#### Single Variable Calculus | Multivariable Mathematics | Learning Proof Writing | More Info

### Single Variable Calculus

The 20-Series: This series can be started at any point in the sequence for those with sufficient background.

**Math 19- Calculus **(3 units) covers properties and applications of limits, continuous functions, and derivatives. Calculations involve trigonometric functions, exponentials, and logarithms, and applications include max/min problems and curve-sketching.

**Math 20- Calculus** (3 units) covers properties and applications of integration, including the Fundamental Theorem of Calculus and computations of volumes, areas, and arc length of parametric curves. An introduction to some basic notions related to differential equations (such as exponential growth/decay and separable equations) is also given.

**Math 21- Calculus** (4 units) covers limits at infinity in the context of integration of infinite sums, including convergence/divergence tests and power series. Taylor series and applications are also covered.

*Please Note: Math 41 and Math 42 have been discontinued and will no longer be offered.

### Courses in Multivariable Mathematics

The department offers 3 sequences in multivariable mathematics.

The 50-Series: See the detailed list of topics

**Math 51- Linear Algebra and Differential Calculus of Several Variables** (5 units) covers the basic geometry and algebra of vectors, matrices, linear transformations, and differential calculus of several variables.

**Math 52- Integral Calculus of Several Variables (5 units) ** covers integral calculus of several variables, and in particular vector analysis. This uses the linear algebra learned in Math 51.

**Math 53- Ordinary Differential Equations with Linear Algebra (5 units) ** integrates further topics in linear algebra with ordinary differential equations. These further topics include using eigenvalues and eigenvectors to solve systems of differential equations.

**Math 52 and Math 53 can be taken in either order.

This series provides the necessary mathematical background for majors in all disciplines, especially for the Natural Sciences, Mathematics, Mathematical and Computational Science, Economics, and Engineering.

#### For those with a strong interest in math and a preference for more conceptual and theoretical understanding we recommend the following two sequences:

The 60CM-Series

**Math 61CM-62CM-63CM- Modern Mathematics: Continuous Methods **(5 units each) This proof-oriented three-quarter sequence covers the material of 51, 52, 53, and additional advanced calculus, higher-dimensional geometry, and ordinary and partial differential equations. This provides a unified treatment of multivariable calculus, linear algebra, and differential equations with a different order of topics and emphasis from standard courses. Students should know single-variable calculus very well and have an interest in a theoretical approach to the subject.

This series provides the necessary mathematical background for majors in all disciplines, especially for the Natural Sciences, Mathematics, Mathematical and Computational Science, Economics, and Engineering.

The 60DM-Series

**Math 61DM-62DM-63DM- Modern Mathematics: Discrete Methods** (5 units each) This proof-oriented three-quarter sequence covers the same linear algebra material as the 60CM-series but focuses on topics in discrete math rather than on the “continuous” methods as in the 50-series and 60CM-series: it covers combinatorics, probability, some basic group theory, and graph theory. Some topological ideas are introduced in Math 63DM for the study of optimization problems and continuous-variable probability involving some basic multivariable calculus notions.

This series provides the necessary mathematical background for majors in Computer Science, Mathematics, Mathematical and Computational Science, and many other disciplines except that **if you plan to major in the Natural Sciences, Economics, or Engineering (outside of Computer Science) then the 60DM-series is not appropriate for you.**

### Learning Proof Writing

Many 100-level mathematics courses assume familiarity with writing proofs, and if you plan to be a Math major then you should learn proof writing as soon as possible. Here is a list of courses to begin learning proof-writing:

- 61CM or 61DM
- Math 104
- Math 110
- Math 113
- Math 115

Please note: Math 104 and Math 113 do not assume background beyond the linear algebra covered in Math 51.

### More Information

For more information about these courses see ExploreCourses for course descriptions and schedule. If you have further questions about which course to take, contact your academic advisor, or our Director of Undergraduate Studies.