Math 20C: Multivariable Calculus Syllabus and Course Structure

Course Overview

This syllabus outlines the structure, policies, and expectations for Math 20C: Multivariable Calculus at UC San Diego. The course covers key topics in vector geometry, vector-valued functions, partial differentiation, optimization, and double integration, providing a foundation for advanced study in calculus and related fields.

Course Content and Topics

• Vector Geometry: Introduction to vectors in two and three dimensions, operations, and geometric interpretations.

• Vector-Valued Functions and Their Derivatives: Study of functions whose outputs are vectors, including differentiation and applications to motion in space.

• Partial Differentiation: Techniques for differentiating functions of several variables, including higher-order partial derivatives.

• Maxima and Minima: Methods for finding local and global extrema of multivariable functions, including the use of the second derivative test and Lagrange multipliers.

• Double Integration: Evaluation of integrals over two-dimensional regions, including applications to area and volume.

Additional info: The course calendar references textbook sections 12.1–15.2, which typically correspond to chapters on vectors, vector-valued functions, partial derivatives, multiple integrals, and applications in standard calculus textbooks.

Prerequisites

• AP Calculus BC score of 4 or 5, or completion of MATH 20B with a grade of C– or better.

Textbook

• Calculus Early Transcendentals: Multivariable with Achieve, Rogawski, 4th edition.

Course Structure and Grading

• Homework: Two types—graded online assignments (via Achieve) and ungraded textbook problems. The lowest three online homework scores are dropped.

• Exams: Two midterms (best score counted) and one comprehensive final exam. No make-up exams except in extreme cases.

• Grading Scheme:

  • Homework: 20%

  • Best Midterm: 35%

  • Final Exam: 45%

Letter Grade Scale

Letter Grade	Score Range (%)
A+	99–100
A	93–99
A-	90–93
B+	87–90
B	83–87
B-	80–83
C+	77–80
C	73–77
C-	70–73
F	0–70

Course Calendar (Topics by Week)

Week	Monday	Tuesday	Wednesday	Thursday	Friday
1	Intro & 12.1	12.1 & 12.2	12.2 & 12.3	Check Dates
2	12.3	12.4	HW Due	12.5
3	13.1	13.2	HW Due	13.2 & 13.3
4	13.5	Review	HW Due	Midterm 1	14.1
5	14.2	14.2	HW Due	14.3
6	14.3 & 14.4	14.4	HW Due	14.5
7	14.5	14.6	HW Due	14.7
8	14.7	Review	HW Due	Midterm 2	14.8
9	Holiday	15.1	HW Due	15.1 & 15.2
10	15.2	Review	HW Due	Review	Final Exam

Additional info: Section numbers (12.1–15.2) correspond to standard multivariable calculus topics such as vectors, vector-valued functions, partial derivatives, multiple integrals, and their applications.

Course Policies and Academic Integrity

• Homework Collaboration: Discussion is allowed, but final write-up must be individual work.

• Academic Integrity: Strict adherence to UCSD's policy; violations include plagiarism, unauthorized collaboration, and use of prohibited aids.

• Makeup Policy: Makeups are rarely granted and only in extreme, documented circumstances.

• Regrades: Requests must be made within a short window after grades are posted on Gradescope.

• Equity and Inclusion: The course adheres to UCSD's Principles of Community, supporting diversity and respect.

Study Suggestions

• Allocate sufficient weekly study time (about 12 hours outside lecture for a 4-unit course).

• Start homework early and read textbook sections before lectures.

• Take notes by hand for better retention.

• Attempt problems before consulting solutions.

• Use mistakes as learning opportunities.

• Teach concepts to others to reinforce understanding.

Key Multivariable Calculus Concepts (Preview)

Vectors and Vector-Valued Functions

• Vector: An object with both magnitude and direction, represented in component form as $\vec{v} = \langle v_1, v_2, v_3 \rangle$.

• Vector-Valued Function: A function whose output is a vector, e.g., $\vec{r}(t) = \langle x(t), y(t), z(t) \rangle$.

• Derivative of a Vector-Valued Function: $\vec{r}'(t) = \langle x'(t), y'(t), z'(t) \rangle$.

• Application: Describes motion in space, where $\vec{r}(t)$ gives the position of a particle at time $t$.

Partial Derivatives

• Partial Derivative: The derivative of a multivariable function with respect to one variable, holding others constant.

• Notation: $\frac{\partial f}{\partial x}$, $\frac{\partial f}{\partial y}$, etc.

• Example: For $f(x, y) = x^2y + y^3$, $\frac{\partial f}{\partial x} = 2xy$ and $\frac{\partial f}{\partial y} = x^2 + 3y^2$.

Double Integrals

• Double Integral: Used to compute the volume under a surface over a region $R$ in the $xy$-plane.

• Notation: $\iint_R f(x, y) \, dA$

• Example: $\iint_R x^2 + y^2 \, dA$ over a circular region $R$.

Optimization (Maxima and Minima)

• Critical Point: A point where all first partial derivatives are zero or undefined.

• Second Derivative Test: Used to classify critical points as local maxima, minima, or saddle points.

• Lagrange Multipliers: Technique for constrained optimization.

Conclusion

This syllabus provides the framework for Math 20C, including course topics, policies, grading, and study strategies. Mastery of these foundational multivariable calculus concepts is essential for success in advanced mathematics, engineering, and the sciences.