Multivariable Calculus Open Source Study Guide

Setting Up Triple Integrals Without Going Crazy

I spent way too many hours last semester watching students fail their vector calculus midterm because they couldn't set up the bounds correctly on a triple integral over a solid bounded by paraboloids and cones. Not because they didn't know the methods, but because they had no single reference they trusted to work through the messy edge cases. That's what drove me to build a Multivariable Calculus Open Source Study Guide focused on exactly those friction points. The study guide is just a collection of openly licensed notes, worked examples, and problem sets organized by topic. No paywall, no registration wall, no "subscribe to unlock Chapter 7." It lives on GitHub and is maintained by a small group of grad students and TAs who actually teach the course each semester. The primary repo is at multicalc-study-guide.org, and the actual source lives on GitHub under the MIT license. You can fork it, clone it, or just read it in your browser.

Multivariable Calculus Open Source Study Guide

Here's the thing about multivariable calculus that textbooks don't emphasize enough: partial differentiation and vector fields behave like totally separate subjects until you force yourself to practice them together. I used to teach Calc III at a state university, and the pattern was always the same. Students could compute a gradient fine. They could evaluate a line integral fine. The moment you asked them to verify Stokes' theorem on a non-planar surface by computing both sides independently, half the class broke down. The guide addresses this by structuring its worked examples around theorems themselves rather than around computation types. Instead of a section called "Computing Curl," there's a section called "Stokes' Theorem: Verifying Both Sides" that walks through the entire pipeline. You compute the surface integral. You compute the line integral. You show they match. The example includes the full setup, the integral evaluation, and then a discussion of where a sign error would have come from and how to catch it before the exam. The most common failure mode in this course is orienting surfaces correctly. Right-hand rule problems. People memorize the rule but can't apply it when the surface is tilted or when the boundary curve is parameterized counter to what the orientation demands. I have a specific memory from 2022 where a student came to office hours with a Green's theorem problem where the region had a hole in it. She applied the theorem directly to the outer boundary and got the wrong answer by exactly the flux through the inner boundary. She'd missed that Green's theorem on a region with holes requires oriented boundaries in opposite directions for each component. The guide has a dedicated subsection on multiply-connected regions with a diagram that shows the boundary orientation visually, not just the formula. That visual was the thing that finally made it click for her.

What's Actually Inside

The guide is roughly 340 pages of content split into six units. The first three cover single-variable review with a focus on what you actually need (implicit differentiation, substitution, basic area calculations). The next two handle vector calculus core material: gradient fields, conservative vector fields, line integrals, surface integrals, and the big three theorems. The final unit is applied: fluid flow, Maxwell's equations in vector form, and optimization with constraints using Lagrange multipliers. Each chapter has a difficulty rating, estimated time to complete, and a list of prerequisite sections. If you're skipping around, the prereq links will prevent you from reading about divergence theorem without having seen the definition of divergence first. The problems are tiered into routine, intermediate, and competition level. The routine problems are textbook-adjacent. The competition problems are the kind that show up on exams when the professor is trying to distinguish who actually understands the material from who just memorized the algorithm. The LaTeX source files are all available. If you want to print it, compile it yourself, and add your own annotations, go ahead. I've had students do that and the compiled PDF comes out cleanly at about 450 pages when you include the appendix with formula sheets and change-of-variables reference tables.

How to Use It Effectively

Don't read it like a novel. You read it, then you do the problems before you move on. The guide assumes you're working through it alongside your course, not replacing your course. The examples are short and the derivations are complete but not exhaustive. If you need more detail on a particular proof, there are links to the OpenStax Calculus Volume 3 chapters that cover the same ground with more rigor. The change-of-variables section is where most people stumble. Jacobians. The absolute value of the determinant. Why it matters. The guide walks through the derivation from first principles using a linear approximation argument, then shows three different coordinate systems: cylindrical, spherical, and a general transformation for ellipsoidal regions. The general transformation part is where I noticed a gap in the publicly available material. Most free resources skip it entirely because the algebra gets ugly. I added a worked example with an ellipsoid mapped from a sphere using a diagonal scaling matrix, showing how the Jacobian factors separate into a product of the individual scale factors. That example alone saved at least a dozen students per semester from losing points on a standard exam problem.

The Parts That Don't Work Well

The guide is not complete. It doesn't cover differential forms, which some upper-level courses require. It doesn't cover numerical methods for integration beyond basic Monte Carlo approaches. The problem sets are curated, which means there's a known bias toward the types of problems that appear on AP Calculus BC and typical undergraduate midterm/final exams. If your course leans heavily into geometric intuition or proof-heavy analysis, you'll need supplementary material. There's also a maintenance issue. The content is updated each semester by the TAs, but the update cycle isn't synchronized across all the repos. You might pull a version where the line integral section was updated but the surface integral section is still from two years ago. I check the commit history weekly and the README has a version matrix showing which sections were last modified. Use it. The git history is public, so you can compare directly if you're unsure. One more limitation: the guide assumes you're comfortable with basic single-variable calculus. If you're struggling with u-substitution or basic integration techniques, multivariable calculus will feel impossible regardless of how good the guide is. There's a review unit at the front, but it covers maybe eight hours of content. That's usually not enough if your fundamentals are genuinely weak. In that case, work through the relevant OpenStax chapters first, then come back.

Getting It

The source is on GitHub. The compiled PDF is on the website. There's no download limit, no email capture, no tiered access. If you want the problem sets separately for printing, they're in a /problems directory. If you want the solutions, they're in /solutions and are marked clearly so you don't accidentally read them while doing the work. The solutions include full derivations, not just final answers, which is where most free resources fall short. I've been maintaining the vector calculus section since 2019. The project started when I realized the campus bookstore wasn't carrying the textbook our department had adopted, and the open alternatives available at the time were either too verbose or too sparse. This guide sits somewhere in between. It's dense enough to be useful and sparse enough that you're doing the work instead of reading someone else's explanation. That's the point.