Louisiana State University (LSU) Math Tutor & Mastery Lab | Calculus II: MATH 1552/1553; Multivariable & Vector Calculus: MATH 2057/2058; Differential Equations: MATH 2065/2070/2090/4027/4038/4340; Linear Algebra: MATH 2085/2090/2070/4153/4064; Abstract Algebra: MATH 4200/4201/4023; graduate MATH 7210/7211; Real Analysis & Advanced Calculus: MATH 4031/4032/4035; graduate MATH 7311; Galois Theory: MATH 4201; graduate MATH 7211; Number Theory: MATH 4181; graduate MATH 7230 (selected topics)

MATH 1552: Analytic Geometry and Calculus II and honors MATH 1553 are LSU’s main Calculus II routes. LSU’s published outline extends beyond integration and series into vectors, space curves, and introductory partial derivatives.

For a mixed set of integrals, first ask what would simplify the expression. Try a substitution when you can see its derivative; consider parts when a product will become easier after differentiation; examine the denominator before choosing partial fractions. When you review volumes by washers and cylindrical shells, draw the region and the axis before writing the integral.

For a series, write the reason for your test choice and check its hypotheses. Treat the endpoints of a power-series interval as separate problems. In the Mastery Lab, study a complete worked example, cover the solution, and rebuild it while explaining each decision aloud. Bring Woody your syllabus and the first step that stops you so your practice follows the work your LSU section is assigning.

MATH 2057: Multidimensional Calculus and honors MATH 2058 are LSU’s Calculus III routes. The departmental outline includes multivariable differentiation, multiple integrals, and the major vector-calculus theorems.

Begin an integral by describing the region in words. Draw a useful slice, label where it begins and ends, and decide whether Cartesian, cylindrical, or spherical coordinates make the bounds clearer. For an optimization problem, identify the quantity being optimized before separating interior critical points from the boundary work.

For a vector field, decide whether the question concerns work, circulation, or flux. Mark the curve’s direction or the surface normal. Then check the hypotheses of Green’s theorem, Stokes’s theorem, or the divergence theorem before using it. The Mastery Lab gives you a place to study the whole setup with Woody and practice explaining why it works, so a change in the picture does not leave you searching for a memorized formula.

LSU offers MATH 2065: Elementary Differential Equations, MATH 2070: Mathematical Methods in Engineering, and MATH 2090: Elementary Differential Equations and Linear Algebra. The engineering course includes Fourier methods; the combined course makes linear algebra central to the work.

Start by classifying the equation. For a first-order problem, inspect its structure before choosing separation, an integrating factor, or a substitution. For a higher-order equation, separate the homogeneous solution from the forcing response. Use the initial conditions carefully, then differentiate and substitute to check your result. For a system, connect the eigenvalues and eigenvectors to the form of the solution.

Advanced support includes MATH 4027: Differential Equations, MATH 4038: Mathematical Methods in Engineering (also ME 4563), and MATH 4340: Partial Differential Equations. Bring the exact assignment. In a proof, identify what the theorem requires; in a boundary-value problem, write the domain and conditions before separating variables. For 4038, we can work through the connection between series solutions, orthogonal expansions, and the boundary conditions.

MATH 2085: Linear Algebra is LSU’s dedicated introductory course. Linear algebra also appears in MATH 2090 and the engineering-methods course MATH 2070. Later courses include MATH 4153: Finite Dimensional Vector Spaces and MATH 4064: Numerical Linear Algebra.

After row reduction, state what the pivots and free variables tell you. A basis requires both spanning and independence; a diagonalization requires enough independent eigenvectors. Keep the linear transformation separate from the matrix representing it in a particular basis. These distinctions help you turn a page of arithmetic into a complete answer.

For more theoretical work, return to the definitions and explain why each implication holds. For numerical linear algebra, distinguish an exact mathematical solution from the effect of rounding or sensitivity in the computation. Bring Woody the problem and your reasoning so your practice strengthens the step you need, whether that is finding a null space, organizing an eigenvalue argument, or interpreting a least-squares result.

MATH 4200: Abstract Algebra I leads into MATH 4201: Abstract Algebra II. MATH 4023: Applied Algebra follows an applications-oriented route. Graduate references include MATH 7210: Algebra I and MATH 7211: Algebra II.

A proof should tell the reader what is being assumed and why the conclusion follows. To show that a subset is a subgroup, start with arbitrary elements of that subset. To define a map on cosets, check that your answer does not depend on the chosen representative. To use a quotient construction, identify exactly which algebraic property makes the construction valid.

Bring your LSU problem and your attempted proof to the Mastery Lab. Woody can help you find the missing implication, choose a useful example, and organize a clean argument. Then practice a related problem without the solution in front of you. Graduate questions are matched to the assigned material; send the topic and syllabus so the guidance fits the level of your course.

LSU’s sequence includes MATH 4031: Advanced Calculus I, MATH 4032: Advanced Calculus II, and MATH 4035: Advanced Calculus of Several Variables. Graduate MATH 7311: Real Analysis I develops measure and integration.

Start each limit proof by writing the quantifiers in order. If you choose an index or a radius, make clear which quantities it may depend on. For a convergence argument, decide whether one choice must work across the whole domain. Before interchanging a limit with another operation, identify the theorem that permits it and verify each hypothesis.

The Mastery Lab gives you a place to work through the point where a proof stops making sense. Bring the exact statement, your current attempt, and the definition or theorem you are trying to use. Woody can help you turn an intuitive explanation into a logically complete solution. For graduate material, share the assigned problem and syllabus so the discussion matches your course.

LSU explicitly includes Galois theory in MATH 4201: Abstract Algebra II and graduate MATH 7211: Algebra II.

A field-extension problem becomes easier when you keep the objects separate. What is the base field? Which element is being adjoined? Which polynomial is its minimal polynomial, and why is that polynomial irreducible? For a splitting field, check whether adjoining one root also produces others before adding more generators.

When finding automorphisms, determine where each generator can go while preserving its algebraic relations. A permutation of roots is a candidate, not automatically an automorphism. For the Galois correspondence, draw the subgroup and intermediate-field diagrams and check the reversal of inclusions. Bring your assigned polynomial, attempted field diagram, or proof to the Mastery Lab and work through the next step with Woody. For graduate questions, include the precise statement and course context.

MATH 4181: Elementary Number Theory is LSU’s core reference. Graduate MATH 7230: Topics in Number Theory varies by offering; share your syllabus and assigned topic to confirm the appropriate support.

Modular arithmetic has its own rules for division. Before canceling a factor, check whether it is invertible for your modulus. When solving an integer equation, first decide whether a solution can exist, then describe all allowable solutions. A successful numerical example can suggest the method, but it does not replace the proof.

In the Mastery Lab, bring the problem and the first step that feels uncertain. Work with Woody on choosing a useful modulus, explaining a divisibility argument, or turning a calculation into a complete conclusion. For graduate number theory, the actual subject of your section matters more than the course number alone. Send the definitions and results your instructor is using so the discussion starts in the right setting.