Mathematics – Study Materials

Based On

MATH 2210 MATH 2210 - Multivariable Calculus and MATH 2250 MATH 2250 – Linear Algebra and Differential Equations .

The official textbooks for the three calculus classes are open access and freely available at the following links:

The official textbook for MATH 2250 is “Differential Equations and Linear Algebra” External Link Icon by Strang. This book is neither open access nor freely available. The author does have a series of videos, book samples, and solutions that are freely available.

Prerequisites

  • MATH 1210 (Calculus 1).
  • MATH 1220 (Calculus 2).

Format

The exam consists of 20+ questions. A passing score is at least 80%. Each problem is worth 2.5 points or 5 points. Problems that are worth 2.5 points are not graded with partial credit. Problems that are worth 5 points are graded with partial credit. Possible scores on such problems are 0 points (for minimal understanding), 2.5 points (for setting up the problem), and 5 points (for setup and execution). The reason for having many small problems is to help you. You can receive zero credit on 8 small problems or 4 large problems and still pass the exam.

The exam is paper and pencil only. No calculators, phones, watches, or other devices are allowed.

The exam contains a mix of conceptual and calculation problems. Most of the conceptual problems can be solved with a single sentence or equation. With a solid conceptual understanding, many of the calculation problems can be solved in 3-5 steps. In my experience as a teacher, and as the writer of the mathematics exam, it is common to see students filling a page with calculations when only a few lines are needed. For the most part, numbers are chosen to simplify your calculations.

If you find yourself filling a page with calculations, pause and think about whether such calculations are essential to answering the problem.

The formula sheet at the end of this document will be provided with the exam. I believe it contains all the formulas one would need to make a 100 on the exam. However, you and I may have internalized different topics, and you are welcome to also generate an additional formula sheet. This additional formula sheet is limited to front and back of an 8.5 x 11 in paper. The formulas may be handwritten, typed, or borrowed from other sources.

Recognize that the provided formula sheet does not have any information on differential equations and linear algebra. Items such as characteristic equations, eigenvalues/vectors, column space, null space, etc. are not on the formula sheet. I do not view these items as formulas to memorize but rather concepts to internalize. You may feel differently or be at a different level of internalization. If so, these would be items (among others) that you are welcome to include in your formula sheet.

Focused Study Materials

The exam covers all of MATH 2210 and MATH 2250, and it is best for you to study the associated books completely. Recognizing that you may be short on time or interested in short cutting the study process, the following selection of chapters is provided.

Multivariable Calculus

  • C3, Ch. 2, all sections
  • C3, Ch. 3, all sections
  • C3, Ch. 4, all sections
  • C3, Ch. 5, sections 5.1-5.3
  • C3, Ch. 6, all sections

Differential Equations

  • C2, Ch. 4, all sections
  • C3, Ch. 7, all sections
  • DELA, Ch. 2, all sections

Linear Algebra

  • C3, Ch. 2, all sections
  • DELA, Ch. 4, all sections
  • DELA, Ch. 5, all sections
  • DELA, Ch. 6, all sections

This selection of chapters does not mean that concepts and problems from other chapters are not allowed. It is provided only to help if you do not have time or energy to study completely.

Advice

Unlike the other qualifying exams, the mathematics exam is required and includes topics from multiple courses. The reference calculus books each contain more than 800 pages, and the other book contains more than 400 pages. In total, the reference material contains about 3000 pages.

To study 3000 pages is monumental. The assumption, however, is that you have already internalized the most important concepts. This is particularly true of the concepts in Calculus 1 and 2, which are prerequisites for Calculus 3, Differential Equations, and Linear Algebra.

For example, the Calculus 1 book discusses trigonometric, exponential, and logarithmic functions; the definition of a derivative; the chain rule; maxima and minima; the definition of an integral; etc. These topics should already be internalized through years of practice.

If you struggled through Calculus 1 and 2 (or if it has been many years since taking those classes), then a monumental effort may be required.

It is important to know that memorization of “complicated” derivatives and integrals is not expected, nor is the use of derivative and integral tables. It is expected, however, that you know how to differentiate and integrate the most common functions: polynomials, exponentials, logarithms, and the standard trigonometric functions.

Once Calculus 1 and 2 are internalized, Calculus 3 is their extension in multiple dimensions. Some of these concepts are partial derivatives, the chain rule, maxima and minima; etc. These topics should already be internalized, too. Also introduced in Calculus 3 are more geometric ideas: vectors, dot products, cross products, tangent planes, directional derivatives, curvature, etc. Depending on your undergraduate emphasis or graduate interests, some of these may require you to memorize formulas, or for the exam, include them on your formula sheet.

With these topics behind you, the major concepts of Calculus 3 are line integrals, Green’s Theorem, Stokes’ Theorem, divergence, curl, etc. These major concepts bring together all of Calculus 1, 2, and 3 – to understand and compute successfully you must understand differentiation, integration, and vectors. Again, these major concepts may require you to memorize formulas, or for the exam, include them on your formula sheet.

In this respect, Chapter 6 of the Calculus 3 book might be the place to begin your study. If the topics of this chapter are easy for you, then you have probably internalized Calculus and are prepared for this aspect of the exam. If the topics of this chapter are difficult for you, then you probably need to spend considerable time studying Calculus 1, 2, and 3.

Differential equations are discussed in Chapter 4 of the Calculus 2 book and Chapter 7 of the Calculus 3 book. Differential equations on the exam are limited to first and secondorder linear ones. These two chapters contain almost all the important concepts: direction fields, stability, characteristic equations, solution techniques for nonhomogeneous equations, etc. Absent from these two chapters are Laplace transforms and their use in solving differential equations. If you do not know about Laplace transforms, then you need to find a different source such as the official text by Strang. The requisite Laplace transform formulas are given on the provided formula sheet.

Linear algebra is not discussed in the Calculus books. You need to find a different source such as Chapters 4-7 of the official text by Strang. There are many books on the subject, however you must take care to find a book that balances well theory and computation. One such book is “Linear Algebra and Its Application” by Lay. A more theoretical book is “Linear Algebra Done Right” by Axler. A more computational book is “Introduction to Applied Linear Algebra” by Boyd.

Some linear algebraic concepts such as matrix multiplication and matrix determinants should already be internalized. Chapter 5 of the book by Strang contains the most important linear algebraic concepts: column space, null space, solution of linear equations, independence, basis, dimension, subspaces, etc. Eigenvalues and eigenvectors are also of fundamental importance