University of North Carolina at Chapel Hill (UNC) Math Help & Mastery Lab | Calculus II: MATH 232; Calculus III: MATH 233; Differential Equations: MATH 383/524; Linear Algebra: MATH 347/577; Advanced Calculus & Real Analysis: MATH 521/522; Abstract Algebra: MATH 534/578; Advanced Algebra & Galois Theory: MATH 676/677; Number Theory: MATH 533; Mathematical Methods: MATH 528/529; Complex Analysis: MATH 523; Nonlinear Dynamics: MATH 594; Discrete Mathematics & Proofs: MATH 381

MATH 232: Calculus of Functions of One Variable II combines integration methods and applications with sequences, series, and approximation. The course also introduces elementary differential equations.

Practice recognizing when substitution, integration by parts, trigonometric substitution, or partial fractions fits. For volumes by disks, washers, and cylindrical shells, sketch the region and axis before choosing the integral.

For an infinite series, separate the question of convergence from the question of its sum or approximation. Check the conditions of the test and the endpoints of a power-series interval. Use the Taylor series visual lesson to connect polynomial approximations with the function they represent.

MATH 233: Calculus of Functions of Several Variables develops vector algebra, spatial geometry, partial derivatives, and multiple integrals.

Start a multiple-integral problem by describing the region in words and inequalities. Compare coordinate systems before calculating. For a gradient or directional derivative, identify the direction and what is changing.

Woody also supports optimization, vector fields, and line or surface integrals when they appear in your assigned work. Use your current MATH 233 syllabus to prioritize the topics and theorems your section covers.

MATH 383: First Course in Differential Equations studies first- and higher-order equations, applications, and systems. Linear algebra enters as needed, making matrix and eigenvalue skills useful companions to calculus.

MATH 524: Elementary Differential Equations follows MATH 383 and develops further work with linear equations, series solutions, transforms, and numerical approaches.

MATH 528: Mathematical Methods for the Physical Sciences I connects transform methods, Fourier analysis, and Sturm–Liouville problems. MATH 529: Mathematical Methods for the Physical Sciences II moves into boundary-value problems, classical PDEs, special functions, and complex-variable methods.

For a boundary problem, identify the domain and conditions before choosing a representation. For an ODE, substitute the result back into the equation and check the initial values.

MATH 347: Linear Algebra for Applications emphasizes matrix methods and applications, including solving systems, determinants, orthogonalization, and eigenvalues.

MATH 577: Linear Algebra takes a more abstract approach. Work with vector spaces and maps develops into decomposition, canonical forms, inner products, and spectral reasoning.

These courses need different levels of explanation. Bring the exact assignment so Woody can help with the row reduction, the conceptual interpretation, or the proof. Linear algebra is a core part of the teaching portfolio, including its connections to differential equations, algebra, physics, and data science.

MATH 534: Elements of Modern Algebra studies algebraic operations and structures through groups, quotient ideas, rings, and polynomials.

MATH 578: Algebraic Structures connects algebra with permutations, matrices, symmetries, linear maps, and concrete fields. Bring your course number so the study plan follows its particular emphasis.

Before using an isomorphism theorem, define the map and verify its properties. Before forming a quotient, check the required normal-subgroup or ideal condition. Use concrete examples to understand the definition, then explain why the proof works beyond those examples.

MATH 521: Advanced Calculus I treats one-variable calculus rigorously, including the real-number system, continuity, differentiation, series, and integration.

MATH 522: Advanced Calculus II extends the analysis to several variables. The derivative becomes a linear transformation, and inverse and implicit function results become important tools alongside multiple integration.

These courses build on the reasoning developed in MATH 381. Practice identifying what must be chosen, what it may depend on, and which estimate or theorem finishes the proof. Counterexamples help explain why the hypotheses matter.

MATH 676: Modules, Linear Algebra, and Groups develops the structural algebra background for the next course. MATH 677: Groups, Representations, and Fields explicitly includes Galois theory in UNC’s graduate course listings.

Woody supports the field-theory and Galois work you bring from this sequence or an independent study. Build from irreducible polynomials and extension degrees to splitting fields, automorphisms, fixed fields, and the Galois correspondence.

Share the precise problem and the results you are allowed to use. That keeps the explanation at the right level and helps you see which structural fact makes the next step possible.

MATH 533: Elementary Theory of Numbers connects arithmetic, congruences, and proof writing, with topics extending into residue classes, number-theoretic functions, continued fractions, and Gaussian integers.

Practice moving between a divisibility statement and a modular equation. Check coprimality before using an inverse, and explain the conditions under which a system of congruences can be solved.

The habits developed in MATH 381 remain useful here: precise statements, induction, contradiction, and complete cases. Woody helps you turn numerical exploration into a readable mathematical argument.