# Difference between Calculus I,II,III?

• Calculus
What is the difference between Calculus I,II,III? Which subjects distinguish them?

Which self-study books would you recommend for each of these?

(I already have Calculus by Spivak, but I don't know if that covers Calculus I,II or III)

gfd43tg
Gold Member
If you don't know whats in Calculus, then you are not ready for Spivak.

jtbell
Mentor
What is the difference between Calculus I,II,III?

It depends on the school. Go to a few university web sites and look up the course descriptions.

A common pattern is:

1 - limits and basic differentiation and integration
2 - more sophisticated integration techniques, and infinite series
3 - multivariable calculus a.k.a. vector calculus

• lonely_nucleus
If you don't know whats in Calculus, then you are not ready for Spivak.
OK, I get it. Thanks.

The book I am currently reading is Engineering Mathematics by K.A.Stroud. I haven't seen this one mentioned in the Physics Forums yet so I'm not sure if it is good or not. What I am trying to figure out is whether it includes all the calculus I will need. There is a follow up book called Advanced Engineering Mathematics by Stroud that I haven't bought and I wondered if that one covered Calculus III.

Note that I am self studying and my previous knowledge is roughly A-Level standard (UK) but it was a long time ago when I last studied.

Pretty much any introductory (read: 1000 page long) calculus text will cover the material presented in a three or four term calculus sequence. For example, my school uses Stewart's book for four terms. The difference is often professorial preference and how expensive the book the is. Things like Spivak are meant as "advanced" textbooks that one uses to extend all these things to more complicated mathematical structures (the manifold in Spivak's case).

Think of it like learning algebra before calculus, or arithmetic before algebra, except this time the gross subject matter is still calculus.

jtbell
Mentor
Engineering Mathematics by K.A.Stroud

A Google search turned up the publisher's web page for this book, which includes a table of contents. I see some Calculus III topics (partial differentiation and multiple integrals), but this apparently does not include vector calculus (gradient, divergence, curl, line integrals, divergence theorem, Stokes's theorem).

According to the publisher's page, this includes "Vector Analysis, Parts 1, 2 & 3" which I suspect covers the vector calculus topics I listed above, but of course I can't say for sure.

QuantumCurt
If you don't know whats in Calculus, then you are not ready for Spivak.

^^This.

Spivak is more accurately called an introduction to real analysis. It's very rigorous, and very abstract. I've completed calc I, calc II, differential equations, and linear algebra. Next semester is calc III. I've made my way through some of Spivak, and it is a challenging book.

As jtbell pointed out, calc I is all about limits, derivatives, and applications of derivatives (graphing, optimization, related rates etc). The basic forms of integration are introduced at the end of calc I. Calc II is all about different methods of integration, and typically ends with infinite series. I found calc I to be a bit more conceptually difficult. Some of the notions that are introduced, such as the Epsilon-Delta definition of a limit are a bit abstract and a little hard to wrap your head around at first. You also see the more formal definitions of a derivative and an integral, which can be complicated and time consuming sometimes. Calc II was, in my opinion, more difficult. It wasn't as conceptually difficult, but it's much more technically involving. There are a TON of methods of integration involved, and they can be easily confused and quite complicated.

• Franco_Carr14
From my experience:
Calculus I: limits, differentiation and integration
Calculus II: Integration and it's applications-volume, surface area, arc length, etc.
Calculus III: Infinite sequences a series, power series, taylor series, vector spaces
Calcuslus IV: More learning about vector spaces, multivariable calculus, introduction to differential equations and more
I would also like to recommend getting the Schaum's outline for Calculus.

• lonely_nucleus
From my experience:
Calculus I: limits, differentiation and integration
Calculus II: Integration and it's applications-volume, surface area, arc length, etc.
Calculus III: Infinite sequences a series, power series, taylor series, vector spaces
Calcuslus IV: More learning about vector spaces, multivariable calculus, introduction to differential equations and more
I would also like to recommend getting the Schaum's outline for Calculus.

However, the Schaum's outline of calculus is dumbed down approach of conveying basic Calculus. I recommend Thomas Calculus with Analytic Geometry 3rd edition used in conjunction with an old version of Stewart Calculus.

What is the difference between Calculus I,II,III? Which subjects distinguish them?

Calculus 1 will be focused on the basic concepts of calculus. Limits, differentiation, and integration. Also some applications like linear approximations and optimization problems.

I would say Calculus 2 is both the direct continuation of that into more advanced techniques for those things but also the point where the concepts of differential equations are introduced. The main conceptual idea is that calculus problems are actually about finding solutions to a differential equation, and some basic methods for solving simple ones. You'll also meet sequences and series. It's also, from my experience, one of the big weed-out classes for engineering majors.

Calculus 3 is multivariable calculus, which extends the methods you learned in the first two semesters to working with vectors, parameterizations, and equations of more than one independent variable.

Sometimes differential equations (may or may not be combined with linear algebra) is called "calculus 4", but since there may be an "elementary" or "applied" version of these courses meant for engineering and physics majors taken after calculus 2 this is not always the case. Linear algebra covers the topics of vectors, vector spaces, and solving linear systems of equations. Differential equations covers methods of dealing with families of equations related by their derivative.

Spivak is a real analysis book, not an introductory calculus book. Real analysis is much more formal than what you need right now. I would go with Stewart's calculus book. It's what I learned with, and there's plenty of old editions floating around that are dirt cheap.

However, the Schaum's outline of calculus is dumbed down approach of conveying basic Calculus. I recommend Thomas Calculus with Analytic Geometry 3rd edition used in conjunction with an old version of Stewart Calculus.

Thanks to everyone who answered my questions about the difference between Calculus I,II.III.

I have in fact ordered both of the books MidgetDwarf has recommended (second hand, older editions to keep the cost down. They haven't arrived yet though).

Can I take it that these two books will cover all the calculus I will need to eventually study subjects like Classical Mechanics, Quantum Mechanics, Electrodynamics and Relativity? i.e. do they cover Calculus I,II, III & IV?

If there is indeed more calculus I need can you recommend books beyond Thomas / Stewart?

(I appreciate that I will probably need to study other maths subject like more Linear Algebra, but for now I want to ensure I have calculus properly studied.)

QuantumCurt
As convention, calculus is typically broken up into I, II, and III. Some schools on quarter systems and the like have a calculus IV, but it's less typical.

After the calculus sequence, the next courses you'll need for physics are differential equations and linear algebra. With Calc I-III, differential equations, and linear algebra, you'll be pretty well prepared for everything you'd encounter in introductory physics. For the most part you really only need Calc I-III though.

OK, I get it. Thanks.

The book I am currently reading is Engineering Mathematics by K.A.Stroud. I haven't seen this one mentioned in the Physics Forums yet so I'm not sure if it is good or not. What I am trying to figure out is whether it includes all the calculus I will need. There is a follow up book called Advanced Engineering Mathematics by Stroud that I haven't bought and I wondered if that one covered Calculus III.

Note that I am self studying and my previous knowledge is roughly A-Level standard (UK) but it was a long time ago when I last studied.
friend, @ZapperZ and @neosoul gave you a straightforward concise answer, if you want to show appreciation like their replies.

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SteamKing
Staff Emeritus
What Schaum's lacks in pedagogical sophistication, it makes up in providing a wealth of problems, some solved, on which the reader can practice to hone his skill at solving calculus problems. 