# Caltech Ma 1 abc Calculus and Linear Algebra: Topics and Practice

> A study guide for Caltech Ma 1 abc: calculus review, series, linear algebra, multiple integrals and the theorems of Green and Stokes, with free practice.

Source: https://practicecalculus.com/courses/caltech-ma-1abc
Updated: 2026-11-08

## At a glance

- **Course code:** Ma 1 abc (three terms: Ma 1 a, Ma 1 b, Ma 1 c)
- **Units:** 9 units (4-0-5) per term; the special section of Ma 1 a is 12 units (5-0-7)
- **Offered:** First, second and third terms (a full academic year)
- **Prerequisites:** High-school algebra, trigonometry and calculus
- **Covers:** Review of calculus, complex numbers, Taylor polynomials, infinite series, linear algebra, derivatives of vector functions, multiple integrals, line and path integrals, theorems of Green and Stokes
- **Next course:** Ma 2 (Differential Equations) and Ma 3 (Probability and Statistics) both list Ma 1 abc as the prerequisite
- **Practice here:** [Calculus 2](/learn/calculus-2) and [Calculus 3](/learn/calculus-3)

## Key takeaways

- Ma 1 abc is Caltech's year-long core mathematics course, taken over the first, second and third terms.
- It assumes high school calculus, reviews it, then covers complex numbers, Taylor polynomials, series, linear algebra and multivariable calculus.
- Ma 1 b and c split into an analytic track and a practical track; students get advice on choosing at the end of the fall term.
- Students who need more calculus take a special 12-unit section of Ma 1 a, followed by Ma 1 d on series.

**Caltech Ma 1 abc**, Calculus of One and Several Variables and Linear Algebra, is the core mathematics course every Caltech first-year student meets. It runs through the first, second and third terms and assumes you arrive with high school calculus. This guide explains what the catalog says it covers and links each calculus topic to free practice with instant feedback.

## What Ma 1 abc covers

The Caltech catalog describes Ma 1 abc as a review of calculus followed by complex numbers, Taylor polynomials and infinite series, a comprehensive presentation of linear algebra, derivatives of vector functions, multiple integrals, line and path integrals, and the theorems of Green and Stokes.

The catalog adds two structural notes. First, Ma 1 b and c are divided into two tracks, analytic and practical, and students get information to help them choose at the end of the fall term. Second, there is a special section of Ma 1 a for students who, because of their background, need more calculus than the regular sequence provides. That section is 12 units instead of 9, and its students take [Ma 1 d](/courses/caltech-ma-1d) on series in the second term.

The catalog does not split the topics term by term, so the order your instructor uses may differ. In terms of content, Ma 1 abc spans the series half of **Calculus 2** and most of **Calculus 3**, plus linear algebra, which this site does not cover.

## Ma 1 abc topic map

| Ma 1 abc topic | Practice | Read |
| --- | --- | --- |
| Review of calculus | [Chain rule in general](/learn/calculus-1/derivative-rules/chain-rule-general) | [Fundamental Theorem of Calculus](/blog/fundamental-theorem-of-calculus) |
| Taylor polynomials | [Taylor polynomials](/learn/calculus-2/power-series-parametric-polar/taylor-polynomials) | [Taylor's theorem](/theorems/taylors-theorem) |
| Infinite series | [Series and partial sums](/learn/calculus-2/sequences-and-series/series-and-partial-sums) | [Convergence tests](/blog/convergence-tests) |
| Power series | [Power series and radius](/learn/calculus-2/power-series-parametric-polar/power-series-and-radius) | [Power series](/blog/power-series) |
| Vectors (used throughout linear algebra) | [Dot product](/learn/calculus-3/vectors-and-space/dot-product) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Derivatives of vector functions | [Curves and motion](/learn/calculus-3/vectors-and-space/curves-and-motion) | [Partial derivatives](/blog/partial-derivatives) |
| Partial derivatives and the gradient | [Gradient and directional derivatives](/learn/calculus-3/partial-derivatives/gradient-and-directional-derivatives) | [Gradient guide](/blog/gradient-and-directional-derivatives) |
| Multiple integrals | [Iterated integrals](/learn/calculus-3/multiple-integrals/iterated-integrals) | [Double integrals](/blog/double-integrals) |
| Change of variables | [Change of variables](/learn/calculus-3/multiple-integrals/change-of-variables) | [Cylindrical and spherical](/learn/calculus-3/multiple-integrals/cylindrical-spherical) |
| Line and path integrals | [Line integrals of vector fields](/learn/calculus-3/vector-calculus/line-integrals-of-vector-fields) | [Conservative fields](/learn/calculus-3/vector-calculus/conservative-fields) |
| Green's theorem | [Green's theorem](/learn/calculus-3/vector-calculus/greens-theorem) | [Green's theorem](/theorems/greens-theorem) |
| Stokes' theorem | [Stokes and divergence theorems](/learn/calculus-3/vector-calculus/stokes-and-divergence-theorems) | [Stokes' theorem](/theorems/stokes-theorem) |

Complex numbers and linear algebra are also part of Ma 1 abc, but they are outside the calculus lessons on this site.

## The Integrated Core version

The current catalog also lists IC/Ma 1 abc, Integrated Core: Linear Algebra and Multivariable Calculus, offered at 6, 9 or 3 units across the three terms. It requires concurrent enrollment in the other Integrated Core classes and instructor permission, and it orders its topics around the applications that come up in the Integrated Core. Its listed content overlaps the multivariable part of Ma 1 abc: derivatives of vector functions, multiple integrals, line and surface integrals, and the theorems of Green and Stokes.

## Example problems at the Ma 1 abc level

**1. A Taylor polynomial.** The degree 2 Taylor polynomial of $\ln(1 + x)$ at $0$ is $x - \frac{x^2}{2}$.

**2. A convergent series.** $\sum_{n=1}^{\infty} \frac{n}{2^n}$ converges by the [ratio test](/rules/ratio-test), since $\frac{a_{n+1}}{a_n} = \frac{n+1}{2n} \to \frac{1}{2} < 1$.

**3. A line integral.** For $\mathbf{F} = \langle y, x \rangle$, a gradient field with potential $f = xy$, the work along any path from $(0, 0)$ to $(1, 2)$ is $f(1, 2) - f(0, 0) = 2$.

## How to do well in Ma 1 abc

- **Use the calculus review to find gaps.** The course moves quickly past single-variable calculus, so practice any shaky topic right away, especially [integration techniques](/learn/calculus-2/integration-techniques/choosing-a-strategy).
- **Treat series as a skill to drill.** Work through [choosing a test](/learn/calculus-2/sequences-and-series/choosing-a-test) until you can classify a series in seconds.
- **Link linear algebra and multivariable calculus.** Gradients, Jacobians and change of variables all rely on matrix ideas from the same course.
- **Practice the big theorems with examples.** Green's and Stokes' theorems stick once you have computed both sides of a few examples yourself.

## Frequently asked questions

### What is Caltech Ma 1?

Ma 1 abc, Calculus of One and Several Variables and Linear Algebra, is Caltech's core mathematics sequence. It runs through all three terms of the first year at 9 units per term.

### What are the prerequisites for Ma 1 abc?

The Caltech catalog lists high-school algebra, trigonometry and calculus. The course starts with a review of calculus rather than teaching it from scratch.

### What topics does Ma 1 abc cover?

The catalog lists a review of calculus, complex numbers, Taylor polynomials, infinite series, a comprehensive presentation of linear algebra, derivatives of vector functions, multiple integrals, line and path integrals, and the theorems of Green and Stokes.

### What are the analytic and practical tracks?

The catalog says Ma 1 b and c are divided into two tracks, analytic and practical. Students receive information to help them choose at the end of the fall term.

### What is the special section of Ma 1 a?

It is a 12-unit (5-0-7) section for students whose background means they need more calculus than the regular Ma 1 a provides. Those students take Ma 1 d, a short course on series, in the second term.