# Calculus III

## Course Overview

Calculus 3 is a comprehensive Calculus course designed to cover the Calculus of Multivariable and Vector Calculus. Because the sheer volume of material is so great, the course is designed to take an entire school year and is intended for students with a proven record of accomplishment and interest in mathematics. Specific topics covered at the beginning of the course are included below. Students begin the course by covering vector operators such as dot and cross products, as well as sketching common quadric surfaces. These are considered the “mother graphs” of many of the surfaces we use later in the course. Students then move on to vector values functions, space curves and derivatives and integrals of vector valued functions. The course then proceeds to what could be considered the Differential Calculus of Multivariable Functions. Students explore the ideas of Limit, Continuity, Partial Derivatives, Tangent Planes, Chain Rule. After learning these fundamentals, students Use Directional Derivatives and Gradients to optimize functions. Lastly, students learn Lagrange Multipliers as a potentially easier and more elegant method for finding maxima and minima. Next, students move on to the Integral Calculus of Multivariable Functions. Students learn Double and Triple Integrals, as well as method such as integrating with Spherical, Cylindrical, and Polar Coordinates. Lastly, Surface Area of solids is covered.

Students then move on to the concepts of Vector Calculus. They are introduced to Vector Fields, Line Integrals and the Fundamental Theorem of Line Integrals. More advanced topics including Green’s Theorem, Curl and Divergence, Parametric Surfaces, Surface Integrals, Stokes’ Theorem, and the Divergence Theorem conclude the course.

This course is designed to expose students to the more advanced topics of Calculus, and provide them with a strong computational and theoretical understanding of content that will be used subsequently in both advanced mathematical physics courses.

## Course Content

### Unit 1: Vectors and the Geometry of Space

Because understanding spacial relationships of points, lines, planes and functions of multiple variables is crucial to understanding vector and multivariable calculus, this course takes the time to review Trigonometric and Geometric concepts, including Vectors. Students need to be proficient at defining lines, planes and functions in space. They must also be proficient with dot and cross products of vectors. The following topics are covered in Unit 1:

• Three Dimensional Coordinate System
• Vectors
• The Dot Product
• The Cross Product
• Equations of Lines and Planes

### Unit 2: Vector Functions

This unit provides students with the skills to take the derivatives, integrals, and sketch the behavior of vector valued functions. It is a generalization from Cartesian functions, but using vectors, instead, to define curves in space. The following topics are covered in Unit 2:

• Vector Functions and Space Curves
• Derivatives and Integrals of Vector Functions
• Arc Length and Curvature
• Motion in Space: Velocity and Acceleration

### Unit 3: Partial Derivatives

Students learn to take derivatives so they can apply them to solve real-world and theoretical problems. General extrapolation of derivatives into higher dimensions requires the use of Partial Derivatives, Directional Derivatives and Gradients. The unit covers the first part of the chapter. This unit covers the following topics:

• Functions of Several Variables
• Limits and Continuity
• Tangent Planes and Linear Approximation
• Partial Derivatives

### Unit 4: More Partial Derivatives and an Introduction to Multiple Integrals

This unit simply concludes the material from Partial Derivatives, as well as introduces students to Multiple Integrals. This unit covers the following topics:

• Directional Derivatives and the Gradient Vector
• Maximum and Minimum Values of Multivariable Functions
• Lagrange Multipliers
• Double Integrals over Rectangles
• Iterated Integrals

### Unit 5: Multiple Integrals

This unit picks up where Unit 4 left off, exploring additional techniques of integrating in 3-space, as well as surface areas of multivariable functions. This unit covers the following topics:

• Double Integrals over General Regions
• Double Integrals in Polar Coordinates
• Applications of Double Integrals
• Surface Area

### Unit 6: More Multiple Integrals

This unit concentrates on more advanced techniques of multiple integration. This unit covers the following topics:

• Triple Integrals
• Triple Integrals in Cylindrical Coordinates
• Triple Integrals in Spherical Coordinates

### Unit 7: Vector Calculus

This unit rounds out calculus of two-dimensional spaces, introducing students to using calculus on less traditional functions such as parametric and polar functions. The unit also includes and introduction to differential equations. The following topics are included in the unit:

• Vector Fields
• Line Integrals
• The Fundamental Theorem of Line Integrals
• Green’s Theorem
• Curl and Divergence
• Parametric Surfaces and Their Areas

### Unit 8: More Vector Calculus

This is the final unit in the course and can be the most challenging for students, as it is probably the most theoretical. The unit explores all aspects of convergence and divergence of sequences and series, and concludes with writing functions as power series using Taylor and Maclaurin polynomials. The unit covers the following topics:

• Surface Integrals
• Stokes’ Theorem
• The Divergence Theorem