# Math 314: Calculus of Several Variables

## Contents

## Catalog Information

### Title

Calculus of Several Variables.

### (Credit Hours:Lecture Hours:Lab Hours)

(3:3:0)

### Offered

F, W, Sp, Su

### Prerequisite

Math 113; 313 or concurrent enrollment.

### Description

Partial differentiation, the Jacobian matrix, and integral theorems of vector calculus.

## Desired Learning Outcomes

This course is aimed at students majoring in mathematical and physical sciences, and engineering, and students minoring in mathematics or mathematical education. Calculus is the foundation for most of the mathematics studied at the university level. The mastery of calculus requires well-developed manipulative skills, clear conceptual understanding, and the ability to model phenomena in a variety of settings. Calculus of several variables extends the concepts of limit, integral, and derivative from one dimension to higher dimensional settings and is therefore fundamental for many fields of mathematics. This course contributes to all the expected learning outcomes of the Mathematics BS (see [[1]]).

### Prerequisites

Students are expected to have completed Math 113, and Math 313 or concurrent registration in Math 313.

### Minimal learning outcomes

Students should achieve mastery of the topics below. This means that they should know all relevant definitions, full statements of the major theorems, and examples of the various concepts. Further, students should be able to solve non-trivial problems related to these concepts, and prove simple theorems in analogy to proofs given by the instructor.

- Vectors and vector functions
- Compute vector operations
- Describe lines and planes in space
- Identify standard quadratic surfaces
- Describe parametric curves as vector functions
- Compute and interpret the derivative of a vector function
- Apply vector functions to curvilinear motion

- Derivatives of functions of several variables
- Compute and interpret partial derivatives
- Use the gradient to find directional derivatives and normal vectors
- Find local and global extrema
- Solve optimization problems involving several variables
- Use the method of Lagrange to find extrema under constraints

- Multiple integrals
- Know and apply Fubini’s theorem to express multiple integrals as iterated integrals
- Change the order of integration
- Transform integrals to polar, cylindrical, and spherical coordinates
- Use multiple integrals to find area, volume, mass, center of mass, and other applications
- Use the Jacobian of a coordinate change to transform integrals

- Line and surface integrals
- Evaluate line integrals
- Use line integrals to compute work and circulation
- Use surface integrals to compute surface area and flux
- Set up and use integrals over parametric surfaces

- Divergence and curl
- Explain and interpret divergence and curl
- Know and apply the divergence theorem
- Know and apply Stokes’ theorem
- Use the divergence and curl tests to describe properties of vector fields

### Additional topics

These are at the instructor's discretion as time allows; applications to physical problems are particularly helpful.

### Courses for which this course is prerequisite

This course is required for Math 303, Math 352, Math 410, Math 465, Math 480, Math 541, Math 543, and Math 547.