Rule and the Gradient
[LIST]
[*] The chain rule
[*] Tangent plane
[*] Directional derivative
[*] Conservation law
[/LIST]
[*] Potential Functions and Curve Integrals
[LIST]
[*] Potential functions
[*] Differentiating under the integral
[*] Local existence of potential functions
[*] Curve integrals
[*] Dependence of the integral on the path
[/LIST]
[*] Higher Derivatives
[LIST]
[*] Repeated partial derivatives
[*] Partial differential operators
[*] Taylor's formula
[*] Integral expressions
[/LIST]
[*] Maximum and Minimum 
[LIST]
[*] Critical points
[*] The quadratic form
[*] Boundary points
[*] Lagrange multipliers
[/LIST]
[/LIST]
[*] Matrices, Linear Maps, and Determinants
[LIST]
[*] Matrices
[LIST]
[*] Matrices
[*] Multiplication of matrices
[/LIST]
[*] Linear Mappings
[LIST]
[*] Mappings
[*] Linear mappings
[*] Geometric applications
[*] Composition and inverse of mappings
[/LIST]
[*] Determinants 
[LIST]
[*] Determinants of order 2
[*] Determinants of order 3
[*] Additional properties of determinants
[*] Independence of vectors
[*] Determinant of a product
[*] Inverse of a matrix
[/LIST]
[/LIST]
[*] Mappings from Vectors to Vectors
[LIST]
[*] Applications to Functions of Several Variables
[LIST]
[*] The derivative as a linear map
[*] The Jacobian matrix
[*] The chain rule
[*] Inverse mappings and implicit functions
[/LIST]
[/LIST]
[*] Multiple Integration
[LIST]
[*] Multiple Integrals
[LIST]
[*] Double integrals
[*] Repeated integrals
[*] Polar coordinates
[*] Triple integrals
[*] Center of mass
[/LIST]
[*] The Change of Variables Formula
[LIST]
[*] Determinants as area and volume
[*] Dilations
[*] Change of variables formula in two dimensions
[*] Change of variables formula in three dimensions
[/LIST]
[*] Green's Theorem
[LIST]
[*] Statement of the theorem
[*] Application to the change of variables formula
[/LIST]
[*] Surface Integrals
[LIST]
[*] Parametrization, tangent plane, and normal vector
[*] Surface area
[*] Surface integrals
[*] Curl and divergence of a vector field
[*] Divergence theorem
[*] Stokes' theorem
[/LIST]
[*] Appendix: Fourier Series 
[LIST]
[*] General scalar products
[*] Computation of Fourier series
[/LIST]
[/LIST]
[*] Answers
[*] Index
[/LIST]
 
Last edited by a moderator:
  • #12
Astronuc
Staff Emeritus
Science Advisor
18,823
2,034

Table of Contents:

Code:
1. Setting the Stage. 

Euclidean Spaces and Vectors. 
Subsets of Euclidean Space. 
Limits and Continuity. 
Sequences. 
Completeness. 
Compactness. 
Connectedness. 
Uniform Continuity.

2. Differential Calculus. 

Differentiability in One Variable. 
Differentiability in Several Variables. 
The Chain Rule. 
The Mean Value Theorem. 
Functional Relations and Implicit Functions: A First Look. 
Higher-Order Partial Derivatives. 
Taylor's Theorem. 
Critical Points. 
Extreme Value Problems. 
Vector-Valued Functions and Their Derivatives.

3. The Implicit Function Theorem and Its Applications. 

The Implicit Function Theorem. 
Curves in the Plane. 
Surfaces and Curves in Space. 
Transformations and Coordinate Systems. 
Functional Dependence.

4. Integral Calculus. 

Integration on the Line. 
Integration in Higher Dimensions. 
Multiple Integrals and Iterated Integrals. 
Change of Variables for Multiple Integrals. 
Functions Defined by Integrals. 
Improper Integrals. 
Improper Multiple Integrals. 
Lebesgue Measure and the Lebesgue Integral.

5. Line and Surface Integrals; Vector Analysis. 

Arc Length and Line Integrals. 
Green's Theorem. 
Surface Area and Surface Integrals. 
Vector Derivatives. 
The Divergence Theorem. 
Some Applications to Physics. 
Stokes's Theorem. 
Integrating Vector Derivatives. 
Higher Dimensions and Differential Forms.

6. Infinite Series. 

Definitions and Examples. 
Series with Nonnegative Terms. 
Absolute and Conditional Convergence. 
More Convergence Tests. 
Double Series; Products of Series.

7. Functions Defined by Series and Integrals. 

Sequences and Series of Functions. 
Integrals and Derivatives of Sequences and Series. 
Power Series. 
The Complex Exponential and Trig Functions. 
Functions Defined by Improper Integrals. 
The Gamma Function. 
Stirling's Formula.

8. Fourier Series. 

Periodic Functions and Fourier Series. 
Convergence of Fourier Series. 
Derivatives, Integrals, and Uniform Convergence. 
Fourier Series on Intervals. 
Applications to Differential Equations. 
The Infinite-Dimensional Geometry of Fourier Series. 
The Isoperimetric Inequality.

APPENDICES. 

A. Summary of Linear Algebra. 

Vectors. 
Linear Maps and Matrices. 
Row Operations and Echelon Forms. 
Determinants. Linear Independence. 
Subspaces; Dimension; Rank. 
Invertibility. 
Eigenvectors and Eigenvalues.

B. Some Technical Proofs. 

The Heine-Borel Theorem. 
The Implicit Function Theorem. 
Approximation by Riemann Sums. 
Double Integrals and Iterated Integrals. 
Change of Variables for Multiple Integrals. 
Improper Multiple Integrals. 
Green's Theorem and the Divergence Theorem.

Answers to Selected Exercises. 

Bibliography. 

Index.
For undergraduate courses in Advanced Calculus and Real Analysis.

This book presents a unified view of calculus in which theory and practice reinforces each other. It is about the theory and applications of derivatives (mostly partial), integrals, (mostly multiple or improper), and infinite series (mostly of functions rather than of numbers), at a deeper level than is found in the standard calculus books. Chapter topics cover: Setting the Stage, Differential Calculus, The Implicit Function Theorem and Its Applications, Integral Calculus, Line and Surface Integrals—Vector Analysis, Infinite Series, Functions Defined by Series and Integrals, and Fourier Series. For individuals with a sound knowledge of the mechanics of one-variable calculus and an acquaintance with linear algebra.
http://www.pearsonhighered.com/educator/product/Advanced-Calculus/9780130652652.page

Publisher: Pearson; 1 edition (December 31, 2001)
 
Last edited by a moderator:
  • #13
153
3
''Visual calculus by Mamikon Mnatsakanian (known as Mamikon) is an approach to solving a variety of integral calculus problems. Many problems that would otherwise seem quite difficult yield to the method with hardly a line of calculation, often reminiscent of what Martin Gardner calls "aha! solutions'' or Roger Nelsen a proof without words.'' Is this method better than Cavalieri's principle?does give you a better perspective?there 's a new book by Tom Apostol and Mamikon Mnatsakanian about it,you might review it here https://www.physicsforums.com/showthread.php?t=704972
 
  • #14
23
0
I am studying calculus, real analysis, and abstract algebra. Sometimes while solving problems in books such as Apostol's calculus I get stuck because I need to refresh my preliminaries. The preliminaries I am referring to include:

1)inequalities involved summation upto n terms,

2)properties of polynomials and their roots,

And more importantly how factorials behave. As in how to solve limits involving factorials and exponentials such as e^x/x! and such kind of ratios. I wish to know how these functions such as e^x , x! , x^x etc grow.

Could you suggest a basic book - like a pre calculus book which deals with this ? Even if it is not exactly pre-calculus its fine. Just that it should make these properties lucid. Thanks.
 
  • #15
208
1
I am looking for references and free online books of solved problems in these topics: applications of differential equations in geometry, percentage calculus, physics and geometric analysis.

By applications in geometry I mean something like this: http://tutorial.math.lamar.edu/Classes/CalcI/IntAppsIntro.aspx

article about applications of integrals (something of that difficulty).

By applications in physics I mean simple classical mechanics (velocity, acceleration,...), combined with geometric representation.

Applications in percentage calculus should also include geometric representation.

Could someone suggest some good references and free online books with solved problems?
 
  • #16
BiGyElLoWhAt
Gold Member
1,570
116
Hi,
I am more or less just wanting to mess around, but what are some good resources to get in touch with my fractional calculus side?

Bonus points for free/cheap resources.
 

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