University of Maryland (UMD) Math Tutor & Mastery Lab | Calculus II: MATH 141; Calculus III & Vector Calculus: MATH 241/340; Differential Equations: MATH 246/243/341; Linear Algebra: MATH 240/243/341/401/405/461; Abstract Algebra: MATH 402/403; Real Analysis & Advanced Calculus: MATH 410/411; Galois Theory: MATH 404; Number Theory: MATH 406

MATH 141: Calculus II is Maryland’s second-semester calculus course. Work through integration, applications, and infinite series with a method you can explain and repeat.

For volumes of revolution, practice both shells and disks/washers. Draw the region and rotation axis, choose the slice, and label the radius and height before writing the integral. For a convergence question, decide which feature of the series points toward a useful test.

Bring your current UMD assignment or practice exam into the Lab. Use a complete worked example to understand the decisions, then try a similar problem without looking. The goal is to know how to begin when the next question looks unfamiliar.

MATH 241: Calculus III is UMD’s standard multivariable and vector-calculus course. The honors sequence begins with MATH 340: Multivariable Calculus, Linear Algebra and Differential Equations I (Honors) and continues in MATH 341.

The two-semester honors route connects the material from MATH 240, MATH 241, and MATH 246 with greater depth. Bring your actual course number so the examples match the route you are taking.

With Woody, make the setup do the hard work: sketch the solid, compare coordinate systems, and explain the bounds. For circulation or flux, check the boundary and orientation before choosing a direct integral or Green’s, Stokes’s, or the divergence theorem.

MATH 246: Differential Equations for Scientists and Engineers is the standard UMD ODE course. Support also fits the differential-equations material in MATH 341: Multivariable Calculus, Linear Algebra, Differential Equations II (Honors).

MATH 243: Introduction to Linear Algebra and Differential Equations combines the two subjects for applications. It has a different scope and does not satisfy prerequisites that require MATH 240 or MATH 246; use your syllabus to select the right practice.

Work with Woody on the reasoning behind a separable equation, integrating factor, second-order method, or Laplace transform. In a systems question, connect eigenvalues and initial conditions to the solution and phase portrait. Interpret a numerical or software result instead of accepting its output without a check.

MATH 240: Introduction to Linear Algebra and MATH 461: Linear Algebra for Scientists and Engineers are introductory routes with different emphases. Linear algebra also appears in the combined MATH 243 course and the MATH 340–341 honors sequence.

MATH 401: Applications of Linear Algebra uses matrix ideas in models such as networks and Markov chains. MATH 405: Linear Algebra develops the abstract theory and proof work; it is a further course, not another name for MATH 240.

With Woody, explain what row reduction tells you about a solution space. Distinguish a transformation from its matrix in a chosen basis. In advanced work, build the argument from definitions and justify claims about invariant subspaces, eigenstructure, and operators.

MATH 403: Introduction to Abstract Algebra is UMD’s more theoretical undergraduate algebra route. MATH 402: Algebraic Structures covers related structures with a less theoretical approach. These are alternatives with different expectations, not a required two-course sequence.

For either course, work with Woody on the step where reasoning matters: checking a subgroup condition, showing a map respects an operation, identifying an ideal, or proving that a quotient construction is well-defined.

Practice a definition with examples and counterexamples before writing the general argument. If you are continuing to Field Theory, note that UMD identifies MATH 403 as the prerequisite for MATH 404; MATH 402 does not fill that role.

MATH 410: Advanced Calculus I treats one-variable calculus rigorously. MATH 411: Advanced Calculus II moves that reasoning into several variables.

With Woody, slow down the beginning of a proof: what is given, what must be shown, and which definition turns the goal into a usable condition? Work on epsilon arguments, sequence reasoning, counterexamples, and the difference between a pointwise statement and a uniform one.

In MATH 411, connect the derivative matrix to local behavior and check the assumptions behind inverse or implicit function arguments. Use the exact theorem and notation from your section so your written solution communicates a complete chain of reasoning.

MATH 404: Field Theory explicitly includes Galois theory. It follows MATH 403 and belongs naturally with the Lab’s abstract-algebra support.

Bring the polynomial, the base field, and the exact question. Work with Woody on proving irreducibility, finding a minimal polynomial, identifying a splitting field, and calculating extension degrees before proposing a Galois group.

Draw the intermediate fields and explain what each automorphism fixes. Check the hypotheses before applying the Galois correspondence, then connect the result to a question about constructibility or solvability by radicals. The objective is a justified conclusion, not a guessed group name.

MATH 406: Introduction to Number Theory is UMD’s dedicated undergraduate course in this subject. It connects integer calculations with the reasoning that makes them valid.

With Woody, decide whether a question is best approached through factorization, a congruence, or a gcd condition. Before canceling modulo an integer or applying a theorem, explain why the required inverse or hypothesis is available.

Use examples to test an idea, then write the general argument. In a quadratic-residue or Diophantine problem, keep the logical goal visible and check that the conclusion answers the original question. Match optional topics to your current section.