A First Course in Calculus by Lang

[micromass]

• Author: Serge Lang
• Title: A First Course in Calculus
• Amazon Link: https://www.amazon.com/dp/0387962018/?tag=pfamazon01-20
• Prerequisities: High-school mathematics
• Level: Undergrad

Table of Contents:

[LIST]
[*] Review of Basic Material
[LIST]
[*] Numbers and Functions
[LIST]
[*] Integers, rational numbers, and real numbers
[*] Inequalities
[*] Functions
[*] Powers
[/LIST]
[*] Graphs and Curves
[LIST]
[*] Coordinates
[*] Graphs
[*] The straight line 
[*] Distance between two points
[*] Curves and equations
[*] The circle
[*] The parabola. Changes of coordinates
[*] The hyperbola
[/LIST]
[/LIST]
[*] Differentiation and Elementary Functions 
[LIST]
[*] The Derivative
[LIST]
[*] The slope of a curve
[*] The derivative
[*] Limits
[*] Powers
[*] Sums, products, and quotients
[*] The chain rule
[*] Higher derivatives
[*] A Rate of change
[/LIST]
[*] Sine and Cosine
[LIST]
[*] The sine and cosine functions
[*] The graphs
[*] Addition formula
[*] The derivatives
[*] Two basic limits
[/LIST]
[*] The Mean Value Theorem
[LIST]
[*] The maximum and minimum theorem
[*] The mean value theorem
[*] Increasing and decreasing functions
[/LIST]
[*] Sketching Curves
[LIST]
[*] Behavior as x becomes very large
[*] Curve sketching
[*] Convexity
[*] Polar coordinates
[*] Parametric curves
[/LIST]
[*] Inverse Functions
[LIST]
[*] Definition of inverse functions
[*] Derivative of inverse functions
[*] The arcsine
[*] The arctangent
[/LIST]
[*] Exponents and Logarithms
[LIST]
[*] The logarithm
[*] The exponential function
[*] The general exponential function
[*] Order of magnitude
[*] Some applications
[/LIST]
[/LIST]
[*] Integration
[LIST]
[*] Integration
[LIST]
[*] The indefinite integral
[*] Continuous functions
[*] Area
[*] Fundamental theorem
[*] Upper and lower sums
[*] The basic properties
[*] Integrable functions
[/LIST]
[*] Properties ot the Integral
[LIST]
[*] Further connection with the derivative
[*] Sums
[*] Inequalities
[*] Improper integrals
[/LIST]
[*] Techniques of Integration
[LIST]
[*] Substitution
[*] Integration by parts
[*] Trigonometric integrals
[*] Partial fractions
[/LIST]
[*] Some Substantial Exercises
[LIST]
[*] An estimate for (n!)^{l/n}
[*] Stirling's formula
[*] Wallis' product
[/LIST]
[*] Applications of Integration
[LIST]
[*] Length of curves
[*] Area in polar coordinates
[*] Volumes of revolution
[*] Work
[*] Density and mass
[*] Probability
[*] Moments
[/LIST]
[/LIST]
[*] Series
[LIST]
[*] Taylor's Formula
[LIST]
[*] Taylor's formula
[*] Estimate for the remainder
[*] Trigonometric functions
[*] Exponential function
[*] Logarithm
[*] The arctangent
[*] The binomial expansion
[*] Uniqueness theorem
[/LIST]
[*] Series
[LIST]
[*] Convergent series
[*] Series with positive terms
[*] The ratio test
[*] The integral test
[*] Absolute and alternating convergence
[*] Power series
[*] Differentiation and integration of power series
[/LIST]
[/LIST]
[*] Miscellaneous
[LIST]
[*] Complex Numbers
[LIST]
[*] Definition
[*] Polar form
[*] Complex valued functions
[/LIST]
[*] Appendix: \varepsilon and \delta
[LIST]
[*] Least upper bound
[*] Limits
[*] Points of accumulation
[*] Continuous functions
[/LIST]
[*] Appendix: Induction
[*] Appendix: Sine and Cosine
[*] Appendix: Physics and Mathematics
[/LIST]
[*] Functions of Several Variables
[LIST]
[*] Vectors
[LIST]
[*] Definition of points in n-space
[*] Located vectors
[*] Scalar product
[*] The norm of a vector
[*] Lines and planes
[/LIST]
[*] Differentiation of Vectors
[LIST]
[*] Derivative
[*] Length of curves
[/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]
[/LIST]
[*] Answers
[*] Index
[/LIST]

 

[mathwonk]

i rather liked these elementary but clear explanations of calculus, but as with his algebra book, probably one should supplement it with a book with many more problems. lang's problems are excellent, so do them, but there are too few of them.

 

[theoristo]

Great book to get a somewhat easy somewhat rigourous introdution to single variable calculus,I use it with Courant's and Fritz's book to undrstand the hard parts and it's great .

 

[lurflurf]

The book is very good. I like the care taken in the explanations that really benefit students. The book is not comprehensive. That is an advantage compared to some bloated monstrosities in that one can focus on the most important concepts. The disadvantage is one will want to read other books in addition which is always wise. For this reason the book is not a good reference, though perhaps that is to be expected.

 

[QuantumCurt]

I'm planning on picking this book up soon. I'm starting Calculus in a couple weeks, and from the sounds of it, this is going to be a great supplement. I'm also considering picking up the Kline book.

I have a question regarding this book though. From what I can tell, this book is geared more towards the material from Calc I and II, and just briefly touches on the material from Calc III(multivariable etc.) at the end. Is Lang's book, Calculus of Several Variables a good choice too? I won't need it for a while, but I'm just looking ahead a little bit, so I can snag it up if I find a good deal on it.

 

[micromass]

I'm planning on picking this book up soon. I'm starting Calculus in a couple weeks, and from the sounds of it, this is going to be a great supplement. I'm also considering picking up the Kline book.

I have a question regarding this book though. From what I can tell, this book is geared more towards the material from Calc I and II, and just briefly touches on the material from Calc III(multivariable etc.) at the end. Is Lang's book, Calculus of Several Variables a good choice too? I won't need it for a while, but I'm just looking ahead a little bit, so I can snag it up if I find a good deal on it.

Yes, Lang's books are usually very good, and so is his calculus of several variables.
Another good book on that topic is: https://www.amazon.com/dp/0130414085/?tag=pfamazon01-20 I highly recommend it. It's a bit more abstract than Lang though.

 

[QuantumCurt]

Yes, Lang's books are usually very good, and so is his calculus of several variables.
Another good book on that topic is: https://www.amazon.com/dp/0130414085/?tag=pfamazon01-20 I highly recommend it. It's a bit more abstract than Lang though.

That sounds like a really good one. I like the fact that it mixes in some introductory Linear Algebra as well. That's a topic that I'm planning on doing some self study in after I've gotten a little calculus under my belt. My community college doesn't offer linear algebra anymore, because apparently nobody ever signs up for it. That's concerned me, because one of the first courses I'll be taking after transferring will likely be a 400 level abstract linear algebra course.

This book sounds fantastic! Thanks for the recommendation!

 

[micromass]

That sounds like a really good one. I like the fact that it mixes in some introductory Linear Algebra as well. That's a topic that I'm planning on doing some self study in after I've gotten a little calculus under my belt. My community college doesn't offer linear algebra anymore, because apparently nobody ever signs up for it. That's concerned me, because one of the first courses I'll be taking after transferring will likely be a 400 level abstract linear algebra course.

This book sounds fantastic! Thanks for the recommendation!

If you're comfortable with basic matrix arithmetics (such as multiplying matrices, determinants, finding inverses, solving systems of equations, etc.), then an abstract linear algebra course should be doable. Still, self-studying it is never a bad idea.

Lang has two linear algebra books that are very good. So you can pick those up as well.

 

[QuantumCurt]

If you're comfortable with basic matrix arithmetics (such as multiplying matrices, determinants, finding inverses, solving systems of equations, etc.), then an abstract linear algebra course should be doable. Still, self-studying it is never a bad idea.

Lang has two linear algebra books that are very good. So you can pick those up as well.

After the College Algebra course I took spring semester, matrices and systems of equations are a cakewalk. My professor loved matrices and their applications. I like to self study new material when I'm on breaks between semesters though too, so I figure linear algebra will be a good choice. I've looked at the Lang books for linear algebra too, they sound really good.

Is it safe to say that basically all of the Lang books are a good choice? I'm planning on picking up Lang's basic mathematics book as well, because the reviews for it are glowing. I've read a lot of good things about it around here too. I figure it'll be good to have on hand to reinforce some of the basics.

 

[micromass]

After the College Algebra course I took spring semester, matrices and systems of equations are a cakewalk. My professor loved matrices and their applications. I like to self study new material when I'm on breaks between semesters though too, so I figure linear algebra will be a good choice. I've looked at the Lang books for linear algebra too, they sound really good.

Is it safe to say that basically all of the Lang books are a good choice? I'm planning on picking up Lang's basic mathematics book as well, because the reviews for it are glowing. I've read a lot of good things about it around here too. I figure it'll be good to have on hand to reinforce some of the basics.

Basic Mathematics is a very nice book. I'm sure you'll enjoy it a lot.

Not all of Lang's books are good though, but most of them are. His differential geometry text is simply awful, for example. I also don't like his algebra text, even though it's very popular in grad classes. His undergrad analysis text and complex analysis text are ok, but there are better texts.

 

[QuantumCurt]

Basic Mathematics is a very nice book. I'm sure you'll enjoy it a lot.

Not all of Lang's books are good though, but most of them are. His differential geometry text is simply awful, for example. I also don't like his algebra text, even though it's very popular in grad classes. His undergrad analysis text and complex analysis text are ok, but there are better texts.

I'll keep that in mind. Those are all classes that I'll be taking down the line, so I'll have to do some shopping around when I get to that point. I've still got a couple years before I have to start worrying about that though.

There are too many math books that I want. There are 4-5 that I'm wanting to get right now, aside from the required texts for my classes. My friends look at me like I'm a lunatic sometimes...lol

 

[micromass]

I'll keep that in mind. Those are all classes that I'll be taking down the line, so I'll have to do some shopping around when I get to that point. I've still got a couple years before I have to start worrying about that though.

There are too many math books that I want. There are 4-5 that I'm wanting to get right now, aside from the required texts for my classes. My friends look at me like I'm a lunatic sometimes...lol

I understand that. I'm addicted to buying math books. It's basically the only thing I spend money on except the basics such as food and stuff. Not many people understand it though.

 

[QuantumCurt]

I understand that. I'm addicted to buying math books. It's basically the only thing I spend money on except the basics such as food and stuff. Not many people understand it though.

I haven't really spent much money on anything aside from books lately. I also just got done buying all of my textbooks for this coming semester, which was incredibly painful, but I'm getting ready to order several supplementary math books too. My friends think I'm crazy...lol

Thanks for all the input. :)

 

[Deinosuchus735]

It's actually been kind of difficult funding this book online for me. What would you guys consider a reasonable price to get it?

 

[jbunniii]

Where do you live? Amazon in the US is selling used copies for as cheap as $9.94. The price for a new copy is $53.81, which is not an unusual price for a math book.

https://www.amazon.com/dp/0387962018/?tag=pfamazon01-20

 

[Deinosuchus735]

Ah I see. I actually tend to use more eBay than Amazon. Was just wondering about any other outlets

 

[jbunniii]

I see some copies as cheap as $3.34 here:

http://www.alibris.com/A-First-Course-in-Calculus-Serge-Lang/book/2337968