# Multivariable Calculus

## Course Overview

In this course, students learn to differentiate and integrate functions of several variables. They extend the Fundamental Theorem of Calculus to multiple dimensions and the course culminates in Green’s, Stokes’, and Gauss’ Theorems.

The course opens with a unit on vectors, which introduces students to this critical component of advanced calculus. They then move on to study partial derivatives, double and triple integrals, and vector calculus in both two and three dimensions. Students are expected to develop fluency with vector and matrix operations.

Understanding parametric curves as a trajectory described by a position vector is an essential concept, which allows us to break free from one-dimensional calculus and investigate paths, velocities, and other applications of science that exist in three-dimensional space. Students study derivatives in multiple dimensions and use the ideas of the gradient and partial derivatives to explore optimization problems with multiple variables as well as consider constrained optimization problems using Lagrangians.

After studying differentials in multiple dimensions, the course moves to integral calculus. Students use line and surface integrals to calculate physical quantities especially relevant to mechanics, electricity, and magnetism, such as work and flux. They employ volume integrals for calculations of mass and moments of inertia and conclude with the major theorems (Green’s, Stokes’, Gauss’) of the course, applying each to some physical applications that commonly appear in calculus-based physics.

Prerequisite: The equivalent of a college year of single-variable calculus, including integration techniques, such as trigonometric substitution, integration by parts, and partial fractions. Completion of the AP Calculus BC curriculum with a score of 4 or 5 on the AP Exam would be considered adequate preparation.

NCAA-approved course

UC-approved course

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