Princeton University Math Tutor & Mastery Lab | Calculus II: MAT 104/210; Calculus III & Vector Calculus: MAT 201/203/216/218; Differential Equations: MAT 322/APC 350, MAT 391/MAE 305; Linear Algebra: MAT 202/204/216/217/218; Abstract Algebra: MAT 345/346; Real Analysis & Advanced Calculus: MAT 215/216/218/300/320/425; Galois Theory: MAT 346 (selected topics); Number Theory: MAT 214/419

MAT 104: Calculus II develops integration and infinite-series methods, with problems that can require several techniques. MAT 210: One-Variable Calculus with Proofs is Princeton’s more theoretical alternative at this stage, combining calculations with an introduction to rigorous arguments.

Study substitutions, integration by parts, trigonometric substitutions, partial fractions, and improper integrals with Woody. For a volume problem, draw the region and the axis of rotation before choosing washers or cylindrical shells. Write the radius, height, and bounds before integrating.

For series, separate the question of convergence from the question of approximation. Practice selecting a test, checking its assumptions, and controlling a Taylor remainder. In MAT 210, add the definitions and proof steps that justify the calculation. Use independent practice to make the process your own.

MAT 201: Multivariable Calculus is the standard third-semester course. MAT 203: Advanced Vector Calculus covers related material with greater abstraction and more demanding reasoning. The integrated MAT 216: Multivariable Analysis and Linear Algebra I and MAT 218: Multivariable Analysis and Linear Algebra II sequence treats the foundations through rigorous proofs; it is not simply another name for MAT 201.

Work with Woody on gradients, directional derivatives, constrained optimization, and multiple integrals. Begin by describing the domain. Then choose Cartesian, polar, cylindrical, or spherical coordinates and explain the bounds and any Jacobian.

For circulation and flux, identify the curve or surface and its orientation. Decide whether direct integration or a theorem gives a cleaner route. In the theoretical sequence, connect the calculation to differentiability, hypotheses, and the argument that makes it valid.

MAT 322 / APC 350: Introduction to Differential Equations connects ordinary and partial differential equations with scientific models, theory, and approximations. MAT 391 / MAE 305: Mathematics in Engineering I, also cross-listed as CBE 305 / EGR 305, emphasizes differential-equation methods and engineering applications.

With Woody, begin by identifying the independent variable, the order, and whether the equation is linear or nonlinear. Practice separation, integrating factors, second-order equations, and systems in the context appropriate to your syllabus. Use the initial or boundary conditions at the correct stage, then substitute back to check the answer.

For the engineering route, connect eigenvalues, Laplace transforms, series solutions, and phase portraits to the question being asked. When your course introduces partial differential equations, distinguish those methods from ordinary differential equations and work from the model and its conditions.

MAT 202: Linear Algebra with Applications develops the standard tools. MAT 204: Advanced Linear Algebra with Applications adds depth and more abstract work. MAT 217: Honors Linear Algebra emphasizes proofs. Linear algebra is also integrated into MAT 216/218, alongside rigorous analysis.

Work with Woody on row reduction, span, independence, bases, dimension, and linear transformations. Keep a distinction between an abstract map and the matrix representing it in a chosen basis. Explain what the kernel, image, and rank tell you before proceeding with an algorithm.

Practice eigenvalue questions, orthogonal projection, and changes of basis with a reason for each step. For MAT 204 or MAT 217, match the level of justification to your section, including spectral arguments or canonical forms when assigned as study topics. Linear algebra is one of the Lab’s core subjects.

MAT 345: Algebra I emphasizes symmetry, group theory, representation theory, rings, and modules. MAT 346: Algebra II continues into further algebraic structures; its detailed emphasis varies with the offering.

Bring the definitions and a practice question to Woody. For a subgroup, homomorphism, quotient, or group action, identify exactly what needs to be shown. Separate an example that illustrates the idea from an argument that establishes the general result.

Work through kernels and images, normality, orbit and stabilizer arguments, and Sylow reasoning as appropriate to your section. In ring and module problems, state the ambient object and the hypotheses you use. Reconstruct the reasoning after studying an example so that you can adapt it when the next question changes.

MAT 215: Single Variable Analysis with an Introduction to Proofs develops rigorous one-variable reasoning. MAT 216/218 combines analysis and linear algebra in a theoretical sequence. MAT 300: Multivariable Analysis I extends the work to differentiable maps, manifolds, and differential forms.

MAT 320: Introduction to Real Analysis develops measure and Lebesgue integration. MAT 425: Analysis III: Integration Theory and Hilbert Spaces provides another advanced analysis route. These courses require different preparation and are not interchangeable levels of the same class.

With Woody, unpack the quantifiers before estimating. Distinguish pointwise from uniform convergence, identify where compactness or completeness is needed, and check the conditions for interchanging limits and integrals. Practice constructing examples and counterexamples, then write a proof whose logical steps are visible to the reader.

MAT 346: Algebra II has included Galois theory—for example, its Spring 2025 offering focused on local fields and their Galois theory. The course’s subject matter varies, so this is support for relevant topics when covered, not a claim that every MAT 346 offering is a dedicated Galois course.

For field-theory practice, start with the base field and the polynomial. Work with Woody on irreducibility, minimal polynomials, extension degrees, and splitting fields. Explain how an automorphism acts on generators and why those choices respect the field relations.

Check normality and separability before using the Galois correspondence. Connect intermediate fields to subgroups with a diagram and a written justification. Share your syllabus early if your course moves into local fields or another specialized topic, so the study plan matches the material you actually need.

MAT 214: Numbers, Equations, and Proofs is Princeton’s introduction to classical number theory. MAT 419: Topics in Geometry and Number Theory is an advanced topics course whose focus depends on the offering; it is not a fixed continuation of the MAT 214 syllabus.

With Woody, learn to decide whether a question calls for a divisibility argument, a congruence, or a factorization. Practice the Euclidean algorithm and the Chinese Remainder Theorem. Before canceling a factor modulo an integer, check whether that step is justified.

Use small examples to investigate a claim, then develop the proof. In quadratic-residue, reciprocity, or Gaussian-integer questions, keep the relevant definitions visible. For MAT 419, begin with your current syllabus and review the algebraic and arithmetic tools needed for its selected topic.