California Institute of Technology (Caltech) Math Tutor & Mastery Lab | Calculus II Topics & Series: Ma 1 a; Calculus III & Vector Calculus: Ma 1 c, IC/Ma 1 abc; Differential Equations: Ma 2/102, ACM 95/100 ab, Ma/ACM 142 ab; Linear Algebra: Ma 1 b, ACM/IDS 104, CMS/ACM/IDS 107 a; Abstract Algebra: Ma 5/105 abc, Ma 120 abc; Real Analysis & Advanced Calculus: Ma 108 ab, Ma 110 a/c, CMS/ACM/IDS 107 a; Galois Theory: Ma 5/105 c, Ma 120 abc; Number Theory: Ma 7/107, Ma 160 a

Ma 1 abc: Calculus of One and Several Variables and Linear Algebra is a three-term sequence. Its Ma 1 a term develops one-variable calculus, with Taylor polynomials and infinite series in the regular section. It is not a separate course officially named “Calculus II.” Your section determines the pace and emphasis.

Use the Mastery Lab’s Calculus 2 lessons to practice the integration and series skills you need. For an integral, inspect the structure before trying substitutions or integration by parts. For an application, sketch the region and define the quantities before calculating. When reviewing volumes, compare washers and cylindrical shells and explain the radius, height, and bounds.

For a series, distinguish convergence from approximation. Choose a test whose hypotheses you can check; for a Taylor approximation, explain the center, degree, and remainder. After studying a solution with Woody, rebuild it from memory and try a new example to test whether you can choose the method independently.

Ma 1 c is the multivariable and vector-calculus stage of the Ma 1 abc sequence. Caltech lists practical and analytical tracks. The separate IC/Ma 1 abc: Integrated Core: Linear Algebra and Multivariable Calculus route organizes related mathematics around applications across the Integrated Core; its topic order is different.

With Woody, work from a sketch to partial derivatives, gradients, constrained optimization, or a multiple integral. State what changes and what remains fixed. Before changing coordinates, explain the domain, the bounds, and the Jacobian rather than relying on a remembered setup.

For line and surface integrals, identify the object being integrated over and choose its orientation. Then decide whether direct computation, Green’s theorem, Stokes’s theorem, or a flux argument is appropriate. Match the level of proof to your current track and use your syllabus to choose the next practice topic.

Ma 2/102: Differential Equations introduces ordinary differential equations and their applications. Caltech’s current schedule lists practical and analytical tracks. Work with Woody on exact equations, linear equations, systems, Laplace transforms, series solutions, and nonlinear behavior at the level your track requires.

Begin by identifying the order, linearity, and conditions. Explain why a proposed method applies, carry the constants carefully, and substitute the result back into the equation. For a system, connect eigenvalues to the direction and stability of the motion instead of stopping after a matrix calculation.

For further work, ACM 95/100 ab: Introductory Methods of Applied Mathematics for the Physical Sciences includes differential-equation methods within a broader sequence. Ma/ACM 142 ab: Ordinary and Partial Differential Equations is a more theoretical course. The ODE introduction in Ma 108 b also fits this support. Bring your syllabus so the study plan addresses your equation, its assumptions, and its initial or boundary conditions.

Ma 1 b is the linear-algebra stage of Caltech’s first-year sequence, with practical and analytical tracks. Linear algebra also appears throughout IC/Ma 1 abc. ACM/IDS 104: Applied Linear Algebra develops the subject further through matrix methods and applications.

Work with Woody on span, independence, bases, dimension, and linear transformations. When you reduce a matrix, say what the result tells you about the kernel, image, or solution set. Distinguish the transformation itself from its matrix in a particular basis.

For projections, least squares, eigenvalues, and singular value decomposition, connect each calculation to its geometric meaning. The more advanced CMS/ACM/IDS 107 a: Linear Analysis with Applications moves into normed spaces and operator theory. That material also connects to the analysis support below. Linear algebra is a core Mastery Lab subject, with the study plan matched to your course.

Ma 5/105 abc: Introduction to Abstract Algebra progresses through groups, rings and modules, then fields and Galois theory. Ma 120 abc: Abstract Algebra continues into advanced algebra, including commutative and homological algebra, representation theory, and further field theory.

With Woody, begin by writing exactly what must be proved. For a quotient or homomorphism, identify the kernel, normality, or ideal condition that makes the construction valid. For a classification problem, make the assumptions explicit before applying a structural theorem.

Practice group actions and Sylow arguments, then connect ring and module reasoning to the same habit of checking definitions. At the Ma 120 level, connect semisimple algebras, Wedderburn theory, Jacobson radicals, and Brauer groups to the structures you already know. Use your current topic to decide which earlier results need rebuilding. The goal is a proof you can reconstruct and adapt when the next question changes.

Ma 108 abc: Classical Analysis spans several areas. Ma 108 a develops real-number and metric-space foundations and rigorous differentiation; Ma 108 b includes differential equations, Lebesgue integration, and Fourier analysis. Ma 108 c turns to complex analysis, so the full sequence should not be labeled entirely as real analysis.

Ma 110 a and Ma 110 c, within Ma 110 abc: Analysis, extend into measure, functional, harmonic, and operator-theoretic work. CMS/ACM/IDS 107 a: Linear Analysis with Applications provides another route through normed spaces and operators. These are different levels and emphases; your preparation and syllabus determine the plan.

With Woody, write the claim in quantified form before choosing an estimate. Explain where completeness, compactness, or a convergence hypothesis enters. Build examples and counterexamples to separate similar definitions, and check the conditions before interchanging a limit, derivative, sum, or integral.

Ma 5/105 c, the third term of Introduction to Abstract Algebra, explicitly includes field theory and Galois theory. The advanced Ma 120 abc: Abstract Algebra sequence includes further topics such as infinite Galois theory and Kummer theory. The term-by-term emphasis of advanced work should be matched to your syllabus.

Work with Woody on irreducibility, minimal polynomials, extension degrees, and splitting fields. Before proposing an automorphism, determine where the generators may go and whether those choices preserve the defining relations.

For the Galois correspondence, check normality and separability and connect the subgroup lattice to intermediate fields. Use a carefully labeled example to see the correspondence, then write the general argument. When the course moves beyond finite extensions, identify which finite-extension facts still apply and which require an additional hypothesis.

Ma 7/107: Number Theory for Beginners develops elementary number-theory ideas through examples and problems. Ma 160 a: Number Theory is an advanced algebraic number-theory course. Caltech lists only part a of the Ma 160 sequence for 2026–27; parts b and c are not offered that year.

For elementary work, practice the Euclidean algorithm, congruences, prime factorization, and Diophantine equations with Woody. Test a conjecture with small examples, but separate that evidence from a proof. Before canceling modulo an integer, explain why the factor is invertible.

At the algebraic level, distinguish element factorization from ideal factorization and keep the ring or field explicit. Rebuild the relevant algebra before working on ideals, class groups, local fields, or ramification. Bring the exact topic from your current course so the next step is focused and useful.