Unit 4: Vector Calculus

This unit covers the theory, techniques, and applications of vector calculus. Vector calculus studies vector fields -- vectors that involve functions defined throughout the plane or space -- and their calculus, which includes integrating and differentiating vector fields. These operations give us information about how vector fields change and how they accumulate. Vector fields are ubiquitous in science, so we will discuss plenty of applications of vector fields throughout this unit.

We begin in Lesson 13 by introducing the notion of a line integral, which generalizes the definite integral to handle integration over a curve and not just an interval [a,b] along an axis.

Lesson 14 extends the perspective introduced at the end of Lesson 13 -- which views line integrals in terms of vector-valued functions -- to surfaces and introduces two-variable vector-valued functions, shows how to parametrize surfaces using them, and how to calculate tangent planes to surfaces and surface areas using two-variable vector-valued functions.

Lesson 15 generalizes the surface area work in Lesson 14 to help us integrate functions defined over a surface. We then pivot to studying vector fields, one of the foundational concepts in vector calculus. We close the lesson with a brief study of three important types of vector fields and vector field measures: gradient fields, the curl of a vector field, and the divergence of a vector field.

Lesson 16 is a capstone lesson that brings together almost every concept discussed in the course. After extending the scalar line integral content to the setting of vector fields -- yielding what we call vector line integrals, in the sense that they are line integrals whose integrands depend on vector fields -- we then discuss the circulation and flux of vector fields, and Green's Theorems. These theorems relate those new concepts to the curl and divergence, respectively, of a vector field, and also relate line integrals to double integrals.

Much of the content of this unit involves multiple integration, where the integrands do or do not involve vector functions. As such, this unit is capstone-like in that we will be drawing on the main topics discussed in the first three units -- vectors, multivariable differentiable, and multiple integrals -- and combining them in new ways. And, as we combine those ingredients to generate new concepts, we will also generate new insights and interpretations into those old concepts. This unit, therefore, both introduces new concepts and provides new takes on old concepts.