University of Michigan Math Help & Mastery Lab | Calculus II: MATH 116/156/186; Multivariable & Vector Calculus: MATH 205/215/285; Differential Equations: MATH 216/286/316/404; Linear Algebra: MATH 214/217/417/419/420; Abstract Algebra: MATH 312/412/493/494; Real Analysis & Advanced Calculus: MATH 351/451/452; Galois Theory: MATH 494; Number Theory: MATH 475

MATH 116: Calculus II is the standard course. MATH 156: Applied Honors Calculus II and MATH 186: Honors Calculus II offer honors alternatives with different emphases.

The public MATH 116 overview emphasizes interpreting graphs, tables, formulas, and words. With Woody, translate the information into a mathematical setup, select a technique, and explain what the result means. For a series, first identify its structure; for an integral, inspect the integrand before choosing a substitution.

Practice both shells and disks/washers for volumes of revolution. Sketch the region and axis, choose a slice, and label the radius and height before writing the integral. Match the exercises and notation to your current Michigan syllabus.

MATH 215: Multivariable & Vector Calculus completes the standard sequence. Michigan also lists MATH 205: Calculus of Several Variables and MATH 285: Honors Multivariable & Vector Calculus (IBL).

These routes overlap but differ in scope and approach. In particular, do not assume every MATH 215 vector-calculus topic appears in MATH 205. Use the course number and syllabus to choose the right practice.

Work with Woody on the decision before the computation: Cartesian or polar coordinates? A direct integral or a theorem? For flux and circulation, check the surface, boundary, orientation, and hypotheses before applying Green’s, Stokes’s, or the divergence theorem.

MATH 216: Introduction to Differential Equations emphasizes applications. MATH 286: Honors Differential Equations is the honors route. MATH 316: Differential Equations builds on linear algebra and includes more theoretical reasoning.

Laplace methods appear in Michigan’s descriptions for MATH 216 and MATH 286. The MATH 316 description instead foregrounds solution methods, existence and uniqueness, linear systems, and qualitative behavior. Follow your actual syllabus rather than assuming the three courses are identical.

With Woody, organize the work: classify the equation, solve, apply the initial conditions, then substitute to verify. In a systems problem, connect eigenvalues to the behavior of solutions instead of treating the matrix calculation as an isolated step.

MATH 214: Applied Linear Algebra focuses on applications. MATH 217: Linear Algebra also develops proof writing. MATH 417: Matrix Algebra I and MATH 419: Linear Spaces and Matrix Theory are other routes; MATH 420: Advanced Linear Algebra develops the theory further.

With Woody, turn elimination into an explanation about consistency, independence, or span. For a transformation, distinguish the map from its matrix in a chosen basis. In proof work, write the defining condition and show that each hypothesis is used.

For advanced work, bring the exact statement involving dual spaces, operators, diagonalization, or canonical forms. These course numbers represent different pathways, not a checklist of classes every student should take.

MATH 312: Applied Modern Algebra and MATH 412: Introduction to Modern Algebra (IBL) offer different introductions to algebra. The advanced honors sequence is MATH 493: Honors Algebra I followed by MATH 494: Honors Algebra II.

Michigan’s inquiry-based approach in MATH 412 makes communicating your reasoning especially important. Practice stating what a claim means before reaching for a theorem. For a homomorphism, identify its domain and codomain; for a quotient, check that the construction is well-defined.

With Woody, work through a definition, test examples and counterexamples, and build the proof from the permitted results. In honors work, bring the exact group, ring, field, or module question so the guidance matches your section.

MATH 451: Advanced Calculus I develops rigorous calculus. MATH 351: Principles of Analysis (IBL) is an alternative introduction with an inquiry-based format. MATH 452: Advanced Calculus II extends rigorous reasoning to several variables.

Work with Woody on how to begin an epsilon argument, where an assumption enters, and which estimate produces the conclusion. Distinguish a limit that depends on a point from one bound that works throughout the domain.

For multivariable proofs, identify the linear approximation, the relevant neighborhood, and the hypotheses of the theorem. Explain why a conclusion follows; a correct-looking calculation alone may leave the main reasoning unfinished.

MATH 494: Honors Algebra II is the Michigan course to check for field extensions and Galois theory. The department lists Galois theory among selected topics, so its depth and timing depend on the syllabus.

Bring the polynomial, the base field, and the exact question. Work with Woody on irreducibility, minimal polynomials, splitting fields, and automorphisms in the order the problem requires.

Draw the field inclusions, justify the degrees, and check the hypotheses before using a correspondence theorem. The aim is to explain why a proposed Galois group or intermediate field is correct, with every step connected.

MATH 475: Elementary Number Theory provides the dedicated Michigan number-theory route. It combines integer arithmetic with proof-based reasoning.

With Woody, identify the obstacle before calculating. Does a modular inverse exist? Can a gcd condition settle solvability? Should the argument use congruences, factorization, or a carefully chosen contradiction?

Practice moving between a numerical example and the general statement. Keep each congruence tied to its modulus, justify a cancellation, and distinguish evidence for a conjecture from a proof. Bring your current topic to match the support to your course.