Calculus III Resources | Brian Mulholland

Course Overview

These resources are designed to support students in Calculus III through a combination of lecture notes, companion videos, and interactive visualization tools focused on multivariable thinking.

A major emphasis of the course is developing geometric intuition and the ability to visualize mathematical objects in three dimensions.

Lecture Notes

Structured lecture notes covering vectors, surfaces, partial derivatives, multiple integrals, vector fields, and vector calculus.

Vectors and Geometry of Space Space Curves and Multivariable Functions Optimization Multiple Integrals and Applications Vector Calculus

View Lecture Notes

Companion Videos

Supplemental videos designed to accompany lecture material, reinforce conceptual understanding, and provide worked examples.

Worked Examples Visualization Conceptual Review Exam Preparation

Coming Soon

AR Visualizer

A beta augmented reality tool for exploring Calculus III objects such as surfaces, curves, vectors, and other three-dimensional structures.

Developed in collaboration with Notre Dame’s Office of Digital Learning to help students build geometric intuition beyond the static page.

Beta AR Visualization ODL Collaboration

Open AR Visualizer

Lecture Notes

Student lecture notes and corresponding solution versions organized by unit. Click a unit below to expand the topic list.

Unit 1: Vectors and Geometry of Space

Topic 01: 3D Coordinates and Vectors Student Version Solutions

Topic 02: Dot and Cross Products Student Version Solutions

Topic 03: Lines in Space Student Version Solutions

Topic 04: Planes in Space Student Version Solutions

Topic 05: Common Surfaces Student Version Unavailable Solutions

Unit 2: Space Curves and Multivariable Functions

Topic 06: Space Curves Student Version Solutions

Topic 07: Calculus of Space Curves Student Version Solutions

Topic 08: Arc Length and Curvature Student Version Solutions

Topic 09: Motion in Space Student Version Solutions

Topic 10: Functions of Several Variables Student Version Solutions

Topic 11: Partial Derivatives Student Version Solutions

Topic 12: Directional Derivatives Student Version Solutions

Topic 13: Applications of the Gradient Student Version Solutions

Unit 3: Optimization

Topic 14: Local Maximums and Minimums Student Version Solutions

Topic 15: Absolute Maximums and Minimums Student Version Solutions

Topic 16: Lagrange Multipliers Student Version Solutions

Topic 17: Lagrange Multipliers with Two Constraints Student Version Solutions

Unit 4: Multiple Integrals and Applications

Topic 18: Double Integrals over Rectangular and General Regions Student Version Solutions

Topic 19: Double Integrals in Polar Coordinates Student Version Solutions

Topic 20: Introduction to Triple Integrals Student Version Solutions

Topic 21: Triple Integrals in Cylindrical Coordinates Student Version Solutions

Topic 22: Triple Integrals in Spherical Coordinates Student Version Solutions

Topic 23: Applications with Mass Student Version Unavailable Solutions

Topic 24: Change of Variables Student Version Solutions

Unit 5: Vector Fields and Vector Calculus Theorems

Topic 25: Vector Fields and Line Integrals Student Version Solutions

Topic 26: Fundamental Theorem of Line Integrals Student Version Solutions

Topic 27: Green’s Theorem Student Version Solutions

Topic 28: Curl and Divergence Student Version Solutions

Topics 29–30: Parametric Surfaces Student Version Solutions

Topic 31: Surface Integrals Student Version Solutions

Topic 32: Flux Integrals Student Version Solutions

Topic 33: Stokes’ Theorem Student Version Solutions

Topic 34: The Divergence Theorem Student Version Solutions