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Guidance on Math 51 and 52 following “Calculus III”

chalk board of math equations

Photo by Dan Cristian Pădureț on Unsplash

Introduction 

Many incoming Stanford students are unsure with which math course they should begin. If you have taken a course commonly called “Calculus III” in your high school or at another college or in some other format, this page is intended to provide you with some guidance. (If you did not take such a course or are considering taking either Math 61CM or Math 61DM, this document is not relevant to your plans.)

Calculus III as taught elsewhere typically covers topics such as: geometry of lines and planes in space, partial differentiation, the Chain Rule for functions of 2 or 3 variables, tests for maxima and minima for functions of 2 or 3 variables, and integration of functions of 2 or 3 variables. The latter usually includes part of what is commonly called vector calculus: line integrals, Green’s theorem, and Stokes’ theorem. You may have also taken a separate course on linear algebra (possibly on its own, or as part of a course on differential equations).

These topics are all covered in Math 51 and 52, but (for reasons explained below) the treatment of them here is substantially different than you are likely to have encountered elsewhere. Furthermore, our curriculum was carefully worked out in collaboration with many of the Stanford departments that have classes for which Math 51 or 52 are prerequisites.

This means that these departments will either be assuming, or at least be much happier, if you know this material in the way it is presented in these courses. That is what we discuss below, so you can determine if you should take or skip one or both of these courses, and if skipping then whether there are some background gaps you should fill in (and how to do so).

Math 51

This course is a somewhat unique blend of multivariable differential calculus and linear algebra, and presented with a view toward what is relevant to modern applications such as machine learning, natural sciences, “big data” of any type, and engineering. The aspect that makes it unlikely to be comparable to what you have seen elsewhere is that linear algebra is developed in “high dimensions”; i.e., vectors in Rⁿ, not just n = 2, 3.

The consideration of arbitrary n (possibly in the millions or billions) is what is needed in modern data science, large-scale engineering structures (e.g., buildings and circuits), and many other subjects. It is also the correct language in which to express multivariable differential calculus: the multivariable Chain Rule is most naturally expressed (and understood) using multiplication of derivative matrices, a concept rarely seen in treatments elsewhere yet also making the result more uniform (for any number of variables) and less mysterious.

In the Math 51 approach, multivariable differential calculus is formulated in a way that allows for a modern treatment of optimization for functions of n variables for arbitrary n (necessary for contemporary applications). The formulation of both the multivariable Chain Rule and multivariable second derivative test in treatments elsewhere tend to rely upon ad hoc features of 2 or 3 variables that do not work for n > 3.

If some or all of this sounds unfamiliar to you, that may be a good sign that you should seriously consider taking Math 51. Here is a question to help you make that decision:

Question: Have you taken a linear algebra course that includes the following topics: vectors in Rⁿ, dot products, orthogonality in Rⁿ, subspaces of Rⁿ, projection, linear independence, dimension of subspaces, matrices of arbitrary size and shape, matrix multiplication, matrix inversion, null spaces, column and row rank, eigenvalues, and eigenvectors? (It is important to have learned and be comfortable with this not just for vectors in R² and R³!)

If your answer to this is resolutely “yes”, and you took a “Calculus III” course elsewhere too, then you can probably comfortably skip Math 51, with a crucial caveat (see below). If not, then it is likely that you will learn a lot of useful and necessary material in Math 51.

What is the caveat? If you learned Calculus III and linear algebra separately somewhere else then it is highly likely that you may not be conversant with the important and deep connections between these subjects. If that is the case, then you can still consider enrolling in Math 51 to reinforce your understanding and solidify your knowledge of how these subjects interact with one another, or else skip Math 51 and teach yourself by reading in the Math 51 book (and doing exercises in the relevant chapters) about the following:

  1. (i) Derivative matrices: Section 13.5 and exercises that mention derivative matrices.
  2. (ii) Relationship between matrices and linear transformations R R: Chapter 14.
  3. (iii) Multivariable Chain Rule via derivative matrices for functions of any number of variables: Chapter 17.
  4. (iv) Matrix factorizations (LU and QR), a very important topic used to solve systems of linear equations efficiently, and useful for many other things too: Chapter 22.
  5. (v) Multivariable 2nd derivative test for functions of any number of variables by applying the Spectral Theorem to the symmetric Hessian: Chapters 25 and 26 (and Chapter 24 if the Spectral Theorem and quadratic forms were not in your linear algebra course).

Please note that items (i)–(iii) are needed in Math 53 and in further math and its applications. If all you are missing is (iv), then you should read Chapter 22 and not take Math 51.

Math 52

This is a standard course on multivariable integration and vector calculus. Math 52 follows Chapters 5 and 6 of Volume 3 of the OpenStax calculus books. Math 52 is not a prerequisite to Math 53 (on differential equations).

The vector calculus material in Math 52 (line integrals, surface integrals, Stokes’ theorem) is covered in Chapter 6 of the OpenStax book. It is of greatest relevance if you intend to go on to study physics, math, or certain physics-oriented engineering subjects (e.g., mechanical or aerospace engineering). So if any of those subjects are important for your future, you should only consider skipping Math 52 if you are comfortable with using Green’s and Stokes’ theorems; otherwise do not skip Math 52.

In some fields that don’t use vector calculus, Math 52 can be relevant for the first part of the course: integrating functions over regions in Rⁿ and R³ such as boxes, balls, and other domains, as well as some familiarity with the Change of Variables formula (e.g., computing integrals using spherical or cylindrical coordinates). This material is covered in Chapter 5 of the OpenStax book. In addition to its role in vector calculus, this is important in probability (e.g., relations among random variables) as well further topics in math and applied math. If these skills are relevant to your future and you don’t plan to pursue a field that uses vector calculus, only skip Math 52 if you are comfortable with the content of Chapter 5 of the OpenStax book linked above (and see Section E.6 in the Math 51 book for the connection of Change of Variables to derivative matrices).