Dave4Math / Courses / Vector Analysis

Vector Analysis

Master vector analysis. With this course, you'll become a pro with vector fields, line integrals, Green's Theorem, surface integrals, the Divergence Theorem, and Stokes' Theorem.


January 11, 2021

Created by Dave

This course begins Summer 2021.
Current Status
Not Enrolled
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A famous mathematician once said, “The art of doing mathematics consists in finding that special case which contains all the germs of generality.” In other words, examples matter. Introducing Dave’s online math course: Vector Analysis. Not only will you see the exciting conclusion to calculus 1, 2, and 3, but with Dave’s teaching, you’ll better understand how examples make a difference.

Who This Course is For

  • Students in college who want to learn more mathematics.
  • Anyone interested in vector analysis.
  • Students who are enrolled in calculus 3 and want to improve their grade.
  • Anyone interested in getting prepared for linear algebra.
  • Anyone wanting to learn about the divergence and curl of a vector field, line integrals and surface integrals, Green’s Theorem, the Divergence Theorem, and Stokes’ Theorem.


There is no required textbook, though you will need an up-to-date web browser, paper, and pen. A hand-held scientific calculator of your choice is required.

What You’ll Learn in Vector Analysis

  • Vector Field in Two-Dimensional Space
  • Vector Field in Three-Dimensional Space
  • Conservative Vector Fields
  • Divergence
  • Curl
  • Line Integrals
  • Line Integrals w.r.t Coordinate Variables
  • Line Integrals in Space
  • Line Integrals of Vector Fields
  • Work Done on a Particle by a GF
  • Fundamental Theorem for Line Integrals
  • Line Integrals Along Closed Paths
  • Independence of Path and Conservative Vector Fields
  • Determining Whether a Vector Field Is Conservative
  • Finding a Potential Function
  • Conservation of Energy
  • Green’s Theorem for Simple Regions
  • Green’s Theorem for More General Regions
  • Why We Use Parametric Surfaces
  • Finding Parametric Representations of Surfaces
  • Tangent Planes to Parametric Surfaces
  • Area of a Parametric Surface
  • Surface Integrals of Scalar Fields
  • Oriented Surfaces
  • Surface Integrals of Vector Fields
  • Parametric Surfaces
  • The Divergence Theorem
  • Interpretation of Divergence
  • Understanding Stokes’ Theorem
  • Interpretation of Curl
  • Summary of Line and Surface Integrals
  • Summary of Major Theorems Involving Line Integrals and Surface Integrals

Course Description

This course is the exciting conclusion to the calculus 1, calculus 2, and calculus 3 subjects. Here we bring together limits and continuity and the theories of differentiation and integration into some fantastic applications.

We begin moving from vector functions of one variable to vector functions of several variables, which we call vector fields. We study different types of vector fields and operations on them, including the divergence, curl, and gradient of a vector field.

Next, we have a detailed examination of line integrals and then surface integrals. In both cases, we motivate the underlying concepts and work through many theorems and examples. In particular, we derive the Law of Conservation of Energy.

After a detailed study of parametric surfaces, we study surface integrals of vector fields. In the end, we provide an overwhelming and thorough summary of the beauty that lies in the Fundamental Theorem of Line Integrals, Green’s Theorem, Divergence Theorem, and Stoke’s Theorem. Finally, we elaborate on and illustrate Maxwell’s equations for electromagnetism.

Recommended Prerequisites for Vector Analysis

I recommend the prerequisite course Multiple Integrals. If you’re not sure if this course is for you, checkout the course contents below or find out more by checking out my free calculus 3 articles.

Course Content

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David A. Smith

Mathematics Educator

I sincerely believe that the potential in every student can be unlocked. As an accomplished and dedicated instructor, I have firm confidence that I can provide significant value to your studies.

I relish teaching and use it as a channel for my creativity as I seek to deliver real-life and relevant connections between the mathematics that I teach and my students. Promoting an active learning environment and excitement about mathematics is my dream come true.

David Smith (Dave) has a B.S. and M.S. in Mathematics and has enjoyed teaching precalculus, calculus, linear algebra, and number theory at both the junior college and university levels for over 20 years. David is the founder and CEO of Dave4Math.