Many people on this site refer to undergraduate courses called "Calculus I" or "Calculus II".

In my country (Australia) there is no standard naming convention for university maths courses, and courses with the same name at different universities have different topics. For example, at our University the first-year courses are called "Mathematics 1A" and "Mathematics 1B" and contain both linear algebra and calculus.

Is there a standard list of topics in courses with these names in the USA, and if so, what are these lists of topics?

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    $\begingroup$ There's no standard list, but you can get an idea by googling an assortment of words together, such as: "topics" "covered" "mathematics" "calculus" "linear algebra" "differential equations" "precalculus". (The inclusion of "precalculus" should give you mostly U.S. math department hits.) $\endgroup$ Jul 8 '14 at 21:09
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    $\begingroup$ Interesting, then, that people refer to "Calculus I" as if everyone knows what's in it. $\endgroup$ Jul 8 '14 at 23:04
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    $\begingroup$ When I meant there is no standard list, I meant that there is no official or governmental type standard (or even state standards). However, that said, the content of calculus 1 and calculus 2 is fairly uniform in the U.S. across community colleges, 4-year colleges (small and medium size, public and private), and universities (public and private). The only exceptions tend to be in converting from quarter systems to semester systems, and a handful of universities (Caltech, MIT, etc.) whose first year college calculus is understandably quite different from the usual. $\endgroup$ Jul 9 '14 at 13:37
  • $\begingroup$ On the other hand, linear algebra is less uniform and differential equations even more so. Precalculus also -- usually it includes trigonometry, but in some cases assumes trigonometry (skipping elementary trig., but probably still covering polar coordinates and De Moivre's formula and inverse trig functions). Matt F.'s answer pretty much fits the standard topics for semester-based calculus. The end of Calculus I usually consists of simple antiderivatives (e.g. $u$-substitution, but not parts or trig substitutions), calculating simple areas, and the fundamental theorem of calculus. $\endgroup$ Jul 9 '14 at 13:42

You can get a good idea of American standards by looking at the most popular calculus textbooks.

An Amazon search for "calculus textbook" suggests Larson & Edwards and Stewart as the most popular. They largely agree on the chapter titles and their ordering. If we divide each book into three parts, the agreement between parts is nearly complete. Here's what we get, which corresponds to my experience of these courses:

Calculus I

  • Preparation for Calculus
  • Limits
  • Derivatives
  • Applications of differentiation
  • Integrals

Calculus II

  • Applications of integration
  • Transcendental functions
  • Techniques of integration
  • Differential equations
  • Polar coordinates and parametric equations
  • Infinite sequences and series

Calculus III

  • Vectors and the geometry of space
  • Vector functions
  • Partial derivatives
  • Multiple integrals
  • Vector analysis
  • $\begingroup$ Thanks. interestingly, Calculus II and partial derivatives correspond roughly to the calculus content of Maths 1A and Maths 1B here at my uni. $\endgroup$ Jul 8 '14 at 23:07
  • $\begingroup$ fwiw, this answer represents my meaning for Calculus I, II and III. $\endgroup$ Jul 9 '14 at 3:27
  • $\begingroup$ I'd like to add that for AP (Advanced Placement) courses, "Calc AB" covers Calc I in one year for high school students, and "Calc BC" covers both Calc I and Calc II in one year. $\endgroup$
    – Qaz
    Jul 11 '14 at 0:31

In the United States,

Calculus I typically covers differential calculus (in one variable), plus related topics such as limits.
Calculus II typically covers integral calculus in one variable.
Calculus III is the term for multivariate calculus, and is an introduction to vector calculus.
Advanced Calculus is advanced vector calculus, usually incorporates elements of real analysis, and includes topics in differentiation such as the implicit and inverse function theorems, and theorems of comparable importance in integration.
Linear Algebra is considered a separate topic from calculus, and is mostly taught separately. Many calculus books will have one or two chapters of linear algebra (mostly matrix theory), to "support" their teaching of calculus.

  • $\begingroup$ I wish I saw one or two chapters of matrix theory in the popular calculus texts I've seen. I am genuinely curious, to which texts do you refer? $\endgroup$ Jul 11 '14 at 1:35
  • $\begingroup$ Examples would be Lang's "Calculus of Several Variables" or Kaplan's "Advanced Calculus" $\endgroup$
    – Tom Au
    Jul 11 '14 at 1:40
  • $\begingroup$ Ok +1, I do understand the comment if you throw in advanced calculus texts, but the mainline texts, I think I've only seen it in Gilbert Strang's text. Although, I do not yet possess Spivak, I wouldn't be suprised to see it there. $\endgroup$ Jul 11 '14 at 2:03

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