Calculus 3

Course Description

Calculus 3 extends the concepts of differentiation and integration to functions of several variables. This course explores multivariable calculus, including vector-valued functions, partial derivatives, multiple integrals, and vector calculus. Students will develop a deeper understanding of mathematical tools used in physics, engineering, economics, and other fields that require modeling in three dimensions and beyond.

Course Objectives

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

• Understand and visualize three-dimensional coordinate systems
• Analyze vector functions and their applications
• Compute partial derivatives and interpret their meaning
• Apply multivariable optimization techniques
• Evaluate multiple integrals in various coordinate systems
• Understand vector fields and their applications
• Apply theorems of vector calculus (Green's, Stokes', Divergence)
• Solve problems using mathematical software

Course Topics

• Three-dimensional space: Coordinates, vectors, dot product, cross product
• Vector-valued functions: Differentiation, integration, arc length, curvature
• Functions of several variables: Limits, continuity, partial derivatives
• Gradients and directional derivatives: Tangent planes, extrema, Lagrange multipliers
• Multiple integrals: Double integrals, triple integrals, change of variables
• Vector calculus: Line integrals, surface integrals, fundamental theorems
• Applications: Physical applications in fluid dynamics, electromagnetism, and optimization

Textbook

Instructor Information

• Name: Dr. Sarah Johnson
• Office: Mathematics Building, Room 304
• Email: sarah.johnson@university.edu
• Office Hours: Mondays and Wednesdays, 2:00pm - 4:00pm (via Zoom)

Class Schedule

• Days: Tuesdays and Thursdays
• Time: 10:00am - 11:30am
• Location: Virtual (via Zoom)
• Zoom Link: https://university.zoom.us/j/calculus3sp2021

Tentative Course Schedule

Week	Topic(s)	Textbook Sections
1	Three-dimensional coordinate systems and vectors	12.1-12.4
2	Dot and cross products, equations of lines and planes	12.5-12.6
3	Vector functions and calculus of vector-valued functions	13.1-13.3
4	Arc length and curvature, motion in space	13.4-13.5
5	Midterm 1 review and exam
6	Functions of several variables	14.1-14.2
7	Partial derivatives and tangent planes	14.3-14.4
8	The chain rule, directional derivatives, and gradient	14.5-14.6
9	Maximum and minimum values, Lagrange multipliers	14.7-14.8
10	Double integrals and applications	15.1-15.4
11	Midterm 2 review and exam
12	Triple integrals in different coordinates	15.5-15.9
13	Vector fields, line integrals, and fundamental theorem	16.1-16.3
14	Green's theorem, curl and divergence	16.4-16.5
15	Stokes' theorem and divergence theorem	16.6-16.8
16	Final exam review and exam

Grading Policy

• Homework: 20%
• Midterm Exams (2): 25% each (50% total)
• Final Exam: 25%
• Class Participation: 5%

Assignments and Exams

• Homework: Weekly assignments due on Sundays at 11:59pm via the online portal
• Midterm 1: February 25, 2021 (Chapter 12-13)
• Midterm 2: April 8, 2021 (Chapter 14-15)
• Final Exam: May 13, 2021, 10:00am-12:00pm (Comprehensive)

Course Policies

• Academic Integrity: All work submitted must be your own. Collaboration on homework is encouraged, but everyone must write up their own solutions.
• Attendance: While lectures will be recorded, regular attendance (either live or through recorded sessions) is essential for success in this course.
• Late Assignments: Late homework will be accepted with a 10% penalty per day late, up to 3 days.
• Make-up Exams: Make-up exams will only be given with prior approval or documented extenuating circumstances.
• Technology: Students are expected to have reliable internet access and a working device with a webcam for exams.

Resources

• Online Resources: Additional practice problems and video tutorials are available on the course website.
• Tutoring: Free tutoring is available through the Mathematics Learning Center (virtually during Spring 2021).
• Mathematical Software: Students will be introduced to Mathematica, MATLAB, or Python for visualization and calculation.
• Disability Services: Students who need accommodations should contact Disability Services and provide documentation to the instructor.

Learning Outcomes

Students completing Calculus 3 will demonstrate proficiency in:

• Analyzing functions of several variables graphically, numerically, and symbolically
• Applying concepts of limits, continuity, and differentiability to multivariable functions
• Solving constrained and unconstrained optimization problems
• Evaluating multiple integrals and understanding their applications
• Utilizing vector calculus theorems to solve problems in physics and engineering
• Communicating mathematical ideas effectively using appropriate notation and language

Prerequisite Knowledge

Students enrolling in this course should have successfully completed Calculus 2 (or equivalent) with a grade of C or better. Strong knowledge of differentiation, integration, and transcendental functions from Calculus 1 and 2 is essential for success in this course. Familiarity with basic vector concepts will be helpful but will be reviewed at the beginning of the course.

Success Tips for Calculus 3

• Stay current with reading and homework assignments
• Utilize office hours whenever concepts are unclear
• Form study groups to work through challenging problems
• Visualize three-dimensional concepts using physical models or software
• Practice with past exam questions to prepare for assessments
• Connect calculus concepts to real-world applications whenever possible
• Review Calculus 1 and 2 concepts regularly, especially integration techniques