Calculus III

Calculus III is an advanced multivariable calculus course at the University of Notre Dame that builds on the core ideas from Calculus I and II. It extends single-variable calculus into higher dimensions and introduces the main tools of vector calculus used throughout physics, engineering, and related fields.

Key Topics Include:

Vectors and Geometry in 3D: Dot and cross products, lines, planes, and quadratic surfaces.
Vector-Valued Functions and Motion: Parameterized curves, derivatives/integrals of vector-valued functions, and the T–N–B frame.
Multivariable Differentiation: Partial derivatives, the multivariable chain rule, directional derivatives, and local/absolute extrema.
Constrained Optimization: Lagrange multipliers and interpreting constraints geometrically.
Multiple Integration and Coordinate Systems: Double and triple integrals in rectangular coordinates, plus polar, cylindrical, and spherical coordinates.
Vector Fields and Integral Theorems: Line integrals, conservative fields and potential functions, curl/divergence, Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem.

The course is organized around 22 standards (topics) listed below.

last taught: fall 2023

PDF Summary		Standard 01. Dot and Cross Product An introduction to vector arithmetic and its applications.
PDF Summary		Standard 02. Lines An introduction to lines in 3-dimensional space and three ways to write the equation(s) of a line.
PDF Summary		Standard 03. Planes An introduction to planes in 3-dimension space and how planes interact with each other.
PDF Summary		Standard 04. Vector-valued Functions An extensive introduction to common quadratic surfaces and vector valued functions, in order to find the curve of intersection between two quadratic surfaces and parameterize the solution.
PDF Summary		Standard 05. Calculus for Curves An extension our of the single variable definitions of limits, derivatives, and integrals to vector-valued functions.
PDF Summary		Standard 06. T-N-B Frame A deep dive into an application of the derivative of vector-valued functions called the T-N-B frame.
PDF Summary		Standard 07. Partial Derivative An introduction to partial derivatives and an extension of the chain rule for multivariable calculus.
PDF Summary		Standard 08. Directional Derivative An introduction to a second form of partial derivatives, one where we allow both the x and y vary as we look for the rate of change of f(x,y).
PDF Summary		Standard 09. Local Extrema An extension of local extrema into multivariables through critical points and the second derivative test for multivariable functions. Also contains an example of finding absolute extrema on a closed and bounded region using the same techniques.
PDF Summary		Standard 10. Lagrange Multipliers An introduction to a new method of solving for absolute extrema on a closed and bounded region given one or two constraints.
PDF Summary		Standard 11. Double Integrals - Rectangular An introduction to double integration over a rectangular region or a general two-dimensional region.
PDF Summary		Standard 12. Double Integrals - Polar A look into converting double integrals, and the region of integration, using Cartesian (or rectangular) coordinates into polar coordinates.
PDF Summary		Standard 13. Triple Integrals - Rectangular An introduction to triple integration over a rectangular prism or a general three-dimensional region.
PDF Summary		Standard 14. Triple Integrals - Cylindrical and Spherical A look into converting triple integrals, and the region of integration, using Cartesian coordinate into cylindrical and spherical coordinates.
PDF Summary		Standard 15. Change of Variable A generalization of the process to convert integrals using Cartesian coordinates into an alternate coordinate system.
PDF Summary		Standard 16. Line Integrals An introduction to a new type of integral called line integrals, where the independent variables now are defined by curves rather than regions as we have done before.
PDF Summary		Standard 17. Fundamental Theorem of Line Integrals An introduction to the fundamental theorem of calculus for line integrals of vector fields along with more in-depth discussion on conservative vector fields and their potential functions.
PDF Summary		Standard 18. Curl and Divergence An introduction of the curl and divergence of a vector field along and how the curl can be used to find out if a vector field is conservative.
PDF Summary		Standard 19. Green's Theorem An introduction to one of the three main theorems of calculus III, Green's theorem.
PDF Summary		Standard 20. Surface Integrals An introduction to a new type of integral called surface integrals, where the integrand is a function of two variables or more variables and the independent variables are now on the surface of a three-dimensional solid.
PDF Summary		Standard 21. Stokes' Theorem An introduction to the second of the three main theorems of calculus III, Stokes' theorem.
PDF Summary		Standard 22. Divergence Theorem An introduction to the last of the three main theorems (and ana pplication of divergence) of calculus III, Divergence theorem.
PDF		Final Exam Review