# MA103: Multivariable Calculus

## Course Syllabus for "MA103: Multivariable Calculus"

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Multivariable Calculus is an expansion of Single-Variable Calculus in that it extends single variable calculus to higher dimensions.  You may find that these courses share many of the same basic concepts, and that Multivariable Calculus will simply extend your knowledge of functions to functions of several variables.  The transition from single variable relationships to many variable relationships is not as simple as it may seem; you will find that multi-variable functions, in some cases, will yield counter-intuitive results. The structure of this course very much resembles the structure of Single-Variable Calculus I and II.  We will begin by taking a fresh look at limits and continuity.  With functions of many variables, you can approach a limit from many different directions.  We will then move on to derivatives and the process by which we generalize them to higher dimensions.  Finally, we will look at multiple integrals, or integration over regions of space as opposed to intervals. The goal of Multivariable Calculus is to provide you with the tools you need to handle problems with several parameters and functions of several variables and to apply your knowledge of their behavior.  But a more important goal is to gain a geometrical understanding of what the tools and computations mean.

### Learning Outcomes

Upon successful completion of this course, the student will be able to:

• Define and identify vectors.
• Define and compute dot and cross-products.
• Solve problems involving the geometry of lines, curves, planes, and surfaces in space.
• Define and compute velocity and acceleration in space.
• Define and solve Kepler’s Second Law.
• Define and compute partial derivatives.
• Define and determine tangent planes and level curves.
• Define and compute least squares.
• Define and determine boundaries and infinity.
• Define and determine differentials and the directional derivative.
• Define and compute the gradient and the directional derivative.
• Define, determine, and apply Lagrange multipliers to solve problems.
• Define and compute partial differential equations.
• Define and evaluate double integrals.
• Use rectangular coordinates to solve problems in multivariable calculus.
• Use polar coordinates to solve problems in multivariable calculus.
• Use change of variables to evaluate integrals.
• Define and use vector fields and line integrals to solve problems in multivariable calculus.
• Define and verify conservative fields and path independence.
• Define and determine gradient fields and potential functions.
• Use Green’s Theorem to evaluate and solve problems in multivariable calculus.
• Define flux.
• Define and evaluate triple integrals.
• Define and use rectangular coordinates in space.
• Define and use cylindrical coordinates.
• Define and use spherical coordinates.
• Define and correctly manipulate vector fields in space.
• Evaluate surface integrals and relate them to flux.
• Use the Divergence Theorem (Gauss’ Theorem) to solve problems in multivariable calculus.
• Define and evaluate line integrals in space.
• Apply Stokes’ Theorem to solve problems in multivariable calculus.
• Properly apply Maxwell’s Equations to solve problems.

### Course Requirements

In order to take this course, you must:

√    Have a computer.

√    Have continuous broadband Internet access.

√    Have the ability/permission to install plug-ins or software (e.g., Adobe Reader or Flash).

√    Have the ability to download and save files and documents to a computer.

√    Have the ability to open Microsoft files and documents (.doc, .ppt., .xls, etc.).

√    Have competency in the English language.

√    Have read the Saylor Student Handbook.

√    Have completed the following courses from “The Core Program” of the mathematics discipline: MA101 Single-Variable Calculus I and MA102 Single-Variable Calculus II.

### Course Information

Welcome to MA103.  Below, please find general information on the course and its requirements.

Primary Resources: This course is comprised of the following primary materials:

Requirements for Completion: In order to complete this course, you will need to work through each unit and all of its assigned materials. You will also need to complete the following:

- Unit 1 Assignment: Exercise Sets
- Unit 2 Assignment: Exercise Sets
- Unit 3 Assignment: Exercise Sets
- Unit 4 Assignment: Exercise Sets
- The Final Exam

Note that you will only receive an official grade on your Final Exam.  However, in order to adequately prepare for this exam, you will need to work through the exercise sets listed above.  In order to “pass” this course, you will need to earn a 70% or higher on the Final Exam.  Your score on the exam will be tabulated as soon as you complete it.  If you do not pass the exam, you may take it again.

Time Commitment: This course should take you a total of 88.5 hours to complete.  This is only an approximation and may take longer to complete.  Each unit includes a “time advisory” that lists the amount of time you are expected to spend on each subunit.  These should help you plan your time accordingly.  It may be useful to take a look at these time advisories and to determine how much time you have over the next few weeks to complete each unit, and then to set goals for yourself.  For example, Unit 1 should take you about 22 hours to complete.  Perhaps you can sit down with your calendar and decide to complete subunit 1.1 (a total of 7 hours) over three days, for example by completing sub-subunit 1.1.1 (a total of 2 hours) on Monday; sub-subunit 1.1.2 (a total of 1 hour) and about half of sub-subunit 1.1.3 (about 2 hours) on Tuesday; the rest of sub-subunit 1.1.3 (about 2 hours) on Wednesday; etc.

Tips/Suggestions: As noted in the “Course Requirements,” Single-Variable Calculus is a pre-requisite for this course.  If you are struggling with the material as you progress through this course, consider taking a break to revisit MA101 Single Variable Calculus I and MA102 Single Variable Calculus II.  It will likely be helpful to have a graphing calculator on hand for this course.  If you do not own or have access to one, consider using this free graphing calculator.  As you read, take careful notes on a separate sheet of paper.  Mark down any important equations, formulas, and definitions that stand out to you.  It will be useful to use these notes as a review prior to completing the Final Exam.

### Preliminary Information

• Optional Resource: Monroe Community College's “Exploring Multivariable Calculus” Dynamic Visualization Project

Link: Monroe Community College's “Exploring Multivariable Calculus” (HTML and Java) - Dynamic Visualization Project

Instructions: Please note that this is an optional resource.  Click on “Multivariable Calculus Exploration Applet” link on the left side to upload the applet.  This is a dynamic visualization tool for multivariable calculus and can be used throughout the course, especially to visualize three-dimensional objects.